Previous Page
   Next Page
 
Evokation
  
 
Index
  

 

 

 

 

 

 

 

 

A

MAZE

IN

ZAZAZA ENTERS AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
-
7
-
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
-
7
-
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
-
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
-
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
-
9
-
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
-
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
-
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
-
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

Page 250

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY THE ACCOUNT

IS UNASHAMEDLY SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE AS

THOUGH HE WERE WRITING A BOOK. BUT LANGUAGE HAS BEEN ENTIRELY TRANSFORMED AND THE

MESSAGE OF CREATION IS NO LONGER CLEAR EACH LETTER OF THE HEBREW ALPHABET IS GIVEN

A NUMERICAL VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS REARRANGING

THEM IN ENDLESS CONFIGURATIONS THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL

CONNOTATIONS OF WORDS

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

....

 

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
J
K
L
M
N
O
P
Q
R
10
11
12
13
14
15
16
17
18
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
S
T
U
V
W
X
Y
Z
I
19
20
21
22
23
24
25
26
9
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
ME
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
I
ME
I
ME
I
ME
I
ME
I
9
18
9
18
9
18
9
18
9
=
1+8
=
1+8
=
1+8
=
1+8
=
=
9
=
9
=
9
=
9
=
I
ME
I
ME
I
ME
I
ME
1
9
9
9
9
9
9
9
9
9
I
ME
I
ME
I
ME
I
ME
1

 

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

 

LIGHT AND LIFE

Lars Olof Bjorn 1976

Page 197

"By writing the 26 letters of the alphabet in a certain order one may put down almost any message (this book 'is written with the same letters' as the Encyclopaedia Britannica and Winnie the Pooh, only the order of the letters differs). In the same way Nature is able to convey with her language how a cell and a whole organism is to be constructed and how it is to function. Nature has succeeded better than we humans; for the genetic code there is only one universal language which is the same in a man, a bean plant and a bacterium."

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

"FOR THE GENETIC CODE THERE IS ONLY ONE UNIVERSAL LANGUAGE"

 

DNA AND DNA DNA AND DNA DNA AND DNA

DNA AND DNA DNA AND DNA DNA AND DNA

 

 

A QUEST FOR THE BEGINNING AND THE END

Graham Hancock 1995

Chapter 32

Speaking to the Unborn

Page 285

"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 / universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?

A message in the bottle of time

'Of all the other stupendous inventions,' Galileo once remarked,

what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3

If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."

"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"

"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL CONNOTATIONS OF WORDS

 

....

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
-
7
-
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
-
7
-
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
-
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
-
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
-
9
-
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
-
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
-
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
-
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 351 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 126 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 9 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

 

 

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 351 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 126 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 9 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
9
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
57
First Total
626
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+7
Add to Reduce
6+2+6
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
12
Second Total
14
14
15
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+2
Reduce to Deduce
1+4
1+4
 1+5
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL CONNOTATIONS OF WORDS

 

....

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

9
LANGUAGES
87
33
6
3
AND
19
10
1
7
NUMBERS
92
29
2
19
-
198
72
9
1+9
-
1+9+8
7+2
-
10
-
18
9
9
1+0
-
1+8
-
-
1
-
9
9
9

 

 

INDIA I AND I INDIA

 

 

THE GUINNESS ENCYCLOPEDIA

John Foley

1993

ALPHABETOLOGY

SIGNS AND SYMBOLS

Page 22

The most commonly used numerical symbols throughout the modern World; the so-called Arabic numerals

1 2 3 4 5 6 7 8 9

derive ultimately from a system developed by the Hindus in India sometime between the 3rd Century B,C. and 6th Century A.D.

The more rounded Western Arabic numerals were introduced into Spain by the Moors in the 10th Century.

The first European to take serious note of the new numeration was the French scholar Gerbert of Aurilliac (Pope Sylvester II from 999 to 1003) who had studied the system in Spain

The Hindus are also credited with the invention at some unknown date of the symbol for zero, which was first written as a small circle and later reduced to a large dot.

The nine Indian figures are : 9 8 7 6 5 4 3 2 1

With these nine figures and with the sign O any number may be written.

Leonardo of Pisa

Liber abaci

 

 

1234 5 6789

ONE TWO THREE FOUR 5FIVE5 SIX SEVEN EIGHT NINE

1234 5 6789 9876 5 4321

NINE EIGHT SEVEN SIX 5FIVE5 FOUR THREE TEO ONE

9876 5 4321

 

 

-
LOVE
-
-
-
2
L+O
27
9
9
2
V+E
27
9
9
4
LOVE
54
18
18
-
-
5+4
1+8
1+8
4
LOVE
9
9
9
-
-
-
-
-
-
-
-
-
-
-
EVOLVE
-
-
-
2
E+V
27
9
9
2
O+L
27
9
9
2
V+E
27
9
9
6
EVOLVE
81
27
27
-
-
8+1
2+7
2+7
6
EVOLVE
9
9
9

 

 

EVOLVE LOVE EVOLVE

LOVES SOLVE LOVES

EVOLVE LOVE EVOLVE

 

 

 

 

 

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:

Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly, too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10

Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.

Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.

The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.

A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the / Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)

If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.

This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.

Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.

But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he gave figures of 45" to 46" a year, while the truer precessional lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less than the figure of 25,920.

With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, incorrect by a difference of from 2,397.9 years to 3,024.

Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'

AN IMPORTANT POSTSCRIPT

Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.

Page 206

Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12

This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.

Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.

Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.

Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13

Page 207

"...Feniusa Farsa, Graves equates this hero with Dionysus. Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials

We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14

Graves did not know why this number was necessary, but he points out that he understands Frazer's Golden Bough to be a book hinting that 'the secret involves the truth that the Christian dogma, and rituals, are the refinement of a great body of primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15

Frances A. Yates, historian of Renaissance hermetisma tells, us the cabala had 72 angels through which the sephiroth (the powers of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident there are seventy-two of Pico's Cabalist conclusions, for the conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16

In Hamlet's Mill de Santillana adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss (60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)

The Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted mounted in Genesis for the lifetimes of patriarchs from the Flood also contained the needed secret numbers. (He showed that in the 1,656 years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.) In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, multiply this by 2 and again we have the necessary 432 digits.

Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as-the interesting number of AD.432.18

Whatever one may think-of some of these number coincidences, it becomes difficult to escape the suspicion that many signs (number and otherwise) - indicate that early man observed the results of the movement of Precession and that the - transmission of this information was considered of prime importance.

With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would have constituted the construction-of a 'Unified Field Theory' for nothing less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.

But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced, that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':

"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two, four. Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radius ekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up his piece of stripped and striped wallpaper, decorated / Page 209 / with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.

So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

KEEPER OF GENESIS

A QUEST FOR THE HIDDEN LEGACY OF MANKIND

Robert Bauval Graham Hancock 1996

Page 254

"...Is there in any sense an interstellar Rosetta Stone?

We believe there is a common language that all technical civilizations, no matter how different, must have.

That common language is science and mathematics.

The laws of Nature are the same everywhere:..."

 

 

R
=
9
-
7
ROSETTA
98
26
8
S
=
1
-
5
STONE
73
19
1
-
-
10
-
12
Add to Reduce
171
45
9
-
-
1+0
-
1+2
Reduce to Deduce
1+7+1
4+5
-
-
-
1
-
3
Essence of Number
9
9
9

 

 

-
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
6
1
-
-
-
-
-
1
-
6
5
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
15
19
-
-
-
-
-
19
-
15
14
-
+
=
82
8+2
=
10
1+0
1
=
1
-
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
5
2
2
1
-
-
2
-
-
5
+
=
26
2+6
=
8
=
8
=
8
-
-
18
-
-
5
20
20
1
-
-
20
-
-
5
+
=
89
8+9
=
17
1+7
8
=
8
-
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
18
15
19
5
20
20
1
-
19
20
15
14
5
+
=
171
1+7+1
=
9
=
9
=
9
-
-
9
6
1
5
2
2
1
-
1
2
6
5
5
+
=
45
4+5
=
9
=
9
=
9
-
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
-
2
2
-
-
-
2
-
-
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
22
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
23
-
-
12
-
45
-
27
2+2
1+2
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+3
-
-
1+2
-
4+5
-
2+7
4
3
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
5
-
-
3
-
9
-
9
-
-
9
6
1
5
2
2
1
-
1
2
6
5
5
-
-
-
-
-
-
-
-
-
-
4
3
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
5
-
-
3
-
9
-
9

 

 

12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
1
-
-
-
-
-
1
-
6
5
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
15
19
-
-
-
-
-
19
-
15
14
-
+
=
82
8+2
=
10
1+0
1
=
1
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
9
-
-
5
2
2
1
-
-
2
-
-
5
+
=
26
2+6
=
8
=
8
=
8
-
18
-
-
5
20
20
1
-
-
20
-
-
5
+
=
89
8+9
=
17
1+7
8
=
8
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
18
15
19
5
20
20
1
-
19
20
15
14
5
+
=
171
1+7=1
=
9
=
9
=
9
-
9
6
1
5
2
2
1
-
1
2
6
5
5
+
=
45
4+5
=
9
=
9
=
9
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
2
2
-
-
-
2
-
-
-
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
5
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
12
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
23
-
-
12
-
45
-
27
1+2
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+3
-
-
1+2
-
4+5
-
2+7
3
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
5
-
-
3
-
9
-
9
-
9
6
1
5
2
2
1
-
1
2
6
5
5
-
-
-
-
-
-
-
-
-
-
3
R
O
S
E
T
T
A
-
S
T
O
N
E
-
-
5
-
-
3
-
9
-
9

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
First Total
189
90
18
1+8
Add to Reduce
1+8+9
9+0
1+8
9
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
9
Essence of Number
9
9
9

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

FINGERPRINTS OF THE GODS

A QUEST FOR THE BEGINNING AND THE END

Graham Hancock 1995

Chapter 32

Speaking to the Unborn

Page 285

"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 / universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?

A message in the bottle of time"

'Of all the other stupendous inventions,' Galileo once remarked,

what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3

If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320 or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."

"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"

"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"

 

 

W
=
5
-
4
WHAT
52
16
7
O
=
6
-
3
ONE
34
16
7
W
=
5
-
5
WOULD
75
21
3
L
=
3
-
4
LOOK
53
17
8
F
=
6
-
3
FOR
30
21
3
T
=
2
-
9
THEREFORE
100
46
1
W
=
5
-
5
WOULD
75
21
3
B
=
2
-
2
BE
7
7
7
A
=
1
-
1
A
1
1
1
U
=
3
-
9
UNIVERSAL
121
40
4
L
=
3
-
8
LANGUAGE
68
32
5
-
-
42
4
56
First Total
616
238
49
-
-
4+2
-
5+6
Add to Reduce
6+1+6
2+3+8
4+9
-
-
6
-
11
Second Total
13
13
13
-
-
-
-
1+1
Reduce to Deduce
1+3
1+3
1+3
-
-
6
-
2
Essence of Number
4
4
4

 

 

T
=
2
-
3
THE
33
15
6
K
=
2
-
4
KIND
38
20
2
O
=
6
-
2
OF
21
12
3
L
=
3
-
8
LANGUAGE
68
32
5
T
=
2
-
4
THAT
49
13
4
W
=
5
-
5
WOULD
75
21
3
B
=
2
-
2
BE
7
7
7
C
=
3
-
14
COMPREHENSIBLE
144
72
9
T
=
2
-
2
TO
35
8
8
A
=
1
-
3
ANY
40
13
4
T
=
2
-
15
TECHNOLOGICALLY
161
71
8
A
=
1
-
2
ADVANCED
54
27
9
S
=
1
-
7
SOCIETY
96
33
6
I
=
9
-
2
IN
23
14
5
A
=
1
-
3
ANY
40
13
4
E
=
5
-
5
EPOCH
48
29
2
-
-
47
4
87
First Total
931
400
85
-
-
4+7
-
8+7
Add to Reduce
9+3+1
4+0+0
8+5
-
-
11
-
15
Second Total
13
4
13
-
-
1+1
-
1+5
Reduce to Deduce
1+3
-
1+3
-
-
2
-
2
Essence of Number
4
4
4

 

 

S
=
1
-
4
SUCH
51
15
6
L
=
3
-
9
LANGUAGES
87
33
6
A
=
1
-
3
ARE
24
15
6
F
=
6
-
3
FEW
34
16
7
A
=
1
-
3
AND
19
10
1
F
=
6
-
3
FAR
25
16
7
B
=
2
-
7
BETWEEN
74
29
2
B
=
2
-
3
BUT
43
7
7
M
=
4
-
11
MATHEMATICS
112
40
4
I
=
9
-
2
IS
28
10
1
O
=
6
-
3
ONE
34
16
7
O
=
6
-
2
OF
21
12
3
T
=
2
-
4
THEM
46
19
1
-
-
49
4
57
First Total
598
238
58
-
-
4+9
-
5+7
Add to Reduce
5+9+8
2+3+8
5+8
-
-
11
-
12
Second Total
22
13
13
-
-
1+2
-
1+2
Reduce to Deduce
2+2
1+3
1+3
-
-
3
-
3
Essence of Number
4
4
4

 

 

A
=
1
-
1
A
1
1
1
L
=
3
-
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
L
=
3
-
7
LETTERS
99
27
9
A
=
1
-
3
AND
19
10
1
N
=
5
-
7
NUMBERS
73
28
1
-
-
19
4
28
First Total
299
110
20
-
-
1+9
-
2+8
Add to Reduce
2+9+9
1+1+0
2+0
-
-
10
-
10
Second Total
20
2
2
-
-
1+0
-
1+0
Reduce to Deduce
2+0
-
-
-
-
1
-
1
Essence of Number
2
2
2

 

 

W
=
5
-
4
WHAT
52
16
7
O
=
6
-
3
ONE
34
16
7
W
=
5
-
5
WOULD
75
21
3
L
=
3
-
4
LOOK
53
17
8
F
=
6
-
3
FOR
30
21
3
T
=
2
-
9
THEREFORE
100
46
1
W
=
5
-
5
WOULD
75
21
3
B
=
2
-
2
BE
7
7
7
A
=
1
-
1
A
1
1
1
U
=
3
-
9
UNIVERSAL
121
40
4
L
=
3
-
8
LANGUAGE
68
32
5
T
=
2
-
3
THE
33
15
6
K
=
2
-
4
KIND
38
20
2
O
=
6
-
2
OF
21
12
3
L
=
3
-
8
LANGUAGE
68
32
5
C
=
3
-
4
THAT
144
72
9
T
=
2
-
5
WOULD
35
8
8
A
=
1
-
2
BE
40
13
4
T
=
2
-
14
COMPREHENSIBLE
161
71
8
A
=
1
-
2
TO
54
27
9
S
=
1
-
3
ANY
96
33
6
I
=
9
-
15
TECHNOLOGICALLY
23
14
5
A
=
1
-
2
ADVANCED
40
13
4
E
=
5
-
7
SOCIETY
48
29
2
T
=
2
-
2
IN
49
13
4
W
=
5
-
3
ANY
75
21
3
B
=
2
-
5
EPOCH
7
7
7
S
=
1
-
4
SUCH
51
15
6
L
=
3
-
9
LANGUAGES
87
33
6
A
=
1
-
3
ARE
24
15
6
F
=
6
-
3
FEW
34
16
7
A
=
1
-
3
AND
19
10
1
F
=
6
-
3
FAR
25
16
7
B
=
2
-
7
BETWEEN
74
29
2
B
=
2
-
3
BUT
43
7
7
M
=
4
-
11
MATHEMATICS
112
40
4
I
=
9
-
2
IS
28
10
1
O
=
6
-
3
ONE
34
16
7
O
=
6
-
2
OF
21
12
3
T
=
2
-
4
THEM
46
19
1
A
=
1
-
1
A
1
1
1
L
=
3
-
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
L
=
3
-
7
LETTERS
99
27
9
A
=
1
-
3
AND
19
10
1
N
=
5
-
7
NUMBERS
73
28
1
-
-
157
-
228
First Total
2444
986
212
-
-
1+5+7
-
2+2+8
Add to Reduce
2+4+4+4
9+8+6
2+1+2
-
-
13
-
12
Second Total
14
23
5
-
-
1+3
-
1+2
Reduce to Deduce
1+4
2+3
-
-
-
4
-
3
Essence of Number
5
5
5

 

 

MATHEMATICS A LANGUAGE

OF

LETTER AND NUMBER

 

 

A
=
1
-
1
A
1
1
1
L
=
3
-
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
L
=
3
-
6
LETTER
80
26
8
A
=
1
-
3
AND
19
10
1
N
=
5
-
6
NUMBER
73
28
1
S
-
19
4
26
First Total
261
108
18
-
-
1+9
-
2+6
Add to Reduce
2+6+1
1+0+8
1+8
-
-
10
-
8
Second Total
9
9
9
-
-
1+0
-
-
Reduce to Deduce
2+6+1
1+0+8
1+8
-
-
1
-
8
Essence of Number
9
9
9

 

 

 

 

-
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

15
THE RAINBOW LIGHT
-
-
-
-
THE
33
15
6
-
R
18
9
9
-
A
1
1
1
-
I
9
9
9
-
N+B+O+W
54
18
9
-
L
12
3
3
-
I
9
9
9
-
G+H+T
35
17
8
15
THE RAINBOW LIGHT
171
81
54
1+5
-
1+7+1
8+1
5+4
6
THE RAINBOW LIGHT
9
9
9

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
-
7
-
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
-
7
-
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
-
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
-
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
-
9
-
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
-
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
-
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
-
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

 

 

 

 

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:

Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly. too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10

Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.

Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.

The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.

A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the /Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)

If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.

This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.

Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.

But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he gave figures of 45" to 46" a year, while the truer precessional lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less than the figure of 25,920.

With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, Incorrect by a difference of from 2,397.9 years to 3,024.

Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'

AN IMPORTANT POSTSCRIPT

Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.

Page 206

Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12

This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.

Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.

Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.

Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13

Page 207

"...Feniusa Farsa, Graves equates this hero with Dionysus Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials

We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14

Graves did not know why this number was necessary, but he points out that he understands Frazer's Golden Bough to be a a book hinting that 'the secret involves the truth that the Christian dogma, and rituals, are the refinement of a great body of primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15

Frances A. Yates, historian of Renaissance hermetisma tells, us the cabala had 72 angels through which the sephiroth (the powers of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident there are seventy-two of Pico's Cabalist conclusions, for the conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16

In Hamlet's Mill de Santillarta adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss (60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)

Thee Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted
mounted in Genesis for the lifetimes of patriarchs from the Flood also contained the needed secret numbers. (He showed that in the 1,656.years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.) In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, .multiply this by 2 and again we have.the necessary 432 digits.

Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as.- the interest-
ing number of AD.432.18

Whatever one may think-of some of these number coincidences, it becomes. difficult to escape the suspicion that many signs (number and otherwise) -indicate that early man observed the results.. of the movement of Precession . and that the-.transmission of this information was .considered of prime importance.

'With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would
have constituted the construction-of a 'Unified Field Theory' for nothing .less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.

But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced.,that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':

"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two,four.Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radiusekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up. his piece of striped and striped wallpaper, decorated / Page 209 / with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.

So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"

 

 

3
THE
33
15
6
4
NINE
42
24
6
7
BILLION
73
37
1
5
NAMES
52
16
7
2
OF
21
12
3
3
GOD
26
17
8
24
-
247
121
31
2+4
-
2+4+7
1+2+1
3+1
6
-
13
4
4
-
-
1+3
-
-
6
-
4
4
4

 

 

OF TIME AND STARS

Arthur C. Clarke 1972

FOREWORD

"'Into the Comet' and 'The Nine Billion Names of God' both involve computers and the troubles they may cause us. While writing this preface, I had occasion to call upon my own HP 9100A computer, Hal Junior, to answer an interesting question. Looking at my records, I find that I have now written just about one hundred short stories. This volume contains eighteen of them: therefore, how many possible 18-story collections will I be able to put together? The answer ­as I am sure will be instantly obvious to you - is 100 x 99. . . x 84 x 83 divided by 18 x 17 x 16 ... x .2 x 1. This is an impressive number - Hal Junior tells me that it is approximately 20,772,733,124,605,000,000.

Page 15

The Nine Billion Names of God


'This is a slightly unusual request,' said Dr Wagner, with what he hoped was commendable restraint. 'As far as I know, it's the first time anyone's been asked to supply a Tibetan monastery with an Automatic Sequence Computer. I don't wish to be inquisitive, but I should hardly have thought that your - ah - establishment had much use for such a machine. Could you explain just what you intend to do with it?'
'Gladly,' replied the lama, readjusting his silk robes and carefully putting away the slide rule he had been using far currency conversions. 'Your Mark V Computer can carry out any routine mathematical operation involving up to ten digits. However, for our work we are interested in letters, not numbers. As we wish you to modify the output circuits, the machine will be printing words, not columns of figures.'
'I don't quite understand. . .'
'This is a project on which we have been working for the last three centuries - since the lamasery was founded, in fact. It is somewhat alien to your way of thought, so I hope you will listen with an open mind while I explain it.'
'Naturally.'
'It is really quite simple. We have been compiling a list which shall contain all the possible names of God.'
'I beg your pardon?'

Page16

'We have reason to believe,' continued the lama imperturbably, 'that all such names can be written with not more than nine letters in an alphabet we have devised.'
'And you have been doing this for three centuries?'
'Yes: we expected it would take us about fifteen thousand years to complete the task.'
'Oh,' Dr Wagner looked a little dazed. 'Now I see why you wanted to hire one of our machines. But what exactly is the purpose of this project?'
The lama hesitated for a fraction of a second, and Wagner wondered if he had offended him. If so, there was no trace of annoyance in the reply.
'Call it ritual, if you like, but it's a fundamental part of our belief. All the many names of the Supreme Being - God Jehova, Allah, and so on - they are only man-made labels. There is a philosophical problem of some difficulty here, which I do not propose to discuss, but somewhere among all the possible combinations of letters that can occur are what one may call the real names of God. By systematic permutation of letters, we have been trying to list them all.'
'I see. You've been starting at AAAAAAA . . . and working up to ZZZZZZZZ . . .'
'Exactly - though we use a special alphabet of our own. Modifying the electromatic typewriters to deal with this is, of course, trivial. A rather more interesting problem is that of devising suitable circuits to eliminate ridiculous combinations. For example, no letter must occur more than three times in succession.'
,'Three? Surely you mean two.'
'Three is correct: I am afraid it would take too long to explain why, even if you understood our language.' "

Page 68

Into the Comet


"Pickett's fingers danced over the beads, sliding them up and down the wires with lightning speed. There were twelve wires in all, so that the abacus could handle numbers up to 999,999,999,999 - or could be divided into separate sections where several independent calculations could be carried out simultaneously.
'374072,' said Pickett, after an incredibly brief interval of time. 'Now see how long you take to do it, with pencil and paper.'
There was a much longer delay before Martens, who like most mathematicians was poor at arithmetic, called out '375072'. A hasty check soon confirmed that Martens had taken at least three times as long as Pickett to arrive at the wrong answer.
The atronomer's face was a study in mingled chagrin, astonishment, and curiosity.
'Where did you learn that trick?' he asked. 'I thought those things could only add and subtract.'
'Well - multiplication's only repeated addition, isn't it? All I did was to add 856 seven times in the unit column, three times in the tens column, and four times in the hundreds column. You do the same thing when you use pencil and paper. Of course, there are some short cuts, but if you think I'm fast, you should have seen my granduncle. He used to work in a Yokohama bank, and you couldn't see his fingers / Page 69 / when he was going at speed"

 

 

I

SAY

THREAD THAT THREAD

THREAD READ DEATH DEATH READ THREAD

THREAD R DEATH DEATH R THREAD

THREAD READ DEAR DAERHT

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

Circa 1900

FOURTH EDITION

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

FOURTH EDITION

Circa 1900

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

lisle thread: lisle thread

A strong tightly twisted cotton thread (usually made of long-staple cotton) - lisle. Derived forms: lisle threads. Type of: cotton. Nearest ... www.wordwebonline.com/en/LISLETHREAD

 

 

Definition - of Lisle from Dictionary.net

Lisle thread, a hard twisted cotton thread, originally produced at Lisle. Source: Webster's Revised Unabridged Dictionary (1913) ... www.dictionary.net/lisle - 9k

 

 

CASSELL'S ENGLISH DICTIONARY

1974

Lisle thread (lil thred) [ town in France, now Lille], n, A fine, hard thread orig. made at Lille.

 

 

HOLY BIBLE

C

9 V 9

Page 575

JOB

Which maketh Arcturus, Orion, and Pleiades, and the chambers of the south.

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN SOME AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

 

 

-
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
-
189
90
18
1+8
-
1+8+9
9+0
1+8
9
-
18
9
9
-
-
1+8
-
-
9
-
9
9
9

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN SOME AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

-
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

Shakespeare Quotes - Such Stuff as Dreams Are Made on.
www.enotes.com/shakespeare-quotes/we-such-stuff-dreams-made

The Tempest Act 4, scene 1, William Shakespeare

Prospero:
Our revels now are ended. These our actors,
As I foretold you, were all spirits, and
Are melted into air, into thin air:
And like the baseless fabric of this vision,
The cloud-capp'd tow'rs, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And, like this insubstantial pageant faded,
Leave not a rack behind. We are such stuff
As dreams are made on; and our little life
Is rounded with a sleep.

 

 

2
WE
28
10
1
3
ARE
24
15
6
4
SUCH
51
15
6
5
STUFF
72
18
9
2
AS
20
2
2
6
DREAMS
60
24
6
3
ARE
24
15
6
4
MADE
23
14
5
2
ON
29
11
2
3
AND
19
10
1
3
OUR
54
18
9
4
LITTLE
78
24
6
4
LIFE
32
23
5
2
IS
28
10
1
7
ROUNDED
81
36
9
4
WITH
60
24
6
1
A
1
1
1
5
SLEEP
57
21
3
66
First Total
741
291
84
6+6
Add to Reduce
7+4+1
2+9+1
8+4
12
Second Total
12
12
12
1+2
Reduce to Deduce
1+2
1+2
1+2
3
Essence of Number
3
3
3

 

 

F
=
6
3
FOR
56
29
2
E
=
5
5
EVERY
28
19
1
A
=
1
6
ACTION
29
11
2
T
=
2
5
THERE
26
17
8
I
=
9
2
IS
43
7
7
A
=
1
2
AN
26
17
8
E
=
5
5
EQUAL
56
29
2
A
=
1
3
AND
28
19
1
O
=
6
8
OPPOSITE
29
11
2
R
=
9
8
REACTION
26
17
8
-
-
45
47
First Total
550
235
37
-
-
4+5
4+7
Add to Reduce
5+5+0
2+3+5
3+7
Q
-
5
11
Second Total
10
1
10
-
-
-
1+1
Reduce to Deduce
1+0
-
 1+0
-
-
5
2
Essence of Number
1
1
1

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN SOME AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

-
MATHEMATICS
-
-
-
4
MATH
42
15
6
2
ME
18
9
9
2
AT
21
3
3
1
I
9
9
9
2
CS
22
4
4
11
MATHEMATICS
112
40
31
1+1
-
1+1+2
4+0
3+1
2
MATHEMATICS
4
4
4

 

 

THE MAGIC MOUNTAIN


Thomas Mann 1875-1955

Page 417

"I preach mathematics."

 

 

 

-
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
9
-
1
+
=
18
1+8
=
9
=
9
-
-
-
-
-
8
-
-
-
-
9
-
19
+
=
36
3+6
=
9
=
9
-
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
-
4
1
2
-
5
4
1
2
-
3
-
+
=
22
2+2
=
4
=
4
-
-
13
1
20
-
5
13
1
20
-
3
-
+
=
76
7+6
=
13
1+3
4
-
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
13
1
20
8
5
13
1
20
9
3
19
+
=
112
1+1+2
=
4
=
4
-
4
1
2
8
5
4
1
2
9
3
1
+
=
40
4+0
=
4
=
4
-
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
13
11
M
A
T
H
E
M
A
T
I
C
S
-
-
32
-
-
11
-
40
1+3
1+1
-
-
-
-
-
-
-
-
9
-
-
-
-
3+2
-
-
1+1
-
4+0
4
2
M
A
T
H
E
M
A
T
I
C
S
-
-
5
-
-
2
-
4
-
-
4
1
2
8
5
4
1
2
9
3
1
-
-
-
-
-
-
-
-
4
2
M
A
T
H
E
M
A
T
I
C
S
-
-
5
-
-
2
-
4

 

 

11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
9
-
1
+
=
18
1+8
=
9
=
9
-
-
-
-
8
-
-
-
-
9
-
19
+
=
36
3+6
=
9
=
9
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
4
1
2
-
5
4
1
2
-
3
-
+
=
22
2+2
=
4
=
4
-
13
1
20
-
5
13
1
20
-
3
-
+
=
76
7+6
=
13
1+3
4
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
13
1
20
8
5
13
1
20
9
3
19
+
=
112
1+1+2
=
4
=
4
4
1
2
8
5
4
1
2
9
3
1
+
=
40
4+0
=
4
=
4
11
M
A
T
H
E
M
A
T
I
C
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
-
-
-
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
11
M
A
T
H
E
M
A
T
I
C
S
-
-
32
-
-
11
-
40
1+1
-
-
-
-
-
-
-
-
9
-
-
-
-
3+2
-
-
1+1
-
4+0
2
M
A
T
H
E
M
A
T
I
C
S
-
-
5
-
-
2
-
4
-
4
1
2
8
5
4
1
2
9
3
1
-
-
-
-
-
-
-
-
2
M
A
T
H
E
M
A
T
I
C
S
-
-
5
-
-
2
-
4

 

 

MATHEMATICS MINUS I MINUS I MATHEMATICS

 

 

11
M
A
T
H
E
M
A
T
C
S
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
1
+
=
18
1+8
=
9
=
9
-
-
-
-
8
-
-
-
-
-
19
+
=
36
3+6
=
9
=
9
11
M
A
T
H
E
M
A
T
C
S
-
-
-
-
-
-
-
-
-
4
1
2
-
5
4
1
2
3
-
+
=
22
2+2
=
4
=
4
-
13
1
20
-
5
13
1
20
3
-
+
=
76
7+6
=
13
1+3
4
11
M
A
T
H
E
M
A
T
C
S
-
-
-
-
-
-
-
-
13
1
20
8
5
13
1
20
3
19
+
=
112
1+1+2
=
4
=
4
4
1
2
8
5
4
1
2
3
1
+
=
40
4+0
=
4
=
4
11
M
A
T
H
E
M
A
T
C
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
-
-
1
occurs
x
3
=
3
-
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
4
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
2
=
8
-
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
11
M
A
T
H
E
M
A
T
C
S
-
-
32
-
-
11
-
40
1+1
-
-
-
-
-
-
-
-
-
-
-
-
3+2
-
-
1+1
-
4+0
2
M
A
T
H
E
M
A
T
C
S
-
-
5
-
-
2
-
4
-
4
1
2
8
5
4
1
2
3
1
-
-
-
-
-
-
-
-
2
M
A
T
H
E
M
A
T
C
S
-
-
5
-
-
2
-
4

 

 

-
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
=
=
-
-
-
8
-
-
-
-
9
-
9
`-
5
+
=
31
3+1
=
4
=
4
=
4
=
=
-
-
-
8
-
-
-
-
9
-
9
-
14
+
=
40
4+0
=
4
=
4
=
4
-
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
=
=
4
1
2
-
5
4
1
2
-
3
-
1
-
+
=
23
2+3
=
5
=
5
=
5
-
=
13
1
20
-
5
13
1
20
-
3
-
1
-
+
=
77
7+7
=
14
1+4
5
=
5
-
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
-
-
13
1
20
8
5
13
1
20
9
3
9
1
14
+
=
117
1+1+7
=
9
=
9
=
9
-
-
4
1
2
8
5
4
1
2
9
3
9
1
5
+
=
54
5+4
=
9
=
9
=
9
-
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
--
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
--
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
13
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
32
-
-
13
-
54
-
36
1+3
1+3
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
3+2
-
-
1+3
-
5+4
-
3+6
4
4
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
5
-
-
4
-
9
-
9
--
--
4
1
2
8
5
4
1
2
9
3
9
1
5
-
-
-
-
-
--
-
-
-
-
4
4
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
5
-
-
4
-
9
-
9

 

 

13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
=
-
-
-
8
-
-
-
-
9
-
9
`-
5
+
=
31
3+1
=
4
=
4
=
4
=
-
-
-
8
-
-
-
-
9
-
9
-
14
+
=
40
4+0
=
4
=
4
=
4
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
=
4
1
2
-
5
4
1
2
-
3
-
1
-
+
=
23
2+3
=
5
=
5
=
5
=
13
1
20
-
5
13
1
20
-
3
-
1
-
+
=
77
7+7
=
14
1+4
5
=
5
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
-
13
1
20
8
5
13
1
20
9
3
9
1
14
+
=
117
1+1+7
=
9
=
9
=
9
-
4
1
2
8
5
4
1
2
9
3
9
1
5
+
=
54
5+4
=
9
=
9
=
9
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
4
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
13
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
32
-
-
13
-
54
-
36
1+3
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
3+2
-
-
1+3
-
5+4
-
3+6
4
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
5
-
-
4
-
9
-
9
--
4
1
2
8
5
4
1
2
9
3
9
1
5
-
-
-
-
-
--
-
-
-
-
4
M
A
T
H
E
M
A
T
I
C
I
A
N
-
-
5
-
-
4
-
9
-
9

 

 

-
-
-
-
ISAAC NEWTON
-
-
-
I
=
9
5
ISAAC
33
15
6
N
=
5
6
NEWTON
91
28
1
-
-
14
11
ISAAC NEWTON
124
43
7
-
-
4+5
 1+1
-
1+2+4
4+3
-
-
-
5
2
ISAAC NEWTON
7
7
7

 

 

9
PRINCIPIA
95
59
5
11
MATHEMATICA
94
40
4
20
First Total
189
99
9
2+0
Add to Reduce
1+8+9
9+9
-
2
Second Total
18
18
9
-
Reduce to Deduce
1+8
1+8
-
2
Essence of Number
9
9
9

 

 

 

 

SUPER SCIENCE

Michael White

1

999

"The discovery of the theory of universal gravitation took

NEWTON

almost twenty years and did not really take shape until he wrote his great book

THE

PRINCIPIA MATHEMATICA

between 1684 and its publication in 1687."

 

 

20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
+
=
36
9
NINE
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
+
=
8
8
EIGHT
8
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
=
5
5
FIVE
5
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
16
18
9
14
3
9
16
9
1
-
13
1
20
8
5
13
1
20
9
3
1
+
=
189
9
NINE
9
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9
NINE
9
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
-
-
-
1
+
=
4
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
+
=
4
4
FOUR
4
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
+
=
6
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
+
=
8
8
EIGHT
8
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
+
=
10
1
ONE
1
-
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
=
14
5
FIVE
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
+
=
8
8
EIGHT
8
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
+
=
45
9
NINE
9
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
99
45
NINE
9
2+0
7
9
9
5
3
9
7
9
1
--
4
1
2
8
5
4
1
2
9
3
1
-
-
9+9
4+5
-
-
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
18
9
-
-
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1+8
-
-
-
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
9
9
NINE
9
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9
NINE
9
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
9
9
NINE
9

 

 

20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
9
5
-
9
-
9
-
-
-
-
-
8
-
-
-
-
9
-
-
+
=
49
4
FOUR
4
-
-
-
9
14
-
9
-
9
-
-
-
-
-
8
-
-
-
-
9
-
-
+
=
58
4
FOUR
4
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
7
9
-
-
3
-
7
-
1
-
4
1
2
-
5
4
1
2
-
3
1
+
=
50
5
FIFTY
5
-
16
18
-
-
3
-
16
-
1
-
13
1
20
-
5
13
1
20
-
3
1
+
=
131
5
FIFTY
5
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
16
18
9
14
3
9
16
9
1
-
13
1
20
8
5
13
1
20
9
3
1
+
=
189
9
NINE
9
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9
NINE
9
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
-
-
-
1
+
=
4
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
+
=
4
4
FOUR
4
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
+
=
6
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
+
=
8
8
EIGHT
8
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
+
=
10
1
ONE
1
-
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
=
14
5
FIVE
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
+
=
8
8
EIGHT
8
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
+
=
45
9
NINE
9
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
99
45
NINE
9
2+0
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
-
-
9+9
4+5
-
-
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
18
9
-
-
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1+8
-
-
-
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
9
9
NINE
9
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
-
-
-
-
-
-
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
9
9
NINE
9

 

 

-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
+
=
36
-
-
9
NINE
9
NINE
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
+
=
8
-
-
8
EIGHT
8
EIGHT
8
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
=
5
-
-
5
FIVE
5
FIVE
5
-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
16
18
9
14
3
9
16
9
1
-
13
1
20
8
5
13
1
20
9
3
1
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9+9
=
18
1+8
9
=
9
-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
--
-
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
6
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
39
-
-
20
-
99
-
45
-
2+0
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+9
-
-
2+0
-
9+9
-
4+5
6
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
12
-
-
2
-
18
-
9
-
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
-
-
1+2
-
-
-
-
1+8
-
-
6
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
3
-
-
2
-
9
-
9

 

 

-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
9
-
9
-
-
-
-
-
8
-
-
-
-
9
-
-
+
=
49
4+9
=
13
1+3
4
=
4
-
-
-
-
9
14
-
9
-
9
-
-
-
-
-
8
-
-
-
-
9
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
7
9
-
-
3
-
7
-
1
-
4
1
2
-
5
4
1
2
-
3
1
+
=
50
5+0
=
5
=
5
=
5
-
-
16
18
-
-
3
-
16
-
1
-
13
1
20
-
5
13
1
20
-
3
1
+
=
131
1+3+1
=
5
=
5
=
5
-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
16
18
9
14
3
9
16
9
1
-
13
1
20
8
5
13
1
20
9
3
1
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9+9
=
18
1+8
9
=
9
-
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
--
-
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
6
20
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
39
-
-
20
-
99
-
45
-
2+0
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+9
-
-
2+0
-
9+9
-
4+5
6
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
12
-
-
2
-
18
-
9
-
-
7
9
9
5
3
9
7
9
1
-
4
1
2
8
5
4
1
2
9
3
1
-
-
1+2
-
-
-
-
1+8
-
-
6
2
P
R
I
N
C
I
P
I
A
-
M
A
T
H
E
M
A
T
I
C
A
-
-
3
-
-
2
-
9
-
9

 

 

20
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
9
-
9
-
-
-
-
8
-
-
-
-
9
-
-
+
=
49
4+9
=
13
1+3
4
=
4
-
-
-
9
14
-
9
-
9
-
-
-
-
8
-
-
-
-
9
-
-
+
=
58
5+8
=
13
1+3
4
=
4
20
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
7
9
-
-
3
-
7
-
1
4
1
2
-
5
4
1
2
-
3
1
+
=
50
5+0
=
5
=
5
=
5
-
16
18
-
-
3
-
16
-
1
13
1
20
-
5
13
1
20
-
3
1
+
=
131
1+3+1
=
5
=
5
=
5
20
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
16
18
9
14
3
9
16
9
1
13
1
20
8
5
13
1
20
9
3
1
+
=
189
1+8+9
=
18
1+8
9
=
9
-
7
9
9
5
3
9
7
9
1
4
1
2
8
5
4
1
2
9
3
1
+
=
99
9+9
=
18
1+8
9
=
9
20
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
20
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
39
-
-
20
-
99
-
45
2+0
-
9
9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+9
-
-
2+0
-
9+9
-
4+5
2
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
12
-
-
2
-
18
-
9
-
7
9
9
5
3
9
7
9
1
4
1
2
8
5
4
1
2
9
3
1
-
-
1+2
-
-
-
-
1+8
-
-
2
P
R
I
N
C
I
P
I
A
M
A
T
H
E
M
A
T
I
C
A
-
-
3
-
-
2
-
9
-
9

 

 

-
12
M
A
T
H
E
M
A
T
I
K
O
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
9
-
6
9
+
=
32
3+2
=
5
=
5
=
5
-
-
-
-
-
8
-
-
-
-
9
-
15
9
+
=
41
4+1
=
5
=
5
=
5
-
12
M
A
T
H
E
M
A
T
I
K
O
I
-
-
-
-
-
-
-
-
-
-
-
-
4
1
2
-
5
4
1
2
-
2
-
-
+
=
21
2+1
=
3
=
3
=
3
-
-
13
1
20
-
5
13
1
20
-
11
-
-
+
=
84
8+4
=
12
1+2
3
=
3
-
12
M
A
T
H
E
M
A
T
I
K
O
I
-
-
-
-
-
-
-
-
-
-
-
-
13
1
20
8
5
13
1
20
9
11
15
9
+
=
125
1+2+5
=
8
=
8
=
8
-
-
4
1
2
8
5
4
1
2
9
2
6
9
+
=
53
5+3
=
8
=
8
=
8
-
12
M
A
T
H
E
M
A
T
I
K
O
I
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
2
-
2
-
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
10
12
M
A
T
H
E
M
A
T
I
K
O
I
-
-
35
-
-
12
-
53
-
44
1+0
1+2
-
-
-
-
-
-
-
-
9
-
-
9
-
-
3+5
-
-
1+2
-
5+3
-
4+4
1
3
M
A
T
H
E
M
A
T
I
K
O
I
-
-
8
-
-
3
-
8
-
8
-
-
4
1
2
8
5
4
1
2
9
2
6
9
-
-
-
-
-
-
-
-
-
-
1
3
M
A
T
H
E
M
A
T
I
K
O
I
-
-
8
-
-
3
-
8
-
8

 

 

Nature's Numbers
Ian Stewart 1995

Numerology is the easiest-and consequently the most dangerous-method for finding patterns. It is easy because anybody can do it and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. Here's a case in point. Kepler was fascinated with patterns in nature, and he devoted much of his life to looking for them in the behaviour of the planets. He devised a simple and tidy theory for the existence of precisely six planets (in his time only Mercury, Venus, Earth, Mars, Jupiter, and Saturn were known). He also discovered a very strange pattern relating the orbital period of a / planet- the time it takes to go once around the Sun-to its distance from the Sun. Recall that the square of a number is what you get when you multiply it by itself: for example, the square of 4 is 4 x 4 = 16. Similarly, the cube is what you get when you multiply it by itself twice: for example, the cube of 4 is 4 x 4 x 4 = 64. Kepler found that if you take the cube of the distance of any planet from the Sun and divide it by the square of its orbital period, you always get the same number. It was not an especially elegant number, but it was the same for all six planets.

Which of these numerological observations is the more significant? The verdict of posterity is that it is the second one, the complicated and rather arbitrary calculation with squares and cubes. This numerical pattern was one of the key steps towards Isaac Newton's theory of gravity, which has explained all sorts of puzzles about the motion of stars and planets. In contrast, Kepler's neat, tidy theory for the number of planets has been buried without trace. For a start it must have been wrong, because we now know of nine planets, not six. There could be even more, farther out from the Sun, and small enough to be undetectable But more important, we no longer expect to find a neat, tidy theory for the number of planets. We think that the Solar System condensed from a cloud of gas surrounding the Sun, and the number of planets presumably depended on the amount of matter in the gas cloud, how it was distributed, and how fast and in what directions it was moving. An equally plausible gas cloud could have given us eight planets, or eleven; the number is accidental, depending on the initial conditions of the gas cloud, rather than universal, reflecting a general law of nature"

Page 6

" The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which. For example, there are three stars, roughly equally spaced and in a straight line, in the belt of the constellation Orion. Is that a clue to a significant law of nature?
Here's a similar question. Io, Europa, and Ganymede are three of Jupiter's larger satellites. They orbit the planet in , respectively, 1.77, 3.55, and 7.16 days. Each of these numbers is almost exactly twice the previous one. Is that a significant pattern? Three stars in a row, in terms of orbital period. Which pattern if either, is an important clue..."
    "… In addition to numerical patterns there are geometric ones…"
    "… Until recently the main shapes that appealed to mathematicians were very simple ones: triangles, squares, pen / Page 7 /tagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on. All of these shapes can be found in nature, although some are far more common, or more evident, than others. The rainbow, for example, is a collection of circles, one for each colour. We don't normally see the entire circle just an arc; but rainbows seen from the air can be complete circles. You also see circles in the ripples on a pond, in the human eye, and on butterflies wings.
         Talking of ripples, the flow of fluids provides an inexhaustible supply of nature's patterns. There are waves of many different kinds-surging toward a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outward from an underwater earthquake…"
"…There are swirling spiral whirlpools and tiny vortices. And there is the apparently structureless, random frothing of turbulent flow, one of the great enigmas of mathematics and physics. There are similar patterns in the atmosphere, too, the most dramatic being the vast spiral of a hurricane…"
    "…There are also wave patterns on land. The most strikingly mathematical landscapes on Earth are to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Even when the wind blows steadily in a fixed direction, sand dunes form. The simplest pattern is that of transverse dunes, which-just like ocean waves-line up in parallel straight rows at right angles to the prevailing wind direction. Sometimes the rows themselves become wavy in which case they are called barchanoid ridges; sometimes they break up into / Page 8 / innumerable shield-shaped barchan dunes. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes-shaped like a U, with the rounded end pointing in the direction of the wind. These sometimes occur in clusters, and they resemble the teeth of a rake. If the wind direction is variable, other forms become possible. For example, clusters of sand shaped dunes can form, each having several irregular arms radiating from a central peak. They arrange themselves in a random pattern of spots.

Chapter 6

Page 81

"Nature's symmetries can be found on every scale, from the structure of subatomic particles to that of the entire universe. Many chemical molecules are symmetric. The methane molecule is a tetrahedron - a triangular-sided pyramid - with one carbon atom at its center and four hydrogen atoms at its corners Benzene has the sixfold symmetry of a regular hexagon. The fashionable molecule buckminsterfullerene is a truncated icosahedral cage of sixty carbon atoms. (An icosahedron is a regular solid with twenty triangular faces;
"truncated" means that the corners are cut off.) Its symmetry lends it a remarkable stability, which has opened up new possibilities for organic chemistry.
    On a slightly larger scale than molecules, we find symmetries in cellular structure; at the heart of cellular replication lies a tiny piece of mechanical engineering. Deep within each / Page 82  / living cell, there is a rather shapeless structure known as the centrosome, which sprouts long thin microtubules, basic components of the cell's internal "skeleton", like a diminutive sea urchin. Centrsomes were first discovered in 1887 and play an important role in organizing cell division. How-ever in one respect the structure of the centresome is astonishingly symmetric. Inside it has two structures, known as centrioles, positioned at right angles to each other. Each centriole is cylindrical, made from twenty-seven microtubules fused together along their lengths in threes, and arranged with perfect ninefold symmetry. The microtubules themselves also have an astonishingley regular symmetric form. They are hollow tubes, made from a perfect regular checkerboard pattern of units that contain two distinct proteins, alpha- and betatubulin. One day, perhaps, we will understand why nature chose these symmetric forms. But it is amazing to see symmetric structures at the core of a living cell. "

 

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
58
13
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
-
5
THREE
56
29
2
-
-
2
-
-
-
-
-
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
42
24
6
-
-
-
-
-
-
6
-
-
-
6
-
S
=
1
-
3
SIX
52
16
7
-
-
-
-
-
-
-
7
-
-
7
-
S
=
1
-
5
SEVEN
65
20
2
-
-
2
-
-
-
-
-
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
-
4
-
-
-
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
-
-
6
-
-
-
45
-
-
-
42
-
40
 Add
522
225
45
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
4+2
-
4+0
 Reduce
5+2+2
2+2+5
4+5
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
6
-
4
 Deduce
9
9
9
-
1
4
3
8
5
9
5
8
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ZERO
-
-
-
-
-
-
-
-
-
-
-
-
-
Z
=
8
-
1
Z
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
28
-
4
ZERO
64
28
28
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
16
-
3
ONE
34
16
16
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
13
-
3
TWO
58
13
13
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
29
-
5
THREE
56
29
29
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
24
-
4
FOUR
60
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
24
-
4
FIVE
42
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
X
=
6
-
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
-
-
16
-
3
SIX
52
25
16
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
20
-
5
SEVEN
65
29
20
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
31
-
5
EIGHT
49
31
31
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
24
-
4
NINE
42
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
45
-
40
First Total
522
243
225
-
2
6
3
8
70
42
7
24
63
-
-
4+5
-
4+0
Add to Reduce
5+2+2
2+4+3
2+2+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
-
-
9
-
4
Second Total
9
9
9
-
2
6
3
8
7
6
7
6
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
6
3
8
7
6
7
6
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ZERO
-
-
-
-
-
-
-
-
-
-
-
-
-
Z
=
8
-
1
Z
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
28
-
4
ZERO
64
28
28
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
16
-
3
ONE
34
16
16
7
-
-
-
-
-
-
-
-
-
-
-
-
-
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
13
-
3
TWO
58
13
13
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
29
-
5
THREE
56
29
29
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
24
-
4
FOUR
60
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
24
-
4
FIVE
42
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
X
=
6
-
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
-
-
16
-
3
SIX
52
25
16
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
20
-
5
SEVEN
65
29
20
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
31
-
5
EIGHT
49
31
31
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
24
-
4
NINE
42
24
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
45
-
40
First Total
522
243
225
-
2
6
3
8
70
42
7
24
63
-
-
4+5
-
4+0
Add to Reduce
5+2+2
2+4+3
2+2+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
-
-
9
-
4
Second Total
9
9
9
-
2
6
3
8
7
6
7
6
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
6
3
8
7
6
7
6
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
=
8
-
1
Z
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
X
=
6
-
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
45
-
40
First Total
522
243
225
-
2
6
3
8
70
42
7
24
63
-
-
4+5
-
4+0
Add to Reduce
5+2+2
2+4+3
2+2+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
-
-
9
-
4
Second Total
9
9
9
-
2
6
3
8
7
6
7
6
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
6
3
8
7
6
7
6
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
V
=
4
-
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
W
=
5
-
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
X
=
6
-
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
H
=
8
-
1
Z
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
45
-
40
First Total
522
243
225
-
2
6
3
8
70
42
7
24
63
-
-
4+5
-
4+0
Add to Reduce
5+2+2
2+4+3
2+2+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
-
-
9
-
4
Second Total
9
9
9
-
2
6
3
8
7
6
7
6
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
6
3
8
7
6
7
6
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
58
13
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
-
5
THREE
56
29
2
-
-
2
-
-
-
-
-
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
42
24
6
-
-
-
-
-
-
6
-
-
-
6
-
S
=
1
-
3
SIX
52
16
7
-
-
-
-
-
-
-
7
-
-
7
-
S
=
1
-
5
SEVEN
65
20
2
-
-
2
-
-
-
-
-
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
-
4
-
-
-
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
-
-
6
-
-
-
45
-
-
-
42
-
40
 Add
522
225
45
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
4+2
-
4+0
 Reduce
5+2+2
2+2+5
4+5
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
6
-
4
 Deduce
9
9
9
-
1
4
3
8
5
9
5
8
9

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

 

8
CALCULUS
-
-
-
-
-
C
3
3
-
3
-
A
1
1
1
-
-
L
12
3
-
3
-
C
3
3
-
3
-
U
21
3
-
3
-
L
12
3
-
3
-
U
21
3
-
3
-
S
19
10
1
-
8
CALCULUS
92
29
2
18
-
-
9+2
2+9
-
1+8
8
CALCULUS
11
11
2
9
-
-
1+1
1+1
-
-
8
CALCULUS
2
2
2
9
-
-
-
-
-
-
-
-
-
-
-
-
11
ISAAC NEWTON
124
43
2
7

 

 

8
C
A
L
C
U
L
U
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
=
1
-
-
-
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
8
C
A
L
C
U
L
U
S
-
-
-
-
-
--
-
-
-
-
-
3
1
3
3
3
3
3
-
+
=
19
1+9
=
10
1+0
1
=
1
-
3
1
12
3
21
12
3
-
+
=
73
7+3
=
10
1+0
1
=
1
8
C
A
L
C
U
L
U
S
-
-
-
-
-
--
-
-
-
-
-
3
1
12
3
21
12
3
19
+
=
92
9+2
=
11
1+0
2
=
2
-
3
1
3
3
3
3
3
1
+
=
20
2+0
=
2
=
2
=
2
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
3
-
3
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
20
-
11
-
3
-
3
3
3
3
3
-
-
-
-
-
-
-
-
2+0
-
1+1
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2

 

 

11
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
6
5
+
=
20
2+0
=
1
-
1
-
1
-
-
-
-
-
-
-
-
-
-
9
15
14
+
=
38
3+8
=
11
2
-
2
11
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
1
2
-
-
-
+
=
19
1+9
=
11
2
-
2
-
-
3
1
12
3
21
12
1
20
-
-
-
+
=
73
7+3
=
10
1
-
1
11
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
1
20
9
15
14
+
=
111
1+1+1
=
3
-
3
-
3
-
-
3
1
3
3
3
3
1
2
9
6
5
+
=
39
3+9
=
12
1+2
3
-
3
11
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
-
2
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
-
2
-
2
-
-
3
-
3
3
3
3
-
-
-
-
-
-
-
3
occurs
x
5
-
15
1+5
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
=
-
5
occurs
x
1
-
5
-
5
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
-
6
-
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
-
9
-
9
19
11
C
A
L
C
U
L
A
T
I
O
N
-
-
26
-
-
11
-
39
-
30
1+9
1+1
3
-
3
3
3
3
-
-
-
-
-
-
-
2+6
-
-
1+1
-
3+9
-
3+0
10
2
C
A
L
C
U
L
A
T
I
O
N
-
-
8
-
-
2
-
12
-
3
1+0
-
3
1
3
3
3
3
1
2
9
6
5
-
-
-
-
-
-
-
1+2
-
-
1
2
C
A
L
C
U
L
A
T
I
O
N
-
-
8
-
-
2
-
3
-
3

 

 

11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
6
5
+
=
20
2+0
=
1
-
1
-
1
-
-
-
-
-
-
-
-
-
9
15
14
+
=
38
3+8
=
11
2
-
2
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
1
2
-
-
-
+
=
19
1+9
=
11
2
-
2
-
3
1
12
3
21
12
1
20
-
-
-
+
=
73
7+3
=
10
1
-
1
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
1
20
9
15
14
+
=
111
1+1+1
=
3
-
3
-
3
-
3
1
3
3
3
3
1
2
9
6
5
+
=
39
3+9
=
12
1+2
3
-
3
11
C
A
L
C
U
L
A
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
-
2
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
-
2
-
2
-
3
-
3
3
3
3
-
-
-
-
-
-
-
3
occurs
x
5
-
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
5
=
-
5
occurs
x
1
-
5
-
5
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
-
6
-
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
-
9
-
9
11
C
A
L
C
U
L
A
T
I
O
N
-
-
26
-
-
11
-
39
-
30
1+1
3
-
3
3
3
3
-
-
-
-
-
-
-
2+6
-
-
1+1
-
3+9
-
3+0
2
C
A
L
C
U
L
A
T
I
O
N
-
-
8
-
-
2
-
12
-
3
-
3
1
3
3
3
3
1
2
9
6
5
-
-
-
-
-
-
-
1+2
-
-
2
C
A
L
C
U
L
A
T
I
O
N
-
-
8
-
-
2
-
3
-
3

 

 

I
=
9
-
-
CALCULATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
2
A+T
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3
O+N+S
48
12
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
12
CALCULATIONS
130
40
31
-
1
2
21
4
5
6
7
8
9
-
-
-
-
1+2
-
1+3+0
4+0
3+1
-
-
-
2+1
-
-
-
-
-
-
I
=
9
-
3
CALCULATIONS
4
4
4
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
4
CALCULATIONS
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
10
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
58
First Total
995
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+8
Add to Reduce
9+9+5
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
13
Second Total
23
14
10
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+3
Reduce to Deduce
2+3
1+4
1+0
-
-
-
-
-
-
-
-
-
-
-
-
8
-
4
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

S
=
1
-
-
SPECULATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
2
P+E
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
2
A+T
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3
O+N+S
48
12
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
12
SPECULATIONS
154
46
28
-
1
2
18
4
5
6
7
8
9
-
-
-
-
1+2
-
1+5+4
4+6
2+8
-
-
-
1+8
-
-
-
-
-
-
S
=
1
-
3
SPECULATIONS
10
10
10
-
1
2
9
4
5
6
7
8
9
-
-
-
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
S
=
1
-
3
SPECULATIONS
1
1
1
-
1
2
9
4
5
6
7
8
9

 

 

I
=
9
-
-
ILLUSTRATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
2
S+T
39
12
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
2
A+T
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
2
O+N+S
48
21
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
13
ILLUSTRATIONS
189
72
45
-
1
2
3
4
5
6
7
8
18
-
-
-
-
1+3
-
1+8+9
7+2
4+5
-
-
-
-
-
-
-
-
-
1+8
I
=
9
-
4
ILLUSTRATIONS
18
9
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+8
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
4
ILLUSTRATIONS
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

I
=
9
-
-
ILLUSTRATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
2
I+L
21
12
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
S+T+R
57
21
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
A+T+I
30
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
O+N+S
48
21
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
13
ILLUSTRATIONS
189
63
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+3
-
1+8+9
6+3
1+8
-
-
-
-
-
-
-
-
-
-
I
=
9
-
4
ILLUSTRATIONS
18
9
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+8
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
4
ILLUSTRATIONS
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

I
=
9
-
-
ILLUMINED
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
-
-
-
-
2
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
9
ILLUMINED
99
45
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
9+9
4+5
1+8
-
-
-
-
-
-
-
-
-
-
I
=
9
-
9
ILLUMINED
18
9
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+8
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
9
ILLUMINED
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

I
=
9
-
-
ILLUMINED
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
M+I+N
36
18
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
2
E+D
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
9
ILLUMINED
99
45
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
9+9
4+5
1+8
-
-
-
-
-
-
-
-
-
-
I
=
9
-
9
ILLUMINED
18
9
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+8
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
9
ILLUMINED
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

B
=
2
-
4
BOOK
43
16
7
O
=
6
-
2
OF
21
12
3
T
=
2
-
3
THE
33
15
6
A
=
1
-
6
ABACUS
47
11
2
-
-
11
-
15
First Total
144
54
18
-
-
1+1
-
1+5
Add to Reduce
1+4+4
5+4
1+8
-
-
2
-
6
Second Total
9
9
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
2
-
6
Essence of Number
9
9
9

 

 

-
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
6
-
-
-
8
-
-
-
-
-
-
-
1
+
=
27
2+7
=
9
=
9
=
9
-
-
-
15
15
-
-
15
-
-
-
8
-
-
-
-
-
-
-
19
+
=
72
7+2
=
9
=
9
=
9
-
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
6
-
2
-
5
-
1
2
1
3
3
-
+
=
27
2+7
=
9
=
9
=
9
-
-
2
-
-
11
-
-
6
-
20
-
5
-
1
2
1
3
21
-
+
=
72
7+2
=
9
=
9
=
9
-
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
2
15
15
11
-
15
6
-
20
8
5
-
1
2
1
3
21
19
+
=
144
1+4+4
=
9
=
9
=
9
-
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
+
=
54
5+4
=
9
=
9
=
9
-
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
``-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
6
6
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
NINE
-
-
-
-
-
-
20
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
25
-
-
15
-
54
-
36
2+0
1+5
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2+5
-
-
1+5
-
5+4
-
3+6
2
6
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
6
-
9
-
9
-
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
-
-
-
-
-
-
-
-
-
-
2
6
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
6
-
9
-
9

 

 

15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
6
-
-
-
8
-
-
-
-
-
-
-
1
+
=
27
2+7
=
9
=
9
=
9
-
-
15
15
-
-
15
-
-
-
8
-
-
-
-
-
-
-
19
+
=
72
7+2
=
9
=
9
=
9
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
6
-
2
-
5
-
1
2
1
3
3
-
+
=
27
2+7
=
9
=
9
=
9
-
2
-
-
11
-
-
6
-
20
-
5
-
1
2
1
3
21
-
+
=
72
7+2
=
9
=
9
=
9
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
2
15
15
11
-
15
6
-
20
8
5
-
1
2
1
3
21
19
+
=
144
1+4+4
=
9
=
9
=
9
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
+
=
54
5+4
=
9
=
9
=
9
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
1
occurs
x
3
=
3
=
3
``-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
6
6
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
15
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
25
-
-
15
-
54
-
36
1+5
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2+5
-
-
1+5
-
5+4
-
3+6
6
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
6
-
9
-
9
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
-
-
-
-
-
-
-
-
-
-
6
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
6
-
9
-
9

 

 

Abacus school - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/Abacus_school‎

These schools sprang after the publication of Fibonacci's Book of the Abacus and his introduction of the Hindu-Arabic numeral system. In Fibonacci's viewpoint ...

Liber Abaci

Liber Abaci

From Wikipedia, the free encyclopedia

A page of the Liber Abaci from the Biblioteca Nazionale di Firenze showing (on right) the numbers of the Fibonacci sequence.
Liber Abaci (1202, also spelled as Liber Abbaci) is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.

Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as "Arabic Numerals". By addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the new numerals.

The title of Liber Abaci means "The Book of Calculation". Although it has also been translated as "The Book of the Abacus", Sigler (2002) writes that this is an error: the intent of the book is to describe methods of doing calculations without aid of an abacus, and as Ore (1948) confirms, for centuries after its publication the algorismists (followers of the style of calculation demonstrated in Liber Abaci) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals).

Contents [hide]
1 Summary of sections
2 Fibonacci's notation for fractions
3 Modus Indorum
4 References

Summary of sections[edit]

The first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems.

The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest.

The third section discusses a number of mathematical problems; for instance, it includes (ch. II.12) the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today.

The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots.

The book also includes proofs in Euclidean geometry. Fibonacci's method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam.[1]

Fibonacci's notation for fractions[edit]

In reading Liber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the Egyptian fractions commonly used until that time and the vulgar fractions still in use today. There are three key differences between Fibonacci's notation and modern fraction notation.
1.We generally write a fraction to the right of the whole number to which it is added, for instance \scriptstyle 2\,{\frac 13} for 7/3. Fibonacci instead would write the same fraction to the left, i.e., \scriptstyle {\frac 13}\,2.
2.Fibonacci used a composite fraction notation in which a sequence of numerators and denominators shared the same fraction bar; each such term represented an additional fraction of the given numerator divided by the product of all the denominators below and to the right of it. That is, \scriptstyle {\frac {b\,\,a}{d\,\,c}}={\frac {a}{c}}+{\frac {b}{cd}}, and \scriptstyle {\frac {c\,\,b\,\,a}{f\,\,e\,\,d}}={\frac {a}{d}}+{\frac {b}{de}}+{\frac {c}{def}}. The notation was read from right to left. For example, 29/30 could be written as \scriptstyle {\frac {1\,\,2\,\,4}{2\,\,3\,\,5}}, representing the value \scriptstyle {\frac 45}+{\frac 2{3\times 5}}+{\frac 1{2\times 3\times 5}}. This can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a foot is 1/3 of a yard, and an inch is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and \scriptstyle 7{\frac 34} inches could be represented as a composite fraction: \scriptstyle {\frac {3\ \,7\,\,2}{4\,\,12\,\,3}}\,5 yards. However, typical notations for traditional measures, while similarly based on mixed radixes, do not write out the denominators explicitly; the explicit denominators in Fibonacci's notation allow him to use different radixes for different problems when convenient. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions.
3.Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like \scriptstyle {\frac 14}\,{\frac 13}\,2 would represent the number that would now more commonly be written \scriptstyle 2\,{\frac {7}{12}}, or simply the vulgar fraction \scriptstyle {\frac {31}{12}}. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the visible break in the bar. If all numerators are 1 in a fraction written in this form, and all denominators are different from each other, the result is an Egyptian fraction representation of the number. This notation was also sometimes combined with the composite fraction notation: two composite fractions written next to each other would represent the sum of the fractions.

The complexity of this notation allows numbers to be written in many different ways, and Fibonacci described several methods for converting from one style of representation to another. In particular, chapter II.7 contains a list of methods for converting a vulgar fraction to an Egyptian fraction, including the greedy algorithm for Egyptian fractions, also known as the Fibonacci–Sylvester expansion.

Modus Indorum[edit]

In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals.
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days.There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business.I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters.Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.The nine Indian figures are: 9 8 7 6 5 4 3 2 1With these nine figures, and with the sign 0 ... any number may be written. (Sigler 2003 and Grimm 1973)
In other words, in his book he advocated the use of the digits 0–9, and of place value. Until this time Europe used Roman Numerals, making modern mathematics almost impossible. The Liber demonstrated the great benefits of the new numeral system and its applications to commercial bookkeeping, conversion of weights and measures, the calculation of interests, money-changing, and numerous other applications.

The book thus made an important contribution to the spread of decimal numerals. The spread of the Hindu-Arabic system, however, as Ore writes, was "long-drawn-out", taking many more centuries to spread widely, and did not become complete until the later part of the 16th century, accelerating dramatically only in the 1500s with the advent of printing.

References[edit]

1.Jump up ^ O'Connor, John J.; Robertson, Edmund F., "Abu Kamil Shuja ibn Aslam", MacTutor History of Mathematics archive.

Latin Wikisource has original text related to this article:
Liber abbaci

Grimm, R. E. (1973), "The Autobiography of Leonardo Pisano", The Fibonacci Quarterly 11 (1): 99–104.
Sigler, Laurence E. (trans.) (2002), Fibonacci's Liber Abaci, Springer-Verlag, ISBN 0-387-95419-8.
Ore, Øystein (1948), Number Theory and its History, McGraw Hill. Dover version also available, 1988, ISBN 978-0-486-65620-5.

Categories: 1202 books
Mathematics books
13th-century medieval Latin literature
13th century in science

This page was last modified on 6 December 2013 at 05:03.

 

 

T
=
2
-
3
THE
33
15
6
B
=
2
-
4
BOOK
43
16
7
O
=
6
-
2
OF
21
12
3
T
=
2
-
3
THE
33
15
6
A
=
1
-
6
ABACUS
47
11
2
-
-
13
-
18
First Total
177
69
24
-
-
1+3
-
1+8
Add to Reduce
1+7+7
1+4+2
2+4
-
-
4
-
9
Second Total
15
6
6
-
-
-
-
-
Reduce to Deduce
1+5
-
-
-
-
4
-
9
Essence of Number
6
6
6

 

 

-
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
6
6
-
-
6
-
-
-
8
-
-
-
-
-
-
-
1
+
=
35
3+5
=
8
=
8
=
8
-
-
-
8
-
-
-
15
15
-
-
15
-
-
-
8
-
-
-
-
-
-
-
19
+
=
80
8+0
=
8
=
8
=
8
-
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
2
-
-
2
-
-
6
-
2
-
5
-
1
2
1
3
3
-
+
=
34
3+4
=
7
=
7
=
7
-
-
20
-
5
-
2
-
-
11
-
-
6
-
20
-
5
-
1
2
1
3
21
-
+
=
97
9+7
=
16
1+6
7
=
7
-
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
2
15
15
11
-
15
6
-
20
8
5
-
1
2
1
3
21
19
+
=
177
1+7+7
=
15
1+5
6
=
6
-
-
2
8
5
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
+
=
69
6+9
=
15
1+5
6
=
6
-
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
5
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
6
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
NINE
-
-
-
-
-
-
20
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
25
-
-
18
-
69
-
24
2+0
1+8
2
-
-
-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2+5
-
-
1+8
-
6+9
-
2+4
2
9
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
9
-
15
-
6
-
-
2
8
5
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
-
-
-
-
-
-
-
1+5
-
-
2
9
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
9
-
6
-
6

 

 

18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
6
6
-
-
6
-
-
-
8
-
-
-
-
-
-
-
1
+
=
35
3+5
=
8
=
8
=
8
-
-
8
-
-
-
15
15
-
-
15
-
-
-
8
-
-
-
-
-
-
-
19
+
=
80
8+0
=
8
=
8
=
8
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
2
-
-
2
-
-
6
-
2
-
5
-
1
2
1
3
3
-
+
=
34
3+4
=
7
=
7
=
7
-
20
-
5
-
2
-
-
11
-
-
6
-
20
-
5
-
1
2
1
3
21
-
+
=
97
9+7
=
16
1+6
7
=
7
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
2
15
15
11
-
15
6
-
20
8
5
-
1
2
1
3
21
19
+
=
177
1+7+7
=
15
1+5
6
=
6
-
2
8
5
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
+
=
69
6+9
=
15
1+5
6
=
6
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
1
occurs
x
3
=
3
=
3
``-
2
-
-
-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
5
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
6
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
18
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
25
-
-
18
-
69
-
24
1+8
2
-
-
-
2
-
-
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
2+5
-
-
1+8
-
6+9
-
2+4
9
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
9
-
15
-
6
-
2
8
5
-
2
6
6
2
-
6
6
-
2
8
5
-
1
2
1
3
3
1
-
-
-
-
-
-
-
1+5
-
-
9
T
H
E
-
B
O
O
K
-
O
F
-
T
H
E
-
A
B
A
C
U
S
-
-
7
-
-
9
-
6
-
6

 

 

E
=
5
-
5
ERDOS
61
24
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
D
=
4
-
11
DISCREPANCY
99
54
9
P
=
7
-
7
PROBLEM
81
36
9
-
-
11
-
18
Add to Reduce
180
90
18
-
-
1+1
-
1+8
Reduce to Deduce
1+8+0
9+0
1+8
-
-
2
-
9
Essence of Number
9
9
9

 

 

-
18
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
-
-
-
-
-
-
-
-
-
-
-
9
1
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
+
=
21
2+1
=
3
=
3
=
3
-
-
-
9
19
-
-
-
-
-
14
-
-
-
-
-
15
-
-
-
-
+
=
57
5+7
=
12
1+2
3
=
3
-
18
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
3
9
5
7
1
-
3
7
-
7
9
-
2
3
5
4
+
=
69
6+9
=
15
1+5
6
=
6
-
-
4
-
-
3
18
5
16
1
-
3
25
-
16
18
-
2
12
5
13
+
=
141
1+4+1
=
6
=
6
=
6
-
18
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
-
-
-
-
-
-
-
-
-
-
4
9
19
3
18
5
16
1
14
3
25
-
16
18
15
2
12
5
13
+
=
198
1+9+8
=
18
1+8
9
=
9
-
-
4
9
1
3
9
5
7
1
5
3
7
-
7
9
6
2
3
5
4
+
=
90
9+0
=
9
=
9
=
9
-
18
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
3
-
-
-
-
-
-
3
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
7
-
-
-
7
-
7
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
8
18
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
37
-
-
18
-
90
-
45
-
1+8
-
9
-
-
9
-
-
-
-
-
-
-
-
9
-
2
-
-
-
-
-
3+7
-
-
1+8
-
9+0
-
4+5
8
9
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
8
-
-
9
-
9
-
9
-
-
4
9
1
3
9
5
7
1
5
3
7
-
7
9
6
2
3
5
4
-
-
-
-
-
-
-
-
-
-
8
9
D
I
S
C
R
E
P
A
N
C
Y
-
P
R
O
B
L
E
M
-
-
8
-
-
9
-
9
-
9

 

 

-
PSYCHICS
-
-
-
4
C
3
3
3
7
PHYSICS
99
36
9
16
PSYCHICS-
102
39
12
1+8
-
1+0+2
3+9
1+2
9
-
3
12
3
-
-
-
 1+2
-
9
-
3
3
3

 

 

-
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
1
9
-
1
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
8
-
19
9
-
19
+
=
55
5+5
=
10
1+0
1
=
1
-
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
7
-
7
-
-
3
-
+
=
17
1+7
=
8
=
8
=
8
-
-
16
-
25
-
-
3
-
+
=
44
4+4
=
8
=
8
=
8
-
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
16
8
25
19
9
3
19
+
=
99
9+9
=
18
1+8
9
=
9
-
-
7
8
7
1
9
3
1
+
=
36
3+6
=
9
=
9
=
9
-
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
7
-
7
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-``
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
17
7
P
H
Y
S
I
C
S
-
-
28
-
-
7
-
36
-
27
1+7
-
-
-
-
-
9
-
-
-
-
2+8
-
-
-
-
3+6
-
2+7
8
7
P
H
Y
S
I
C
S
-
-
10
-
-
7
-
9
-
9
-
-
7
8
7
1
9
3
1
-
-
1+0
-
-
-
-
-
-
-
8
7
P
H
Y
S
I
C
S
-
-
1
-
-
7
-
9
-
9

 

 
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
8
-
1
9
-
1
+
=
19
1+9
=
10
1+0
1
=
1
-
-
8
-
19
9
-
19
+
=
55
5+5
=
10
1+0
1
=
1
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
7
-
7
-
-
3
-
+
=
17
1+7
=
8
=
8
=
8
-
16
-
25
-
-
3
-
+
=
44
4+4
=
8
=
8
=
8
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
16
8
25
19
9
3
19
+
=
99
9+9
=
18
1+8
9
=
9
-
7
8
7
1
9
3
1
+
=
36
3+6
=
9
=
9
=
9
7
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
7
-
7
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
7
P
H
Y
S
I
C
S
-
-
28
-
-
7
-
36
-
27
-
-
-
-
-
9
-
-
-
-
2+8
-
-
-
-
3+6
-
2+7
7
P
H
Y
S
I
C
S
-
-
10
-
-
7
-
9
-
9
-
7
8
7
1
9
3
1
-
-
1+0
-
-
-
-
-
-
-
7
P
H
Y
S
I
C
S
-
-
1
-
-
7
-
9
-
9

 

 

-
ELECTRICITY
-
-
-
5
ELECT
45
18
9
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
1
I
9
9
9
2
T+Y
45
9
9
11
ELECTRICITY
129
57
48
1+1
-
1+2+9
5+7
4+8
2
ELECTRICITY
12
12
12
-
-
1+2
1+2
1+2
2
ELECTRICITY
3
3
3

 

 

-
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
-
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
-
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
-
2
7
+
=
39
3+9
=
12
1+2
3
=
3
-
-
5
12
5
3
20
18
-
3
-
20
25
+
=
111
1+1+1
=
3
=
3
=
3
-
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
9
20
25
+
=
129
1+2+9
=
12
1+2
3
=
3
-
-
5
3
5
3
2
9
9
3
9
2
7
+
=
57
5+7
=
12
1+2
3
=
3
-
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
3
-
3
-
-
-
3
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
19
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
26
-
-
11
-
57
-
30
1+9
1+1
-
-
-
-
-
9
9
-
9
-
-
-
-
2+6
-
-
1+1
-
5+7
-
3+0
10
2
E
L
E
C
T
R
I
C
I
T
Y
-
-
8
-
-
2
-
12
-
3
1+0
-
5
3
5
3
2
9
9
3
9
2
7
-
-
-
-
-
-
-
1+2
-
-
1
2
E
L
E
C
T
R
I
C
I
T
Y
-
-
8
-
-
2
-
3
-
3

 

 

11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
-
2
7
+
=
39
3+9
=
12
1+2
3
=
3
-
5
12
5
3
20
18
-
3
-
20
25
+
=
111
1+1+1
=
3
=
3
=
3
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
9
20
25
+
=
129
1+2+9
=
12
1+2
3
=
3
-
5
3
5
3
2
9
9
3
9
2
7
+
=
57
5+7
=
12
1+2
3
=
3
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
3
-
3
-
-
-
3
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
5
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
9
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
11
E
L
E
C
T
R
I
C
I
T
Y
-
-
26
-
-
11
-
57
-
30
1+1
-
-
-
-
-
9
9
-
9
-
-
-
-
2+6
-
-
1+1
-
5+7
-
3+0
2
E
L
E
C
T
R
I
C
I
T
Y
-
-
8
-
-
2
-
12
-
3
-
5
3
5
3
2
9
9
3
9
2
7
-
-
-
-
-
-
-
1+2
-
-
2
E
L
E
C
T
R
I
C
I
T
Y
-
-
8
-
-
2
-
3
-
3

 

 

-
ELECTRIC
-
-
-
5
ELECT
75
18
9
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
8
ELECTRIC
75
39
30
-
-
7+5
3+9
3+0
8
ELECTRIC
12
12
3
-
-
1+2
1+2
-
8
ELECTRIC
3
3
3

 

ELECTRIC ET CIRCLE ET ELECTRIC

 

-
ELECTRIC
-
-
-
5
E
5
5
5
1
L
12
3
3
5
E
5
5
5
5
C
3
3
3
5
T
20
2
2
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
8
ELECTRIC
75
39
30
-
-
7+5
3+9
3+0
8
ELECTRIC
12
12
3
-
-
1+2
1+2
-
8
ELECTRIC
3
3
3

 

 

-
8
E
L
E
C
T
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
=
9
=
9
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
=
9
=
9
-
8
E
L
E
C
T
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
+
=
30
3+9
=
12
1+2
3
=
3
-
-
5
12
5
3
20
18
-
3
+
=
66
1+1+1
=
3
=
3
=
3
-
8
E
L
E
C
T
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
+
=
75
7+5
=
12
1+2
3
=
3
-
-
5
3
5
3
2
9
9
3
+
=
39
3+9
=
12
1+2
3
=
3
-
8
E
L
E
C
T
R
I
C
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
-
3
-
-
-
3
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
26
8
E
L
E
C
T
R
I
C
-
-
19
-
-
8
-
39
-
21
2+6
-
-
-
-
-
-
9
9
-
-
-
1+9
-
-
-
-
3+9
-
2+1
8
8
E
L
E
C
T
R
I
C
-
-
10
-
-
8
-
12
-
3
-
-
5
3
5
3
2
9
9
3
-
-
1+0
-
-
-
-
-
-
-
8
8
E
L
E
C
T
R
I
C
-
-
1
-
-
8
-
3
-
3

 

 

-
CIRCUITRY
--
-
-
1
C
3
3
3
1
I
9
9
9
1
R
18
9
9
1
C
3
3
3
1
U
21
3
3
1
I
9
9
9
3
T+R+Y
63
18
9
9
CIRCUITRY
126
54
45
-
-
1+2+6
5+4
4+5
9
CIRCUITRY
9
9
9

 

 

-
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
+
=
18
1+8
9
-
=
9
=
9
-
-
-
9
-
-
-
9
-
-
-
+
=
18
1+8
9
-
=
9
=
9
-
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
-
2
9
7
+
=
36
3+6
9
-
=
9
=
9
-
-
3
-
18
3
21
-
20
18
25
+
=
108
1+0+8
9
-
=
9
=
9
-
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
9
20
18
25
+
=
126
1+2+6
9
-
=
9
=
9
-
-
3
9
9
3
3
9
2
9
7
+
=
54
5+4
9
-
=
9
=
9
-
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
3
3
-
-
-
-
-
-
3
occurs
x
3
=
0
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
9
-
-
9
-
9
-
-
-
9
occurs
x
4
=
36
3+6
9
24
9
C
I
R
C
U
I
T
R
Y
-
-
21
-
-
9
-
54
-
27
2+4
-
-
9
9
-
-
9
-
9
-
-
-
2+1
-
-
-
-
5+4
-
2+7
6
9
C
I
R
C
U
I
T
R
Y
-
-
3
-
-
9
-
9
-
9
-
-
3
9
9
3
3
9
2
9
7
-
-
-
-
-
-
  
-
-
-
6
9
C
I
R
C
U
I
T
R
Y
-
-
3
-
-
9
-
9
-
9

 

 

9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
+
=
18
1+8
9
-
=
9
=
9
-
-
9
-
-
-
9
-
-
-
+
=
18
1+8
9
-
=
9
=
9
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
-
2
9
7
+
=
36
3+6
9
-
=
9
=
9
-
3
-
18
3
21
-
20
18
25
+
=
108
1+0+8
9
-
=
9
=
9
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
9
20
18
25
+
=
126
1+2+6
9
-
=
9
=
9
-
3
9
9
3
3
9
2
9
7
+
=
54
5+4
9
-
=
9
=
9
9
C
I
R
C
U
I
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
3
3
-
-
-
-
-
-
3
occurs
x
3
=
0
=
9
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
-
-
9
9
-
-
9
-
9
-
-
-
9
occurs
x
4
=
36
3+6
9
9
C
I
R
C
U
I
T
R
Y
-
-
21
-
-
9
-
54
-
27
-
-
9
9
-
-
9
-
9
-
-
-
2+1
-
-
-
-
5+4
-
2+7
9
C
I
R
C
U
I
T
R
Y
-
-
3
-
-
9
-
9
-
9
-
3
9
9
3
3
9
2
9
7
-
-
-
-
-
-
  
-
-
-
9
C
I
R
C
U
I
T
R
Y
-
-
3
-
-
9
-
9
-
9

 

 

-
CIRCUITS
--
-
-
1
C
3
3
3
1
I
9
9
9
1
R
18
9
9
1
C
3
3
3
1
U
21
3
3
1
I
9
9
9
2
T+S
39
12
3
8
CIRCUITS
102
48
39
-
-
1+0+2
4+8
3+9
8
CIRCUITS
3
12
12
-
-
-
1+2
1+2
8
CIRCUITS
3
3
3

 

 

8
ELECTRIC
75
39
3
8
CIRCUITS
102
48
3

 

 

10
ELECTRICAL
88
43
7
8
CIRCUITS
102
48
3
18
First Total
190
91
10
2+6
Add to Reduce
1+9+0
9+1
1+0
8
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
8
Essence of Number
1
1
1

 

 

-
ELECTRICAL
-
-
-
-
-
-
1
E
5
5
5
-
-
-
1
L
12
3
3
-
3
-
1
E
5
5
5
-
-
-
1
C
3
3
3
-
3
-
1
T
20
2
2
-
-
-
1
R
18
9
9
-
-
9
1
I
9
9
9
-
-
9
1
C
3
3
3
-
3
-
1
A
1
1
1
-
-
-
1
L
12
3
3
-
3
-
10
ELECTRICAL
88
43
43
-
-
-
-
CIRCUITS
--
-
-
-
-
-
1
C
3
3
3
-
3
-
1
I
9
9
9
-
-
9
1
R
18
9
9
-
-
9
1
C
3
3
3
-
3
-
1
U
21
3
3
-
3
-
1
I
9
9
9
-
-
9
1
T+S
39
12
3
-
3
-
8
CIRCUITS
102
48
39
-
-
-
-
-
--
-
-
-
-
-
18
First Total
190
91
82
-
-
-
2+6
Add to Reduce
1+9+0
9+1
8+2
-
-
-
8
Second Total
10
10
10
-
-
-
-
Reduce to Deduce
1+0
1+0
1+0
-
-
-
8
Essence of Number
1
1
1
-
-
-

 

 

-
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
9
-
19
+
=
46
4+6
=
10
1+0
1
=
1
-
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
1
3
-
3
-
9
3
3
-
2
-
+
=
54
5+4
=
9
=
9
=
9
-
-
5
12
5
3
20
18
-
3
1
12
-
3
-
18
3
21
-
20
-
+
=
144
1+4+4
=
3
=
9
=
9
-
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
1
12
-
3
9
18
3
21
9
20
19
+
=
190
1+9+0
=
10
1+0
1
=
1
-
-
5
3
5
3
2
9
9
3
1
3
-
3
9
9
3
3
9
2
1
+
=
82
8+2
=
10
1+0
1
=
1
-
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
3
-
3
-
-
-
3
-
3
-
3
-
-
3
3
-
-
-
-
-
3
occurs
x
7
=
21
2+1
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
25
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
20
-
-
18
-
82
-
19
2+5
1+8
-
-
-
-
-
9
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
2+0
-
-
1+8
-
8+2
-
1+9
7
9
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
2
-
-
9
-
10
-
10
-
-
5
3
5
3
2
9
9
3
1
3
-
3
9
9
3
3
9
2
1
-
-
-
-
-
-
-
1+0
-
1+0
7
9
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
1
-
-
9
-
1
-
1

 

 

18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
9
-
19
+
=
46
4+6
=
10
1+0
1
=
1
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
1
3
-
3
-
9
3
3
-
2
-
+
=
54
5+4
=
9
=
9
=
9
-
5
12
5
3
20
18
-
3
1
12
-
3
-
18
3
21
-
20
-
+
=
144
1+4+4
=
3
=
9
=
9
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
1
12
-
3
9
18
3
21
9
20
19
+
=
190
1+9+0
=
10
1+0
1
=
1
-
5
3
5
3
2
9
9
3
1
3
-
3
9
9
3
3
9
2
1
+
=
82
8+2
=
10
1+0
1
=
1
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
3
-
3
-
-
-
3
-
3
-
3
-
-
3
3
-
-
-
-
-
3
occurs
x
7
=
21
2+1
3
-
5
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
9
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
18
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
20
-
-
18
-
82
-
19
1+8
-
-
-
-
-
9
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
2+0
-
-
1+8
-
8+2
-
1+9
9
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
2
-
-
9
-
10
-
10
-
5
3
5
3
2
9
9
3
1
3
-
3
9
9
3
3
9
2
1
-
-
-
-
-
-
-
1+0
-
1+0
9
E
L
E
C
T
R
I
C
A
L
-
C
I
R
C
U
I
T
S
-
-
1
-
-
9
-
1
-
1

 

 

18
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
9
-
19
+
=
46
4+6
=
10
1+0
1
=
1
18
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
3
1
3
3
-
9
3
3
-
2
-
+
=
54
5+4
=
9
=
9
=
9
-
5
12
5
3
20
18
-
3
1
12
3
-
18
3
21
-
20
-
+
=
144
1+4+4
=
3
=
9
=
9
18
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
5
12
5
3
20
18
9
3
1
12
3
9
18
3
21
9
20
19
+
=
190
1+9+0
=
10
1+0
1
=
1
-
5
3
5
3
2
9
9
3
1
3
3
9
9
3
3
9
2
1
+
=
82
8+2
=
10
1+0
1
=
1
18
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
3
-
3
-
-
-
3
-
3
3
-
-
3
3
-
-
-
-
-
3
occurs
x
7
=
21
2+1
3
-
5
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
9
9
-
-
-
-
9
9
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
18
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
20
-
-
18
-
82
-
19
1+8
-
-
-
-
-
9
9
-
-
-
-
9
9
-
-
9
-
-
-
-
2+0
-
-
1+8
-
8+2
-
1+9
9
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
2
-
-
9
-
10
-
10
-
5
3
5
3
2
9
9
3
1
3
3
9
9
3
3
9
2
1
-
-
-
-
-
-
-
1+0
-
1+0
9
E
L
E
C
T
R
I
C
A
L
C
I
R
C
U
I
T
S
-
-
1
-
-
9
-
1
-
1

 

 

3
VAN
37
10
1
2
DE
9
9
9
6
GRAAFF
39
30
3
10
GENERATORS
122
50
5
21
First Total
207
99
18
2+1
Add to Reduce
2+0+7
9+9
1+8
3
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
3
Essence of Number
9
9
9

 

 

6
DYNAMO
72
27
9

 

 

THE PERIODIC KINGDOM

A JOURNEY INTO THE LAND OF THE CHEMICAL ELEMENTS

Peter Atkins 1995

Page 17

"...carbon has established itself as king of the periodic kingdom. Carbon, of course, is the element of organic compounds: the extraordinary and complex property we term "life" stems almost in its entirety from this black northern region of the kingdom.

"Carbon's principal products, living organisms, have struggled over a few billion years to establish mechanisms for the accumulation and dispersal of information (an austere distillation and definition of what we mean by "life"), and silicon has lain in wait, The recent alliance of the two regions, in which carbon-based organisms have developed the use of silicon based artifacts for information technology, has resulted in the enslavement of silicon. However such is the precosity of carbon's organisms that they are steadily developing silicons latent powers, and one day silicon may well overturn the suzerainty of its northern neighbor and assume the dominant role. It certainly has long-term potential, for its metabolism and replication need not be as messy as carbon's. Here we may see one of the most subtle interplays of alliances anywhere in the kingdom, for silicon will not realize its full potential without the burden of development being carried out by carbon."

 

7
CARBONS
72
27
9
6
CARBON
53
26
8
7
SILICON
81
36
9
8
SILICONS
100
37
1
6
OXYGEN
90
36
9

 

 

-
7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
-
`-
-
-
-
-
15
14
19
+
=
48
4+8
=
12
1+2
3
-
-
-
-
-
-
6
5
1
+
=
12
1+2
=
3
-
3
-
7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
-
`-
3
1
18
2
-
-
-
+
=
24
2+4
=
6
-
6
-
-
3
1
9
2
-
-
-
+
=
15
1+5
=
6
-
6
-
7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
-
`-
3
1
18
2
15
14
19
+
=
72
7+2
=
9
-
9
-
-
3
1
9
2
6
5
1
+
=
27
2+7
=
9
-
9
-
7
C
A
R
B
O
N
S
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
4
-
-
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
5
-
-
-
5
occurs
x
1
=
5
-
-
-
-
4
-
6
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
4
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
4
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
19
7
C
A
R
B
O
N
S
-
-
26
-
-
7
-
27
1+9
-
-
-
9
-
-
-
-
-
-
2+6
-
-
-
-
2+7
10
7
C
A
R
B
O
N
S
-
-
8
-
-
7
-
9
1+0
-
3
1
9
2
6
5
1
-
-
-
-
-
-
-
-
1
7
C
A
R
B
O
N
S
-
-
8
-
-
7
-
9

 

 

7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
`-
-
-
-
-
15
14
19
+
=
48
4+8
=
12
1+2
3
-
-
-
-
-
6
5
1
+
=
12
1+2
=
3
-
3
7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
`-
3
1
18
2
-
-
-
+
=
24
2+4
=
6
-
6
-
3
1
9
2
-
-
-
+
=
15
1+5
=
6
-
6
7
C
A
R
B
O
N
S
-
-
-
-
-
--
-
-
`-
3
1
18
2
15
14
19
+
=
72
7+2
=
9
-
9
-
3
1
9
2
6
5
1
+
=
27
2+7
=
9
-
9
7
C
A
R
B
O
N
S
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
1
-
-
1
occurs
x
2
=
2
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
3
-
4
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
5
-
5
-
-
-
5
occurs
x
1
=
5
-
-
-
4
-
6
-
-
-
-
6
occurs
x
1
=
6
9
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
7
C
A
R
B
O
N
S
-
-
26
-
-
7
-
27
-
-
-
9
-
-
-
-
-
-
2+6
-
-
-
-
2+7
7
C
A
R
B
O
N
S
-
-
8
-
-
7
-
9
-
3
1
9
2
6
5
1
-
-
-
-
-
-
-
-
7
C
A
R
B
O
N
S
-
-
8
-
-
7
-
9

 

 

-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
8
-
-
6
+
=
14
1+4
=
5
=
5
-
-
26
-
-
15
+
=
41
4+1
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
-
5
9
-
+
=
14
1+4
=
5
=
5
-
-
-
5
18
-
+
=
23
2+3
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
26
5
18
15
+
=
64
6+4
=
10
1+0
1
-
-
8
5
9
6
+
=
28
2+8
=
10
1+0
1
-
4
Z
E
R
O
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
17
4
Z
E
R
O
-
-
28
-
-
4
-
28
1+7
-
-
-
9
-
-
-
2+8
-
-
-
-
2+8
8
4
Z
E
R
O
-
-
10
-
-
4
-
10
-
-
-
-
-
 -
-
-
1+0
-
-
-
-
1+0
8
4
Z
E
R
O
-
-
1
-
-
4
-
1

 

 

4
Z
E
R
O
-
-
-
-
-
--
-
-
-
8
-
-
6
+
=
14
1+4
=
5
=
5
-
26
-
-
15
+
=
41
4+1
=
5
=
5
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
5
9
-
+
=
14
1+4
=
5
=
5
-
-
5
18
-
+
=
23
2+3
=
5
=
5
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
26
5
18
15
+
=
64
6+4
=
10
1+0
1
-
8
5
9
6
+
=
28
2+8
=
10
1+0
1
4
Z
E
R
O
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
8
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
9
-
-
-
9
occurs
x
1
=
9
4
Z
E
R
O
-
-
28
-
-
4
-
28
-
-
-
9
-
-
-
2+8
-
-
-
-
2+8
4
Z
E
R
O
-
-
10
-
-
4
-
10
-
-
-
-
-
-
-
1+0
-
-
-
-
1+0
4
Z
E
R
O
-
-
1
-
-
4
-
1

 

 

- GEOPHYSICS

-

-

 -
3 GEO

27

18

 9
7 PHYSICS

99

36

 9
10 GEOPHYSICS

126

54

 18
1+0
 - 1+2+6 5+4 1+8
1 GEOPHYSICS

9

9

 9

 

 

-
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
8
-
1
9
-
1
+
=
25
2+5
=
7
=
7
=
7
-
-
-
-
15
-
-
8
-
19
9
-
19
+
=
70
7+0
=
7
=
7
=
7
-
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
-
7
-
7
-
-
3
-
+
=
29
2+9
=
11
1+1
2
=
2
-
-
7
5
-
-
16
-
25
-
-
3
-
+
=
56
5+6
=
11
1+1
2
=
2
-
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
7
5
15
-
16
8
25
19
9
3
19
+
=
126
1+2+6
=
9
=
9
=
9
-
-
7
5
6
-
7
8
7
1
9
3
1
+
=
54
5+4
=
9
=
9
=
9
-
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
--
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
7
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
6
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
37
-
-
10
-
54
-
36
-
1+0
-
-
-
-
-
-
-
-
9
-
-
-
-
3+7
-
-
1+0
-
5+4
-
3+6
6
1
G
E
O
-
P
H
Y
S
I
C
S
-
-
10
-
-
1
-
9
-
9
-
-
7
5
6
-
7
8
7
1
9
3
1
-
-
1+0
-
-
-
-
-
-
-
6
1
G
E
O
-
P
H
Y
S
I
C
S
-
-
1
-
-
1
-
9
-
9

 

 

10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
8
-
1
9
-
1
+
=
25
2+5
=
7
=
7
=
7
-
-
-
15
-
-
8
-
19
9
-
19
+
=
70
7+0
=
7
=
7
=
7
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
7
5
-
-
7
-
7
-
-
3
-
+
=
29
2+9
=
11
1+1
2
=
2
-
7
5
-
-
16
-
25
-
-
3
-
+
=
56
5+6
=
11
1+1
2
=
2
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
7
5
15
-
16
8
25
19
9
3
19
+
=
126
1+2+6
=
9
=
9
=
9
-
7
5
6
-
7
8
7
1
9
3
1
+
=
54
5+4
=
9
=
9
=
9
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
7
-
-
-
7
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
10
G
E
O
-
P
H
Y
S
I
C
S
-
-
37
-
-
10
-
54
-
36
1+0
-
-
-
-
-
-
-
-
9
-
-
-
-
3+7
-
-
1+0
-
5+4
-
3+6
1
G
E
O
-
P
H
Y
S
I
C
S
-
-
10
-
-
1
-
9
-
9
-
7
5
6
-
7
8
7
1
9
3
1
-
-
1+0
-
-
-
-
-
-
-
1
G
E
O
-
P
H
Y
S
I
C
S
-
-
1
-
-
1
-
9
-
9

 

 

GOD'S SECRET FORMULA

Peter Plichta 1995

Deciphering The Riddle Of The Universe And The Prime Number Code

Page 33

"Table 1. The 81 stable chemical elements

Atomic Name number

Hafnium 72

 

7
HAFNIUM
72
36
9

 

--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
8
-
-
5
9
-
-
+
=
22
2+2
=
4
=
4
-
-
8
-
-
14
9
-
-
+
=
31
3+1
=
4
=
4
--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
1
6
-
-
3
4
+
=
14
1+4
=
5
=
5
-
-
-
1
6
-
-
21
13
+
=
41
4+1
=
5
=
5
--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
8
1
6
14
9
21
13
+
=
72
7+2
=
9
=
9
-
-
8
1
6
5
9
3
4
+
=
36
3+6
=
9
=
9
-
7
H
A
F
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
9
7
H
A
F
N
I
U
M
-
-
36
-
-
7
-
36
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
-
-
3+6
9
7
H
A
F
N
I
U
M
-
-
9
-
-
7
-
9

 

 

-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
9
-
-
+
=
14
1+4
=
5
=
5
-
5
-
`-
-
-
-
14
9
-
-
+
=
23
2+3
=
5
=
5
-
5
-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
3
9
1
-
-
3
4
+
=
20
2+0
=
2
=
2
-
2
-
-
21
18
1
-
-
21
13
+
=
74
7+4
=
11
1+1
2
-
2
-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
`-
21
18
1
14
9
21
13
+
=
97
9+7
=
16
1+6
7
-
7
-
-
3
9
1
5
9
3
4
+
=
34
3+4
=
7
=
7
-
7
-
7
U
R
A
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
4
-
-
4
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
4
-
-
4
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
5
-
-
5
occurs
x
1
=
5
=
5
6
-
-
4
-
-
4
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
4
-
-
4
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
4
-
-
4
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
9
-
-
9
9
9
-
-
9
occurs
x
2
=
18
1+8
9
23
7
U
R
A
N
I
U
M
-
-
22
-
-
7
-
33
-
24
2+3
-
9
9
9
9
9
9
-
-
-
2+2
-
-
-
-
3+3
-
2+4
5
7
U
R
A
N
I
U
M
-
-
4
-
-
7
-
6
-
6

 

 
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
9
-
-
+
=
14
1+4
=
5
=
5
-
5
`-
-
-
-
14
9
-
-
+
=
23
2+3
=
5
=
5
-
5
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
3
9
1
-
-
3
4
+
=
20
2+0
=
2
=
2
-
2
-
21
18
1
-
-
21
13
+
=
74
7+4
=
11
1+1
2
-
2
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
`-
21
18
1
14
9
21
13
+
=
97
9+7
=
16
1+6
7
-
7
-
3
9
1
5
9
3
4
+
=
34
3+4
=
7
=
7
-
7
7
U
R
A
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
-
-
1
occurs
x
1
=
1
=
1
-
3
4
-
-
4
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
9
9
-
-
9
9
9
-
-
9
occurs
x
2
=
18
1+8
9
7
U
R
A
N
I
U
M
-
-
22
-
-
7
-
33
-
24
-
9
9
-
9
9
-
-
-
2+2
-
-
-
-
3+3
-
2+4
7
U
R
A
N
I
U
M
-
-
4
-
-
7
-
6
-
6

 

 

-
6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
6
6
-
-
-
5
+
=
17
1+7
=
8
=
8
=
8
-
-
15
24
-
-
-
14
+
=
53
5+3
=
8
=
8
=
8
-
6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
7
7
5
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
-
25
7
5
-
+
=
37
3+7
=
10
1+0
1
=
1
-
6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
15
24
25
7
5
14
+
=
90
9+0
=
9
=
9
=
9
-
-
6
6
7
7
5
5
+
=
36
3+6
=
9
=
9
=
9
-
6
O
X
Y
G
E
N
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
6
6
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
7
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
27
6
O
X
Y
G
E
N
-
-
18
-
-
6
-
36
-
9
2+7
-
-``
-
-
-
-
-
-
-
1+8
-
-
-
-
3+6
-
-
9
6
O
X
Y
G
E
N
-
-
9
-
-
6
-
9
-
9

 

 

6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
6
6
-
-
-
5
+
=
17
1+7
=
8
=
8
=
8
-
15
24
-
-
-
14
+
=
53
5+3
=
8
=
8
=
8
6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
7
7
5
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
25
7
5
-
+
=
37
3+7
=
10
1+0
1
=
1
6
O
X
Y
G
E
N
-
-
-
-
-
--
-
-
-
-
-
15
24
25
7
5
14
+
=
90
9+0
=
9
=
9
=
9
-
6
6
7
7
5
5
+
=
36
3+6
=
9
=
9
=
9
6
O
X
Y
G
E
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
6
6
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
6
O
X
Y
G
E
N
-
-
18
-
-
6
-
36
-
9
-
-``
-
-
-
-
-
-
-
1+8
-
-
-
-
3+6
-
-
6
O
X
Y
G
E
N
-
-
9
-
-
6
-
9
-
9

 

 

-
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
9
5
-
9
5
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
9
14
-
9
14
9
-
-
+
=
55
5+5
=
10
1+0
1
=
1
-
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
2
7
+
=
15
-
=
6
=
6
=
6
-
-
-
-
6
-
-
-
20
25
+
=
51
-
=
6
=
6
=
6
-
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
9
14
6
9
14
9
20
25
+
=
106
1+0+6
=
7
=
7
=
7
-
-
9
5
6
9
5
9
2
7
+
=
52
5+2
=
7
=
7
=
7
-
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
-`
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
16
8
I
N
F
I
N
I
T
Y
-
-
29
-
-
8
-
52
-
25
1+6
-
-
-
-
-
-
-
-
-
-
-
2+9
-
-
-
-
5+2
-
2+5
7
8
I
N
F
I
N
I
T
Y
-
-
11
-
-
8
-
7
-
7
-
-
9
5
6
9
5
9
2
7
-
-
-
-
-
-
-
-
-
-
7
8
I
N
F
I
N
I
T
Y
-
-
2
-
-
8
-
7
-
7

 

 

8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
9
5
-
9
5
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
9
14
-
9
14
9
-
-
+
=
55
5+5
=
10
1+0
1
=
1
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
2
7
+
=
15
-
=
6
=
6
=
6
-
-
-
6
-
-
-
20
25
+
=
51
-
=
6
=
6
=
6
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
9
14
6
9
14
9
20
25
+
=
106
1+0+6
=
7
=
7
=
7
-
9
5
6
9
5
9
2
7
+
=
52
5+2
=
7
=
7
=
7
8
I
N
F
I
N
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-
-
5
-
-
5
-`
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
-
9
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
8
I
N
F
I
N
I
T
Y
-
-
29
-
-
8
-
52
-
25
-
-
-
-
-
-
-
-
-
-
-
2+9
-
-
-
-
5+2
-
2+5
8
I
N
F
I
N
I
T
Y
-
-
11
-
-
8
-
7
-
7
-
9
5
6
9
5
9
2
7
-
-
 1+1
-
-
-
-
-
-
-
8
I
N
F
I
N
I
T
Y
-
-
2
-
-
8
-
7
-
7

 

 

S
=
1
-
12
SCHRODINGERS
139
67
4
C
=
3
-
3
CAT
24
6
6
D
=
4
-
3
DID
17
17
8
C
=
3
-
10
CURIOUSITY
160
52
7
K
=
2
-
4
KILL
44
17
6
T
=
2
-
3
THE
33
15
6
C
=
3
-
3
CAT
24
6
6
-
-
18
4
38
First Total
441
180
19
-
-
1+8
-
3+8
Add to Reduce
4+4+1
1+8+0
6+2
-
-
9
-
11
Second Total
9
10
10
4
4
-
4
1+1
Reduce to Deduce
-
1+0
1+0
Q
-
9
5
2
Essence of Number
9
7
7

 

 

-
COSMOLOGICAL CONSTANT
-
-
-
8
C+O+S+M+O+L+O+G
99
36
9
1
I
9
9
9
3
C+A+L
16
7
7
2
C+O
18
9
9
4
N+S+T+A
54
9
9
2
N+T
34
7
7
20
COSMOLOGICAL CONSTANT
230
77
50
2+0
-
2+3+0
7+7
5+0
2
COSMOLOGICAL CONSTANT
5
14
5
-
-
-
1+4
-
2
COSMOLOGICAL CONSTANT
5
5
5

 

 

-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
-
-
-
-
-
-
-
-
-
-
-
6
1
-
6
-
6
-
9
-
-
-
-
-
6
5
1
-
-
5
-
+
=
45
4+5
=
9
-
9
-
9
-
--
-
15
19
-
15
-
15
-
9
-
-
-
-
-
15
14
19
-
-
14
-
+
=
135
1+3+5
=
9
-
9
-
9
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
4
-
3
-
7
-
3
1
3
-
3
-
-
-
2
1
-
2
+
=
32
3+2
=
5
-
5
-
5
-
-
3
-
-
13
-
12
-
7
-
3
1
12
-
3
-
-
-
20
1
-
20
+
=
95
9+5
=
14
1+4
5
-
5
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
--
-
-
-
-
-
-
-
-
-
-
-
3
15
19
13
15
12
15
7
9
3
1
12
-
3
15
14
19
20
1
14
20
+
=
230
2+3+0
=
5
-
5
-
5
-
-
3
6
1
4
6
3
6
7
9
3
1
3
-
3
6
5
1
2
1
5
2
+
=
77
7+7
=
14
1+4
5
-
5
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
--
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
-
-
-
-
-
-
1
-
-
-
-
-
1
-
1
-
-
-
-
1
occurs
x
4
=
4
-
4
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
2
occurs
x
2
=
4
-
4
-
-
3
-
-
-
-
3
-
-
-
3
-
3
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
5
=
15
1+5
6
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
6
-
6
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
37
-
-
20
-
77
-
41
-
-
3
-
-
-
-
3
-
-
-
3
-
3
-
3
-
-
-
-
-
-
-
-
-
3+7
-
-
2+0
-
7+7
-
4+1
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
10
-
-
2
-
14
-
5
-
-
3
6
1
4
6
3
6
7
9
3
1
3
-
3
6
5
1
2
1
5
2
-
-
 1+0
-
-
-
-
1+4
-
-
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
1
-
-
2
-
5
-
5

 

 

24
SUPERNATURAL SUPERSTITION
351
126
9
25
GENERAL THEORY OF RELATIVITY
315
135
9
26
ABCDEFGHIJKLMNOPQRSTUVWXYZ
351
135
9

 

 

Q
=
8
7
QUANTUM
107
26
8
M
=
4
9
MECHANICS
75
39
3
-
-
12
16
First Total
182
65
11
-
-
1+2
1+6
Add to Reduce
1+8+2
6+5
1+1
-
-
3
7
Second Total
11
11
2
-
-
-
-
Reduce to Deduce
1+1
1+1
-
-
-
3
7
Essence of Number
2
2
2

 

 

-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
8
-
5
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
14
-
-
-
-
-
-
-
8
-
14
9
-
19
+
=
64
6+4
=
10
1+0
1
=
1
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
4
5
3
-
1
-
-
3
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
17
21
1
-
20
21
13
-
13
5
3
-
1
-
-
3
-
+
=
118
1+1+8
=
10
1+0
1
=
1
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
13
5
3
8
1
14
9
3
19
+
=
182
1+8+2
=
11
1+1
2
=
2
-
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
+
=
65
6+5
=
11
1+1
2
=
2
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
``-
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
13
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
32
-
-
16
-
65
-
38
1+3
1+6
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3+2
-
-
1+6
-
6+5
-
3+8
4
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
11
-
11
-
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
-
-
-
-
-
-
-
1+1
-
1+1
4
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
2
-
2

 

 

16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
8
-
5
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
14
-
-
-
-
-
-
-
8
-
14
9
-
19
+
=
64
6+4
=
10
1+0
1
=
1
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
8
3
1
-
2
3
4
-
4
5
3
-
1
-
-
3
-
+
=
37
3+7
=
10
1+0
1
=
1
-
17
21
1
-
20
21
13
-
13
5
3
-
1
-
-
3
-
+
=
118
1+1+8
=
10
1+0
1
=
1
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
17
21
1
14
20
21
13
-
13
5
3
8
1
14
9
3
19
+
=
182
1+8+2
=
11
1+1
2
=
2
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
+
=
65
6+5
=
11
1+1
2
=
2
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
32
-
-
16
-
65
-
38
1+6
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3+2
-
-
1+6
-
6+5
-
3+8
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
11
-
11
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
-
-
-
-
-
-
-
1+1
-
1+1
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
2
-
2

 

 

Q
=
8
7
QUANTUM
107
26
8
E
=
5
6
ENERGY
74
38
2
-
-
13
13
First Total
181
64
10
-
-
1+3
1+3
Add to Reduce
1+8+1
6+4
1+0
-
-
4
4
Second Total
10
10
1
-
-
-
-
Reduce to Deduce
1+0
1+0
-
-
-
4
4
Essence of Number
1
1
1

 

 

Q
=
8
7
QUANTUM
107
26
8
E
=
5
8
ENERGIES
82
46
1
-
-
13
15
First Total
189
72
9
-
-
1+3
1+5
Add to Reduce
1+8+9
7+2
-
-
-
4
6
Second Total
18
9
9
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
6
Essence of Number
9
9
9

 

 

-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
9
-
1
+
=
20
2+0
=
2
-
2
-
2
-
-
-
-
-
14
-
-
-
-
-
14
-
-
-
9
-
19
+
=
56
5+6
=
11
1+1
2
-
2
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
5
-
5
9
7
-
5
-
+
=
52
5+2
=
7
-
7
=
7
-
-
17
21
1
-
20
21
13
-
5
-
5
18
7
-
5
-
+
=
133
1+3+3
=
7
-
7
-
7
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
5
14
5
18
7
9
5
19
+
=
189
1+8+9
=
18
1+8
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
+
=
72
7+2
=
9
-
9
=
9
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
``-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
39
-
-
15
-
72
-
45
=
1+5
-
-
-
-
-
-
-``
-
-
-
-
9
-
9
-
-
-
-
3+9
-
-
1+5
-
7+2
-
4+5
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
12
-
-
6
-
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
-
-
1+2
-
-
-
-
-
-
-
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
3
-
-
6
-
9
-
9

 

 

-
-
-
-
-
RANDOM
-
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
3
A+N+D
19
10
1
-
-
-
-
2
O+M
28
10
1
R
=
9
Q
8
RANDOM
65
29
11
-
-
-
-
-
-
-
-
-
Q
=
8
Q
7
QUANTUM
107
26
8
-
-
-
-
-
-
-
-
-
F
=
6
Q
12
FLUCTUATIONS
161
44
8
-
-
-
-
-
-
-
-
-
-
-
23
-
27
First Total
333
99
27
-
-
2+3
-
2+7
Add to Reduce
3+3+3
9+9
2+7
-
-
5
-
9
Second Total
9
18
9
-
-
-
-
-
Reduce to Deduce
-
1+8
-
-
-
5
Q
9
Essence of Number
9
9
9

 

 

3
THE
33
15
6
7
KINETIC
71
35
8
6
NATURE
79
25
7
2
OF
21
12
3
6
THINGS
77
32
5

 

 

-
KINETICS
--
-
-
-
-
--
-
-
-
-
-
-
-
KINETICS
--
-
-
-
-
1
K
11
2
2
-
2
1
I
9
9
9
9
-
3
N+E+T
39
12
3
-
3
1
I
9
9
9
9
-
2
C+S
22
4
4
-
4
8
KINETICS
-
-
-
-
-

 

 

K
=
2
-
-
KINETICS
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
-
-
-
-
1
S
19
10
1
K
=
2
Q
8
KINETICS
90
45
36
-
-
-
-
-
-
9+0
4+5
3+6
K
=
2
Q
8
KINETICS
9
9
9

 

 

-
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
--
-
9
-
1
+
=
24
2+4
=
6
=
6
=
6
-
-
-
9
14
-
-
9
-
19
+
=
51
5+1
=
6
=
6
=
6
-
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
5
2
-
3
-
+
=
12
1+2
=
3
=
3
=
3
-
-
11
-
-
5
20
-
3
-
+
=
39
3+9
=
12
1+2
3
=
3
-
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
11
9
14
5
20
9
3
19
+
=
90
9+0
=
9
=
9
=
9
-
-
2
9
5
5
2
9
3
1
+
=
36
3+6
=
9
=
9
=
9
-
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
--
--
-
--
--
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
--
-
5
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
8
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
-
--
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
25
8
K
I
N
E
T
I
C
S
-
-
20
-
-
8
-
36
-
18
2+5
-
-
9
-
-
-
9
-
-
-
-
2+0
-
-
-
-
3+6
-
1+8
7
8
K
I
N
E
T
I
C
S
-
-
2
-
-
8
-
9
-
9
-
-
2
9
5
5
2
9
3
1
-
-
-
-
-
-
-
-
-
-
7
8
K
I
N
E
T
I
C
S
-
-
2
-
-
8
-
9
-
9

 

 

8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
9
5
--
-
9
-
1
+
=
24
2+4
=
6
=
6
=
6
-
-
9
14
-
-
9
-
19
+
=
51
5+1
=
6
=
6
=
6
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
2
-
-
5
2
-
3
-
+
=
12
1+2
=
3
=
3
=
3
-
11
-
-
5
20
-
3
-
+
=
39
3+9
=
12
1+2
3
=
3
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
11
9
14
5
20
9
3
19
+
=
90
9+0
=
9
=
9
=
9
-
2
9
5
5
2
9
3
1
+
=
36
3+6
=
9
=
9
=
9
8
K
I
N
E
T
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
--
--
-
--
--
3
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
5
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
9
-
-
--
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
K
I
N
E
T
I
C
S
-
-
20
-
-
8
-
36
-
18
-
-
9
-
-
-
9
-
-
-
-
2+0
-
-
-
-
3+6
-
1+8
8
K
I
N
E
T
I
C
S
-
-
2
-
-
8
-
9
-
9
-
2
9
5
5
2
9
3
1
-
-
-
-
-
-
-
-
-
-
8
K
I
N
E
T
I
C
S
-
-
2
-
-
8
-
9
-
9

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
7+1
3+5
3+5
K
=
2
Q
7
KINETIC
8
8
8

 

 

E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
8+2
5+5
4+6
E
=
5
Q
8
ENERGIES
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
E
=
5
Q
8
ENERGIES
1
1
1

 

 

-
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
9
-
1
+
=
15
1+5
=
6
=
6
=
6
-
-
-
14
-
-
-
9
-
19
+
=
42
4+2
=
6
=
6
=
6
-
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
5
-
5
9
7
-
5
-
+
=
31
3+1
=
4
=
4
=
4
-
-
5
-
5
18
7
-
5
-
+
=
40
4+0
=
4
=
4
=
4
-
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
5
14
5
18
7
9
5
19
+
=
82
8+2
=
10
1+0
1
=
1
-
-
5
5
5
9
7
9
5
1
+
=
46
4+6
=
10
1+0
1
=
1
-
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
23
8
E
N
E
R
G
I
E
S
-
-
22
-
-
8
-
46
-
19
2+3
-
5
5
5
-
-
-
5
-
-
-
2+2
-
-
-
-
4+6
-
1+9
5
8
E
N
E
R
G
I
E
S
-
-
4
-
-
8
-
10
-
10
-
-
5
5
5
9
7
9
5
1
-
-
-
-
-
-
-
1+0
-
1+0
5
8
E
N
E
R
G
I
E
S
-
-
4
-
-
8
-
1
-
1

 

 

8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
9
-
1
+
=
15
1+5
=
6
=
6
=
6
-
-
14
-
-
-
9
-
19
+
=
42
4+2
=
6
=
6
=
6
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
5
-
5
9
7
-
5
-
+
=
31
3+1
=
4
=
4
=
4
-
5
-
5
18
7
-
5
-
+
=
40
4+0
=
4
=
4
=
4
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
5
14
5
18
7
9
5
19
+
=
82
8+2
=
10
1+0
1
=
1
-
5
5
5
9
7
9
5
1
+
=
46
4+6
=
10
1+0
1
=
1
8
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
E
N
E
R
G
I
E
S
-
-
22
-
-
8
-
46
-
19
-
5
5
5
-
-
-
5
-
-
-
2+2
-
-
-
-
4+6
-
1+9
8
E
N
E
R
G
I
E
S
-
-
4
-
-
8
-
10
-
10
-
5
5
5
9
7
9
5
1
-
-
-
-
-
-
-
1+0
-
1+0
8
E
N
E
R
G
I
E
S
-
-
4
-
-
8
-
1
-
1

 

 

E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
8+2
5+5
4+6
E
=
5
Q
8
ENERGIES
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
E
=
5
Q
8
ENERGIES
1
1
1

 

 

K
=
2
Q
7
KINETIC
71
35
8
E
=
5
Q
8
ENERGIES
82
55
1
Q
Q
7
Q
15
Add to Reduce
153
90
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
-
Q
Q
7
Q
6
Essence of Number
9
9
9

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
7+1
3+5
3+5
K
=
2
Q
7
KINETIC
8
8
8

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
-
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
-
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
K
=
2
Q
7
KINETIC
71
35
35
E
=
5
Q
8
ENERGIES
82
55
46
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
Q
Q
7
Q
6
Essence of Number
9
9
9

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
7+1
3+5
3+5
K
=
2
Q
7
KINETIC
8
8
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
8+2
5+5
4+6
E
=
5
Q
8
ENERGIES
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
E
=
5
Q
8
ENERGIES
1
1
1

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
-
-
-
-
-
  
-
-
-
-
1
K
11
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
K
=
2
-
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
4
3
4
30
6
7
8
36
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
3+0
-
-
-
8+1
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
4
3
4
3
6
7
8
9

 

 

K
=
2
Q
7
KINETIC
71
35
8
E
=
5
Q
8
ENERGIES
82
55
1
Q
Q
7
Q
15
Add to Reduce
153
90
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
-
Q
Q
7
Q
6
Essence of Number
9
9
9

 

 

-
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
--
-
9
-
-
-
5
-
-
-
9
-
1
+
=
38
3+8
=
11
1+1
2
=
2
-
-
-
9
14
-
-
9
-
-
-
14
-
-
-
9
-
19
+
=
74
7+4
=
11
1+1
2
=
2
-
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
5
2
-
3
-
5
-
5
9
7
-
5
-
+
=
43
4+3
=
7
=
7
=
7
-
-
11
-
-
5
20
-
3
-
5
-
5
18
7
-
5
-
+
=
79
7+9
=
16
1+6
7
=
7
-
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
11
9
14
5
20
9
3
-
5
14
5
18
7
9
5
19
+
=
153
1+5+3
=
9
=
9
=
9
-
-
2
9
5
5
2
9
3
-
5
5
5
9
7
9
5
1
+
=
81
8+1
=
9
=
9
=
9
-
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
--
--
-
--
--
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
--
-
5
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
6
=
30
3+0
3
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
-
--
9
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
18
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
27
-
-
15
-
81
-
27
1+8
1+5
-
9
-
-
-
9
-
-
5
5
5
-
-
-
5
-
-
-
2+7
-
-
1+5
-
8+1
-
2+7
9
6
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9
-
-
2
9
5
5
2
9
3
-
5
5
5
9
7
9
5
1
-
-
-
-
-
-
-
-
-
-
9
6
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9

 

 

15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
9
5
--
-
9
-
-
-
5
-
-
-
9
-
1
+
=
38
3+8
=
11
1+1
2
=
2
-
-
9
14
-
-
9
-
-
-
14
-
-
-
9
-
19
+
=
74
7+4
=
11
1+1
2
=
2
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
2
-
-
5
2
-
3
-
5
-
5
9
7
-
5
-
+
=
43
4+3
=
7
=
7
=
7
-
11
-
-
5
20
-
3
-
5
-
5
18
7
-
5
-
+
=
79
7+9
=
16
1+6
7
=
7
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
11
9
14
5
20
9
3
-
5
14
5
18
7
9
5
19
+
=
153
1+5+3
=
9
=
9
=
9
-
2
9
5
5
2
9
3
-
5
5
5
9
7
9
5
1
+
=
81
8+1
=
9
=
9
=
9
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
--
--
-
--
--
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
5
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
9
-
-
--
9
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
15
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
27
-
-
15
-
81
-
27
1+5
-
9
-
-
-
9
-
-
5
5
5
-
-
-
5
-
-
-
2+7
-
-
1+5
-
8+1
-
2+7
6
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9
-
2
9
5
5
2
9
3
-
5
5
5
9
7
9
5
1
-
-
-
-
-
-
-
-
-
-
6
K
I
N
E
T
I
C
-
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9

 

 

15
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
9
5
--
-
9
-
-
5
-
-
-
9
-
1
+
=
38
3+8
=
11
1+1
2
=
2
-
-
9
14
-
-
9
-
-
14
-
-
-
9
-
19
+
=
74
7+4
=
11
1+1
2
=
2
15
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
2
-
-
5
2
-
3
5
-
5
9
7
-
5
-
+
=
43
4+3
=
7
=
7
=
7
-
11
-
-
5
20
-
3
5
-
5
18
7
-
5
-
+
=
79
7+9
=
16
1+6
7
=
7
15
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
11
9
14
5
20
9
3
5
14
5
18
7
9
5
19
+
=
153
1+5+3
=
9
=
9
=
9
-
2
9
5
5
2
9
3
5
5
5
9
7
9
5
1
+
=
81
8+1
=
9
=
9
=
9
15
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
--
--
-
--
--
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
5
5
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
9
-
-
--
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
15
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
27
-
-
15
-
81
-
27
1+5
-
9
-
-
-
9
-
5
5
5
-
-
-
5
-
-
-
2+7
-
-
1+5
-
8+1
-
2+7
6
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9
-
2
9
5
5
2
9
3
5
5
5
9
7
9
5
1
-
-
-
-
-
-
-
-
-
-
6
K
I
N
E
T
I
C
E
N
E
R
G
I
E
S
-
-
9
-
-
6
-
9
-
9

 

 

-
-
-
-
-
-
-
-
-
K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
7+1
3+5
3+5
K
=
2
Q
7
KINETIC
8
8
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
8+2
5+5
4+6
E
=
5
Q
8
ENERGIES
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
E
=
5
Q
8
ENERGIES
1
1
1

 

 

5
MESON
-
-
-
-
M+E
18
9
9
-
S+O+N
48
12
3
5
MESON
66
21
12
-
-
6+6
2+1
1+2
5
MESON
12
3
3
-
-
1+2
-
-
5
MESON
3
3
3

 

 

6
MESONS
-
-
-
-
M+E
18
9
9
-
S+O+N+S
67
13
4
6
MESONS
85
22
13
-
-
8+5
2+2
1+3
6
MESONS
13
4
4
8
THIRTEEN
99
45
9

 

 

11
SCHRODINGERS
139
67
4
3
CAT
24
6
6
14
First Total
163
73
10
1+4
Add to Reduce
1+6+3
7+3
1+0
5
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
5
Essence of Number
1
1
1

 

 

W
=
5
4
WAVE
51
15
6
P
=
7
8
PARTICLE
84
39
3
-
-
12
12
Add to Reduce
135
54
9
-
-
 1+2 1+2
Reduce to Deduce
1+3+5
5+4
-
-
-
3
3
Essence of Number
9
9
9

 

 

5
WAVES
70
16
7
9
PARTICLES
103
40
4

 

 

4
WAVE
51
15
6
9
PARTICLES
103
40
4
13
First Total
154
55
10
1+3
Add to Reduce
1+5+4
5+5
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
14
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+4
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

6
ENERGY
74
38
2
8
ENERGIES
82
46
1

 

 

5
FORCE
47
29
2
6
FORCES
66
30
3

 

 

-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
1
-
6
-
-
-
-
-
-
9
-
-
+
=
22
2+2
=
4
=
4
=
4
-
-
-
15
-
-
-
19
-
15
-
-
-
-
-
-
9
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
-
9
3
5
-
-
-
6
-
7
9
1
4
-
2
7
+
=
59
5+9
=
14
1+4
5
=
5
-
-
6
-
18
3
5
-
-
-
6
-
7
18
1
22
-
20
25
+
=
131
1+3+1
=
5
=
5
=
5
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
15
18
3
5
19
-
15
6
-
7
18
1
22
9
20
25
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
6
6
9
3
5
1
-
6
6
-
7
9
1
4
9
2
7
+
=
81
8+1
=
9
=
9
=
9
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-``
-
6
6
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
8
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
37
-
-
15
-
81
-
36
-
1+5
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
3+7
-
-
1+5
-
8+1
-
3+6
8
6
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
10
-
-
6
-
9
-
9
-
-
6
6
9
3
5
1
-
6
6
-
7
9
1
4
9
2
7
-
-
1+0
-
-
-
-
-
-
-
8
6
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
1
-
-
6
-
9
-
9

 

 

4
HEAT
34
16
7
5
LIGHT
56
29
2
9
Add to Reduce
90
45
9
-
Reduce to Deduce
9+0
4+5
-
9
Essence of Number
9
9
9

 

 

4
MIND
40
22
4
6
MATTER
77
23
5
10
Add to Reduce
117
45
9
1+0
Reduce to Deduce
1+1+7
4+5
-
1
Essence of Number
9
9
9

 

 

--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
6
1
-
6
-
-
+
=
14
1+4
=
5
-
5
=
5
-
-
19
-
-
-
-
-
15
19
-
15
-
-
+
=
68
6+8
=
14
1+4
5
=
5
--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
-
7
5
3
2
9
-
-
3
-
7
5
+
=
41
4+1
=
5
-
5
=
5
-
-
-
16
5
3
20
18
-
-
3
-
16
5
+
=
86
8+6
=
14
1+4
5
=
5
--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
19
16
5
3
20
18
15
19
3
15
16
5
+
+
154
1+5+4
=
10
1+0
1
=
1
-
-
1
7
5
3
2
9
6
1
3
6
7
5
+
+
55
5+5
=
10
1+0
1
=
1
-
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
-
--
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
4
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
12
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
33
-
-
12
-
55
-
28
1+2
1+2
-
-
-
-
-
9
-
-
-
-
-
-
-
-
3+3
-
-
1+2
-
5+5
-
2+8
3
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
10
-
10
-
-
1
7
5
3
2
9
6
1
3
6
7
5
-
-
-
-
-
-
-
1+0
-
1+0
3
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
1
-
1

 

 

12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
6
1
-
6
-
-
+
=
14
1+4
=
5
-
5
=
5
-
19
-
-
-
-
-
15
19
-
15
-
-
+
=
68
6+8
=
14
1+4
5
=
5
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
7
5
3
2
9
-
-
3
-
7
5
+
=
41
4+1
=
5
-
5
=
5
-
-
16
5
3
20
18
-
-
3
-
16
5
+
=
86
8+6
=
14
1+4
5
=
5
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
19
16
5
3
20
18
15
19
3
15
16
5
+
+
154
1+5+4
=
10
1+0
1
=
1
-
1
7
5
3
2
9
6
1
3
6
7
5
+
+
55
5+5
=
10
1+0
1
=
1
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
-
--
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
33
-
-
12
-
55
-
28
1+2
-
-
-
-
-
9
-
-
-
-
-
-
-
-
3+3
-
-
1+2
-
5+5
-
2+8
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
10
-
10
-
1
7
5
3
2
9
6
1
3
6
7
5
-
-
-
-
-
-
-
1+0
-
1+0
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
1
-
1

 

 

-
SPECTRUM
-
-
-
5
SPECT
63
18
9
1
R
18
9
9
2
UM
34
7
7
8
SPECTRUM
115
34
9
-
-
1+1+5
3+4
-
8
SPECTRUM
7
7
7

 

 

-
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
-
=
1
=
1
-
-
19
-
-
-
-
-
-
-
+
=
19
1+9
=
10
1+0
1
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
-
7
5
3
2
9
3
4
+
=
33
3+3
=
6
=
6
-
-
-
16
5
3
20
18
21
13
+
=
96
96
=
15
1+5
6
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
19
16
5
3
20
18
21
13
+
+
115
1+1+5
=
7
=
7
-
-
1
7
5
3
2
9
3
4
+
+
34
3+4
=
7
=
7
-
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
14
8
S
P
E
C
T
R
U
M
-
-
33
-
-
8
-
34
1+4
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
3+4
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
8
-
7
-
-
1
7
5
3
2
9
3
4
-
-
-
-
-
-
-
-
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
8
-
7

 

 

8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
-
=
1
=
1
-
19
-
-
-
-
-
-
-
+
=
19
1+9
=
10
1+0
1
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
7
5
3
2
9
3
4
+
=
33
3+3
=
6
=
6
-
-
16
5
3
20
18
21
13
+
=
96
96
=
15
1+5
6
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
19
16
5
3
20
18
21
13
+
+
115
1+1+5
=
7
=
7
-
1
7
5
3
2
9
3
4
+
+
34
3+4
=
7
=
7
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
8
S
P
E
C
T
R
U
M
-
-
33
-
-
8
-
34
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
3+4
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7
-
1
7
5
3
2
9
3
4
-
-
-
-
-
-
-
-
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7

 

 

7
RAINBOW
82
37
1
6
COLOUR
84
30
3

 

 

7
RAINBOW
82
37
1
7
COLOURS
103
31
4

 

 

8
SPECTRUM
115
34
7
2
OF
21
12
3
3
THE
33
15
6
7
RAINBOW
82
37
1

 

 

3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2

 

 

-
RAINBOW
-
-
-
1
BRAIN
44
26
8
4
OW
54
18
9
7
RAINBOW
82
37
28
-
-
8+2
3+7
3+7
7
RAINBOW
10
10
10
-
-
1+0
1+0
1+0
7
RAINBOW
1
1
1

 

 

15
THE RAINBOW LIGHT
-
-
-
-
THE
33
15
6
-
R
18
9
9
-
A
1
1
1
-
I
9
9
9
-
N+B+O+W
54
18
9
-
L
12
3
3
-
I
9
9
9
-
G+H+T
35
17
8
15
THE RAINBOW LIGHT
171
81
54
1+5
-
1+7+1
8+1
5+4
6
THE RAINBOW LIGHT
9
9
9

 

 

-
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
9
5
-
6
-
-
9
-
8
-
+
+
45
4+5
=
9
=
9
=
9
-
-
8
-
-
-
-
9
14
-
15
-
-
9
-
8
-
+
+
63
6+3
=
9
=
9
=
9
-
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
9
1
-
-
2
-
5
-
3
-
7
-
14
+
+
36
3+6
=
9
=
9
=
9
-
-
20
-
5
-
18
1
-
-
2
-
23
-
12
-
7
-
20
+
+
108
1+0+8
=
9
=
9
=
9
-
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
18
1
9
14
2
15
23
-
12
9
7
8
20
+
+
171
1+7+1
=
9
=
9
=
9
-
-
2
8
5
-
9
1
9
5
2
6
5
-
3
9
7
8
2
+
+
81
8+1
=
9
=
9
=
9
-
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
-
4
FOUR
4
-
-
-
-
-
--
--
--
-
5
-
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
4
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
41
-
-
15
-
81
-
45
-
1+5
-
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+5
-
8+1
-
4+5
4
6
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9
-
-
2
8
5
-
9
1
9
5
2
6
5
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
4
6
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9

 

 

15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
9
5
-
6
-
-
9
-
8
-
+
+
45
4+5
=
9
=
9
=
9
-
8
-
-
-
-
9
14
-
15
-
-
9
-
8
-
+
+
63
6+3
=
9
=
9
=
9
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
9
1
-
-
2
-
5
-
3
-
7
-
14
+
+
36
3+6
=
9
=
9
=
9
-
20
-
5
-
18
1
-
-
2
-
23
-
12
-
7
-
20
+
+
108
1+0+8
=
9
=
9
=
9
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
18
1
9
14
2
15
23
-
12
9
7
8
20
+
+
171
1+7+1
=
9
=
9
=
9
-
2
8
5
-
9
1
9
5
2
6
5
-
3
9
7
8
2
+
+
81
8+1
=
9
=
9
=
9
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
--
-
5
-
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
15
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
41
-
-
15
-
81
-
45
1+5
-
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+5
-
8+1
-
4+5
6
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9
-
2
8
5
-
9
1
9
5
2
6
5
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
6
T
H
E
-
R
A
I
N
B
O
W
-
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9

 

 

15
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
9
5
-
6
-
9
-
8
-
+
+
45
4+5
=
9
=
9
=
9
-
8
-
-
-
9
14
-
15
-
9
-
8
-
+
+
63
6+3
=
9
=
9
=
9
15
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
9
1
-
-
2
-
5
3
-
7
-
14
+
+
36
3+6
=
9
=
9
=
9
-
20
-
5
18
1
-
-
2
-
23
12
-
7
-
20
+
+
108
1+0+8
=
9
=
9
=
9
15
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
18
1
9
14
2
15
23
12
9
7
8
20
+
+
171
1+7+1
=
9
=
9
=
9
-
2
8
5
9
1
9
5
2
6
5
3
9
7
8
2
+
+
81
8+1
=
9
=
9
=
9
15
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
-
-
-
2
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
--
-
5
-
-
-
5
-
-
5
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
9
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
15
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
41
-
-
15
-
81
-
45
1+5
-
-
-
9
-
9
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+5
-
8+1
-
4+5
6
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9
-
2
8
5
9
1
9
5
2
6
5
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
6
T
H
E
R
A
I
N
B
O
W
L
I
G
H
T
-
-
5
-
-
6
-
9
-
9

 

 

-
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
5
-
-
-
9
-
8
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
-
-
-
8
-
-
14
-
-
-
9
-
8
-
+
=
39
3+9
=
12
1+2
3
=
3
-
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
2
9
3
2
-
-
1
-
4
-
3
-
7
-
2
+
=
33
3+3
=
6
-
6
=
6
-
-
20
18
21
20
-
-
1
-
4
-
12
-
7
-
20
+
=
123
1+2+3
=
6
-
6
=
6
-
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
20
18
21
20
8
-
1
14
4
-
12
9
7
8
20
+
=
162
1+6+2
=
9
-
9
=
9
-T
-
2
9
3
2
8
-T
1
5
4
-T
3
9
7
8
2
+
=
63
6+3
=
9
-
9
=
9
-
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
18
1+8
9
6
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
39
-
1
12
-
54
-
45
-
1+2
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+9
-
-
1+2
-
5+4
-
4+5
6
3
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
12
-
1
3
-
9
-
9
-
-
2
9
3
2
8
-
1
5
4
-
3
9
7
8
2
-
-
1+2
-
-
-
-
-
-
-
6
3
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
3
-
-
3
-
9
-
9

 

 

12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
5
-
-
-
9
-
8
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
-
-
8
-
-
14
-
-
-
9
-
8
-
+
=
39
3+9
=
12
1+2
3
=
3
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
2
9
3
2
-
-
1
-
4
-
3
-
7
-
2
+
=
33
3+3
=
6
-
6
=
6
-
20
18
21
20
-
-
1
-
4
-
12
-
7
-
20
+
=
123
1+2+3
=
6
-
6
=
6
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
20
18
21
20
8
-
1
14
4
-
12
9
7
8
20
+
=
162
1+6+2
=
9
-
9
=
9
-
2
9
3
2
8
-T
1
5
4
-T
3
9
7
8
2
+
=
63
6+3
=
9
-
9
=
9
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
--
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
18
1+8
9
12
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
39
-
1
12
-
54
-
45
1+2
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+9
-
-
1+2
-
5+4
-
4+5
3
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
12
-
1
3
-
9
-
9
-
2
9
3
2
8
-
1
5
4
-
3
9
7
8
2
-
-
1+2
-
-
-
-
-
-
-
3
T
R
U
T
H
-
A
N
D
-
L
I
G
H
T
-
-
3
-
-
3
-
9
-
9

 

 

12
T
R
U
T
H
A
N
D
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
-
-
9
-
8
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
-
-
8
-
14
-
-
9
-
8
-
+
=
39
3+9
=
12
1+2
3
=
3
12
T
R
U
T
H
A
N
D
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
2
9
3
2
-
1
-
4
3
-
7
-
2
+
=
33
3+3
=
6
-
6
=
6
-
20
18
21
20
-
1
-
4
12
-
7
-
20
+
=
123
1+2+3
=
6
-
6
=
6
12
T
R
U
T
H
A
N
D
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
20
18
21
20
8
1
14
4
12
9
7
8
20
+
=
162
1+6+2
=
9
-
9
=
9
-
2
9
3
2
8
1
5
4
3
9
7
8
2
+
=
63
6+3
=
9
-
9
=
9
12
T
R
U
T
H
A
N
D
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
2
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
--
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
18
1+8
9
12
T
R
U
T
H
A
N
D
L
I
G
H
T
-
-
39
-
1
12
-
54
-
45
1+2
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
3+9
-
-
1+2
-
5+4
-
4+5
3
T
R
U
T
H
A
N
D
L
I
G
H
T
-
-
12
-
1
3
-
9
-
9
-
2
9
3
2
8
1
5
4
3
9
7
8
2
-
-
1+2
-
-
-
-
-
-
-
3
T
R
U
T
H
A
N
D
L
I
G
H
T
-
-
3
-
-
3
-
9
-
9

 

 

-
-
-
-
-
LIGHT
-
-
-
-
-
-
-
4
LIGH
36
27
9
-
-
-
-
1
T
20
2
2
L
=
3
-
5
LIGHT
56
29
11
-
-
1+0
-
-
-
5+6
2+9
1+1
-
-
1
-
9
LIGHT
11
11
7
-
-
1+0
-
-
-
1+1
1+1
-
-
-
1
-
9
LIGHT
2
2
2

 

 

-
5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
-
-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
-
5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
-
-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
-
5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
-
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
--
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
16
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
1+6
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
7
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
7
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
5
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

L
=
3
-
6
LIVING
73
37
1
H
=
8
-
4
HEAT
34
16
7
L
=
3
-
5
LIGHT
56
29
2
-
-
14
-
15
-
163
82
10
-
-
1+4
-
1+5
-
1+6+3
8+2
1+0
-
-
5
-
6
-
10
10
1
-
-
-
-
-
-
1+0
1+0
-
-
-
5
-
6
-
1
1
9

 

 

L
=
3
-
6
LOVING
79
34
7
H
=
8
-
4
HEAT
34
16
7
L
=
3
-
5
LIGHT
56
29
2
-
-
14
-
15
-
169
79
16
-
-
1+4
-
1+5
-
1+6+9
7+9
1+6
-
-
5
-
6
-
16
10
7
-
-
-
-
-
-
1+6
1+0
-
-
-
5
-
6
-
7
7
7

 

 

P
=
7
-
4
PARTICLES
103
40
4
O
=
6
-
1
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
16
-
10
-
180
81
9
-
-
1+6
-
1+0
-
1+8+0
8+1
-
-
-
7
-
1
-
9
9
9

 

 

9
PARTICLES
103
40
4
2
OF
21
12
3
5
LIGHT
56
29
2
16
Add to Reduce
180
81
9
1+6
Reduce to Deduce
1+8+0
8+1
-
7
Essence of Number
9
9
9

 

 

-T
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-T
`-
-
-
-
-
9
-
-
-
1
-
6
-
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
-
`-
-
-
-
-
9
-
-
-
19
-
15
-
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
-
-
6
-
3
-
7
-
2
+
=
48
4+8
=
12
1+2
3
=
3
-
`-
16
1
18
20
-
3
12
5
-
-
-
6
-
12
-
7
-
20
+
=
120
1+2+0
=
3
=
3
=
3
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
`-
16
1
18
20
9
3
12
5
19
-
15
6
-
12
9
7
8
20
+
=
180
1+8+0
=
9
=
9
=
9
-
-
7
1
9
2
9
3
3
5
1
-T
6
6
-T
3
9
7
8
2
+
=
81
8+1
=
9
=
9
=
9
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
--
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
41
-
1
16
-
81
-
45
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+6
-
8+1
-
4+5
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
1
7
-
9
-
9
-
-
7
1
9
2
9
3
-
5
1
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
-
7
-
9
-
9

 

 

16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
-
-
-
9
-
-
-
1
-
6
-
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
`-
-
-
-
-
9
-
-
-
19
-
15
-
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
-
-
6
-
3
-
7
-
2
+
=
48
4+8
=
12
1+2
3
=
3
`-
16
1
18
20
-
3
12
5
-
-
-
6
-
12
-
7
-
20
+
=
120
1+2+0
=
3
=
3
=
3
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
16
1
18
20
9
3
12
5
19
-
15
6
-
12
9
7
8
20
+
=
180
1+8+0
=
9
=
9
=
9
-
7
1
9
2
9
3
3
5
1
-T
6
6
-T
3
9
7
8
2
+
=
81
8+1
=
9
=
9
=
9
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
3
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
--
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
41
-
1
16
-
81
-
45
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+6
-
8+1
-
4+5
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
1
7
-
9
-
9
-
7
1
9
2
9
3
-
5
1
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
-
7
-
9
-
9

 

 

16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
-
-
-
9
-
-
-
1
6
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
`-
-
-
-
-
9
-
-
-
19
15
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
-
6
3
-
7
-
2
+
=
48
4+8
=
12
1+2
3
=
3
`-
16
1
18
20
-
3
12
5
-
-
6
12
-
7
-
20
+
=
120
1+2+0
=
3
=
3
=
3
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
16
1
18
20
9
3
12
5
19
15
6
12
9
7
8
20
+
=
180
1+8+0
=
9
=
9
=
9
-
7
1
9
2
9
3
3
5
1
6
6
3
9
7
8
2
+
=
81
8+1
=
9
=
9
=
9
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
3
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
--
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
41
-
1
16
-
81
-
45
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+6
-
8+1
-
4+5
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
5
-
1
7
-
9
-
9
-
7
1
9
2
9
3
-
5
1
6
6
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
16
P
A
R
T
I
C
L
E
S
O
F
L
I
G
H
T
-
-
5
-
-
7
-
9
-
9

 

 

-
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
-
-
-
-
-
1
+
=
18
1+8
=
9
=
9
=
9
-
`-
-
9
-
8
-
-
-
-
-
-
19
+
=
36
3+6
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
-
3
-
7
-
2
-
5
1
4
5
-
+
=
27
2+7
=
9
=
9
=
9
-
`-
12
-
7
-
20
-
23
1
22
5
-
+
=
90
7+5
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
`-
12
9
7
8
20
-
23
1
22
5
19
+
=
126
1+2+6
=
9
=
9
=
9
-
-
3
9
7
8
2
-T
5
1
4
5
1
+
=
45
4+5
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
6
10
L
I
G
H
T
-
W
A
V
E
S
-
-
39
-
1
12
-
45
-
36
-
1+0
-
9
-
-
-
-
-
-
-
-
-
-
-
3+9
-
-
1+2
-
4+5
-
3+6
6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
12
-
1
3
-
9
-
9
-
-
3
9
7
8
2
-
5
1
4
5
1
-
-
1+2
-
-
-
-
-
-
-
6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
3
-
-
3
-
9
-
9

 

 

10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
-
-
-
-
-
1
+
=
18
1+8
=
9
=
9
=
9
`-
-
9
-
8
-
-
-
-
-
-
19
+
=
36
3+6
=
9
=
9
=
9
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
3
-
7
-
2
-
5
1
4
5
-
+
=
27
2+7
=
9
=
9
=
9
`-
12
-
7
-
20
-
23
1
22
5
-
+
=
90
7+5
=
9
=
9
=
9
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
`-
12
9
7
8
20
-
23
1
22
5
19
+
=
126
1+2+6
=
9
=
9
=
9
-
3
9
7
8
2
-T
5
1
4
5
1
+
=
45
4+5
=
9
=
9
=
9
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
L
I
G
H
T
-
W
A
V
E
S
-
-
39
-
1
12
-
45
-
36
1+0
-
9
-
-
-
-
-
-
-
-
-
-
-
3+9
-
-
1+2
-
4+5
-
3+6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
12
-
1
3
-
9
-
9
-
3
9
7
8
2
-
5
1
4
5
1
-
-
1+2
-
-
-
-
-
-
-
1
L
I
G
H
T
-
W
A
V
E
S
-
-
3
-
-
3
-
9
-
9

 

 

10
L
I
G
H
T
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
-
-
-
-
1
+
=
18
-
=
9
=
9
=
9
`-
-
9
-
8
-
-
-
-
-
19
+
=
38
-
=
9
=
9
=
9
10
L
I
G
H
T
W
A
V
E
S
-
-
-
-
-
-
-
-
-
-
-
3
-
7
-
2
5
1
4
5
-
+
=
27
2+7
=
9
=
9
=
9
`-
12
-
7
-
20
23
1
22
5
-
+
=
90
7+5
=
9
=
9
=
9
10
L
I
G
H
T
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
`-
12
9
7
8
20
23
1
22
5
19
+
=
126
1+2+6
=
9
=
9
=
9
-
3
9
7
8
2
5
1
4
5
1
+
=
45
4+5
=
9
=
9
=
9
10
L
I
G
H
T
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
L
I
G
H
T
W
A
V
E
S
-
-
39
-
1
12
-
45
-
36
1+0
-
9
-
-
-
-
-
-
-
-
-
-
3+9
-
-
1+2
-
4+5
-
3+6
1
L
I
G
H
T
W
A
V
E
S
-
-
12
-
1
3
-
9
-
9
-
3
9
7
8
2
5
1
4
5
1
-
-
1+2
-
-
-
-
-
-
-
1
L
I
G
H
T
W
A
V
E
S
-
-
3
-
-
3
-
9
-
9

 

 

5
WAVES
70
16
7
9
PARTICLES
103
40
4

 

 

4
WAVE
51
15
6
9
PARTICLES
103
40
4
13
First Total
154
55
10
1+3
Add to Reduce
1+5+4
5+5
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

W
=
5
4
WAVE
51
15
6
P
=
7
8
PARTICLE
84
39
3
-
-
12
12
Add to Reduce
135
54
9
-
-
1+2
1+2
Reduce to Deduce
1+3+5
5+4
-
-
-
3
3
Essence of Number
9
9
9

 

 

-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
=
9
=
9
-
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
=
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
=
9
=
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
=
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
14
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+4
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
=
9
=
9
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
=
9
=
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
=
9
=
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
=
9
=
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

12
W
A
V
E
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
=
9
12
W
A
V
E
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
5
1
4
5
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
=
9
=
9
-
23
1
22
5
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
=
9
=
9
12
W
A
V
E
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
23
1
22
5
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
=
9
=
9
-
5
1
4
5
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
=
9
=
9
12
W
A
V
E
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
5
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
W
A
V
E
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+2
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
3
W
A
V
E
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
5
1
4
5
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
3
W
A
V
E
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

S
=
1
-
5
SPEED
49
22
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
10
-
12
-
126
63
9
-
-
1+0
-
1+2
-
1+2+6
6+3
-
-
-
1
-
3
-
9
9
9

 

 

-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
6
-
-
-
9
-
8
-
+
=
24
2+4
=
6
=
6
=
6
-
-
19
-
-
-
-
-
15
-
-
-
9
-
8
-
+
=
51
5+1
=
6
=
6
=
6
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
7
5
5
4
-
-
6
-
3
-
7
-
2
+
=
39
3+9
=
12
1+2
3
=
3
-
-
-
16
5
5
4
-
-
6
-
12
-
7
-
20
+
=
75
7+5
=
12
1+2
3
=
3
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
19
16
5
5
4
-
15
6
-
12
9
7
8
20
+
=
126
1+2+6
=
9
=
9
=
9
--
-
1
7
5
5
4
-T
6
6
-T
3
9
7
8
2
+
=
63
6+3
=
9
=
9
=
9
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
--
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
27
=
9
45
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
45
-
1
12
-
81
-
36
4+5
1+2
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
1+2
-
8+1
-
3+4
9
3
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
9
-
1
3
-
9
-
9
-
-
1
7
5
5
4
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
9
3
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
9
-
-
3
-
9
-
9

 

 

-
12
S
P
E
E
D
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
6
-
-
9
-
8
-
+
=
24
2+4
=
6
=
6
=
6
-
-
19
-
-
-
-
15
-
-
9
-
8
-
+
=
51
5+1
=
6
=
6
=
6
-
12
S
P
E
E
D
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
7
5
5
4
-
6
3
-
7
-
2
+
=
39
3+9
=
12
1+2
3
=
3
-
-
-
16
5
5
4
-
6
12
-
7
-
20
+
=
75
7+5
=
12
1+2
3
=
3
-
12
S
P
E
E
D
O
F
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
19
16
5
5
4
15
6
12
9
7
8
20
+
=
126
1+2+6
=
9
=
9
=
9
--
-
1
7
5
5
4
6
6
3
9
7
8
2
+
=
63
6+3
=
9
=
9
=
9
-
12
S
P
E
E
D
O
F
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
--
-
7
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
27
=
9
45
12
S
P
E
E
D
O
F
L
I
G
H
T
-
-
45
-
1
12
-
81
-
36
4+5
1+2
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
1+2
-
8+1
-
3+4
9
3
S
P
E
E
D
O
F
L
I
G
H
T
-
-
9
-
1
3
-
9
-
9
-
-
1
7
5
5
4
6
6
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
9
3
S
P
E
E
D
O
F
L
I
G
H
T
-
-
9
-
-
3
-
9
-
9

 

 

16
15
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
-
-
1
-
9
-
-
-
-
6
-
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
`-
-
-
-
19
-
9
-
-
-
-
15
-
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
16
15
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
6
5
-
2
-
4
1
3
-
-
6
-
3
-
7
-
2
+
=
39
3+9
=
12
1+2
3
=
3
`-
-
6
5
-
20
-
22
1
12
-
-
6
-
12
-
7
-
20
+
=
111
1+1+1
=
3
=
3
=
3
16
15
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
6
5
19
20
9
22
1
12
-
15
6
-
12
9
7
8
20
+
=
171
1+7+1
=
9
=
9
=
9
-
-
6
5
1
2
9
4
1
3
-T
6
6
-T
3
9
7
8
2
+
=
72
7+2
=
9
=
9
=
9
16
15
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
6
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
15
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
45
-
1
15
-
72
-
54
4+5
1+5
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
1+5
-
7+2
-
5+4
9
6
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
9
-
1
6
-
9
-
9
-
-
6
5
1
2
9
4
1
3
-T
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
9
6
F
E
S
T
I
V
A
L
-
O
F
-
L
I
G
H
T
-
-
9
-
-
6
-
9
-
9

 

 

16
15
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
-
-
1
-
9
-
-
-
6
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
`-
-
-
-
19
-
9
-
-
-
15
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
16
15
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
6
5
-
2
-
4
1
3
-
6
3
-
7
-
2
+
=
39
3+9
=
12
1+2
3
=
3
`-
-
6
5
-
20
-
22
1
12
-
6
12
-
7
-
20
+
=
111
1+1+1
=
3
=
3
=
3
16
15
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
`-
-
6
5
19
20
9
22
1
12
15
6
12
9
7
8
20
+
=
171
1+7+1
=
9
=
9
=
9
-
-
6
5
1
2
9
4
1
3
6
6
3
9
7
8
2
+
=
72
7+2
=
9
=
9
=
9
16
15
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
6
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
--
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
15
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
45
-
1
15
-
72
-
54
4+5
1+5
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
1+5
-
7+2
-
5+4
9
6
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
9
-
1
6
-
9
-
9
-
-
6
5
1
2
9
4
1
3
6
6
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
9
6
F
E
S
T
I
V
A
L
O
F
L
I
G
H
T
-
-
9
-
-
6
-
9
-
9

 

 

3
THE
33
15
6
7
RAINBOW
82
37
1
8
SPECTRUM
115
34
7
2
OF
21
12
3
6
COLOUR
84
30
3

 

 

8
COLOURED
-
-
-
-
C+O
18
9
9
9
L+O
27
9
9
9
U
21
3
3
-
R
18
9
9
9
E+D
9
9
9
8
COLOURED
-
-
-

 

 

6
COLOUR
84
30
3
3
RED
27
18
9

 

 

-
QUANTUM
8
-
-
1
Q
17
8
8
3
UAN
35
9
9
3
TUM
54
9
9
7
QUANTUM
107
26
26
-
-
1+0+7
2+6
2+6
7
QUANTUM
8
8
8

 

 

-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
-
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
6
SIX
-
-
-
-
7
``-
-
-
-
-
-
-
-
-
-
7
SEVEN
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
-
9
NINE
-
-
-
-
22
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
2+2
-
-
-
-
-
-
-
-``
-
-
2+3
-
-
-
-
2+6
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-
-
-
-
-
-``
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-
1
-
2
3
4
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

RA ATUM ATUM RA

 

Q
=
8
-
7
QUANTUM
107
26
8
E
=
5
-
8
ENERGIES
82
46
1
-
-
13
-
27
First Total
189
72
9
-
-
1+3
-
2+7
Add to Reduce
1+8+9
7+2
-
Q
-
4
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
5
9
Essence of Number
9
9
9

 

 

-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
9
-
1
+
=
20
2+0
=
2
-
2
=
2
-
-
-
-
-
14
-
-
-
-
-
14
-
-
-
9
-
19
+
=
56
5+6
=
11
1+1
2
=
2
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
5
-
5
9
7
-
5
-
+
=
52
5+2
=
7
-
7
=
7
-
-
17
21
1
-
20
21
13
-
5
-
5
18
7
-
5
-
+
=
133
1+3+3
=
7
-
7
=
7
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
5
14
5
18
7
9
5
19
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
+
=
72
7+2
=
9
-
9
=
9
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
-
-
-
-
-
-
-
``-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
39
-
-
15
-
72
-
45
=
1+5
-
-
-
-
-
-
-``
-
-
-
-
9
-
9
-
-
-
-
3+9
-
-
1+5
-
7+2
-
4+5
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
12
-
-
6
-
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
-
-
1+2
-
-
-
-
-
-
-
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
3
-
-
6
-
9
-
9

 

QUANTUM THOUGHTS THOUGHTS QUANTUM

 

A
=
1
-
3
AND
19
10
1
Q
=
8
-
7
QUANTUM
107
26
8
I
=
9
-
2
IS
28
10
1
A
=
1
-
2
AS
20
2
2
Q
=
8
-
7
QUANTUM
107
26
8
D
=
4
-
4
DOES
43
16
7
-
-
31
-
25
First Total
324
90
27
-
-
3+1
-
2+5
Add to Reduce
3+2+4
9+0
2+7
Q
-
4
-
7
Second Total
9
9
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
4
5
7
Essence of Number
9
9
9

 

 

 

Q
=
8
7
QUANTUM
107
26
8
T
=
2
6
THEORY
91
37
1
-
-
10
13
First Total
198
63
9
-
-
1+0
1+3
Add to Reduce
1+9+8
6+3
-
-
-
1
4
Second Total
18
9
9
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
1
4
Essence of Number
9
9
9

 

 

-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
8
-
6
-
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
-
-
14
-
-
-
-
-
8
-
15
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
2
-
5
-
9
7
+
=
44
4+4
=
8
-
8
=
8
-
-
17
21
1
-
20
21
13
-
20
-
5
-
18
25
+
=
161
1+6+1
=
8
-
8
=
8
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
20
8
5
15
18
25
+
+
198
1+9+8
=
18
1+8
9
=
9
-
-
8
3
1
5
2
3
4
-
2
8
5
6
9
7
+
+
63
6+3
=
9
-
9
=
9
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
``-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
45
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
45
-
-
13
-
63
-
45
4+5
1+3
-
-
-
-
-
-
-``
-
-
-
-
-
9
-
-
-
4+5
-
-
1+3
-
6+3
-
4+5
9
4
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
9
-
-
4
-
9
-
9
-
-
8
3
1
5
2
3
4
-
2
8
5
6
9
7
-
-
-
-
-
-
-
-
-
-
9
4
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
9
-
-
4
-
9
-
9

 

 

THE EMPERORS'S NEW MIND

CONCERNING COMPUTERS, MINDS, AND THE LAWS OF PHYSICS

Roger Penrose 1989

QUANTUM MAGIC AND QUANTUM MYSTERY

 

EXPERIMENTS WITH PHOTONS: A PROBLEM FOR RELATIVITY?

Page 369

"We must ask whether actual experiments have borne out these astonishing quantum expectations. The precise experiment just described is a hypothetical one which has not actually been performed, but similar experiments have been performed using the polarizations of pairs of photons, rather than the spin of spin-one-half massive particles. Apart from this distinction, these experiments are, in their essentials, the same as the one described above - except that the angles concerned (since photons have spin one rather than one-half) would be just one-half of those for spin-one-half particles. The polarizations of the pairs of photons have been measured in various different combinations of directions, and the results are fully in agreement with the predictions of quantum theory, and inconsistent with any local realistic model!
The most accurate and convincing of the experimental results which have been obtained to date are those of Alain Aspect (1986) and his colleagues in Paris.15 Aspect's experiments had another interesting feature. The 'decisions' as to which way to measure the polarizations of the photons, were made only after the photons were in full flight. Thus, if we think of some non-local 'influence' travelling from one photon detector to the photon at the opposite side, signalling the direction in which it intends to measure the polarization direction of the approaching photon, then we see that this 'influence' must travel faster than light! Any kind of realistic description of the quantum world which is consistent with the facts must apparently be non-causal, in the sense that effects must be able to travel faster than light!

Page 370

" But we saw in the last chapter that, so long as relativity holds true, the sending of signals faster than light leads to absurdities (and conflict with our feelings of 'free will', etc., cf. p. 273). This is certainly true, but the non-local 'influences' that arise in EPR-type experiments are not such that they can be used to send messages ­as one can see, for the very reason that they would lead to such absurdities, if so. (A detailed demonstration that such 'influences' cannot be used to signal messages has been carried out by Ghirardi, Rimini, and Weber 1980.) It is of no use to be told that a photon is polarized 'either vertically or horizontally', (as opposed, say, to 'either at 60° or 150°') until one is informed which of the two alternatives it actually is. It is the first piece of 'information' (i.e. the directions of alternative polarization) which arrives faster than light ('instantaneously'), while the knowledge as to which of these two directions it must actually be polarized in arrives more slowly, via an ordinary signal communicating the result of the first polarization measurement.
Although EPR-type experiments do not, in the ordinary sense of sending messages, conflict with the causality of relativity, there is a definite conflict with the spirit of relativity in our picture of 'physical reality'. Let us see how the realistic view of the state­vector applies to the above EPR-type experiment (involving photons). As the two photons move outwards, the state-vector describes the situation as a photon pair, acting as a single unit. Neither photon individually has an objective state: the quantum state applies only to the two together. Neither photon individually has a direction of polarization: the polarization is a combined quality of the two photons together. When the polarization of one of these photons is measured, the state-vector jumps so that the unmeasured photon now does have a definite polarization. When that photon's polarization is subsequently measured, the probability values are correctly obtained by applying the usual quantum rules to its polarization state. This way of looking at the situation provides the correct answers; it is, indeed, the way that we ordinarily apply quantum mechanics. But it is an essentially non-relativistic view. For the two measurements of polarization are what are called space like-separated, which means that each lies outside the other's light cone, like the points R and Q in Fig. / Page 371 / 5.21. The question of which of these measurements actually occurred first is not really physically meaningful, but depends on the 'observer's' state of motion (see Fig. 6.32). If the 'observer' moves rapidly enough to the right, then he considers the right­hand measurement to have occurred first; and if to the left, then it is the left-hand measurement! But if we regard the right-hand photon as having been measured first we get a completely different picture of physical reality from that obtained if we regard the left-hand photon as having been measured first! (It is a different measurement that causes the non-local 'jump'.) There is an essential conflict between our space-time picture of physical reality - even the correctly non-local quantum-mechanical one ­and special relativity! This is a severe puzzle, which 'quantum realists' have not been able adequately to resolve (cf. Aharonov and Albert 1981). I shall need to return to the matter later.

Fig. 6.32. (omitted) Two different observers form mutually inconsistent pictures of 'reality' in an EPR experiment in which two photons are emitted in opposite directions from a spin-O state. The observer moving to the right judges that the left-hand part of the state jumps before it is measured, the jump being caused by the measurement on the right. The observer moving to the left has the opposite opinion!"

 

 

http://home.btconnect.com/scimah/Quantumphenomena.htm

Spooky action at a distance - EPR

"One of the most vivid illustrations of the interactions of the mind of the observer with a quantum system is given by EPR - the 'Einstein Podolsky Rosen Paradox', or 'Spooky action at a distance' as it is sometimes known. The experimental evidence seems to show that the observer's mind goes to its object unobstructedly and instantaneously, for example through ten kilometres of intervening Geneva city-scape (walls, buildings, railway stations, the lot!) at speeds exceeding that of light.

Nor does the effect diminish with distance. According to the Copenhagen interpretation of quantum theory, the 'spooky action' can affect a particle instantaneously whether it is a metre away from the observer or halfway across the universe.

The observation of 'spooky action' relies on the concept of entanglement. It is possible to obtain pairs of fundamental particles where it is known that their properties will always cancel one another out, even when those properties have not been defined. These pairs are said to be 'entangled' . However the entanglement is conceptual rather than physical and the particles are free to move far apart.

Consider an experiment where we create an entangled pair of magnetic particles. Their polar alignments will always be opposite. We allow them to move far apart. We then place a Stern-Gerlach magnet in the path of one of the particles and observe what happens when it passes through. If it is defected upwards then, according to the 'spooky action' hypothesis, its distant partner would be deflected downwards by a similar magnet. By making the nearby observation we have instantaneously defined the properties of the distant particle.

Note that this is not the same thing as saying 'The near particle was always up but we didn't know until we decided to observe it. So the distant particle must always have been down even though we didn't know at the the time.'

The reason the statement above is incompatible with quantum theory is that we could have equally well decided to align the Stern-Gerlach magnet on a left/right axis instead of up/down. In which case we would have fixed the near particle as, say, left-deflected and the distant particle would instantaneously be known to be right-deflected.

For many years both theoretical and technical difficulties stood in the way of determining whether 'spooky action' does indeed take place. However as a result of the theoretical work of John Bell and the ingenious experimental designs of Alain Aspect strong evidence was obtained that the effect occurred over distances of a few metres. The act of making a decision of what attributes of one member of an entangled pair were to be observed immediately determined what could be observed of the other member.

Since then 'spooky action' has been demonstrated over increasing distances. The current record is 10 km obtained by Nicolas Gisin and his team at the University of Geneva [BUCHANAN 1997]. Starting from near Geneva railway station they sent entangled photons along optical fibres through the city to destinations separated by 10km. They showed that observing the state of one member of the pair instantaneously determined the state of the other."

 

 

MUSIC OF THE MIND

Darryl Reanney 1994

An Adventure Into Consciousness

Page 77

REALITIES WE DO NOT SEE

"This is the strange reality-all thoughts, whatever 'waveband' they occupy, exist in the disembodied reality of the quantum dream. The essence of a quantum ripple is that it exists in an indeterminate, non-localised state, in that uninhabited quantum 'space' where there is neither yesterday nor tomorrow, neither here nor there. Moreover, a thought, once created, no longer needs the physical structure of the brain that made it to sustain it further-it is thereafter just as 'real' as an electron or a stone.
Common sense tends to rebel at this. We have such an ingrained tendency to explain aspects of experience in terms of everyday models that our minds stumble when confronted with a phenomenon for which there is no model. But the quantum state poses the ultimate paradox: it seems so alien because it is so familiar. Our thoughts are, in essence, quantum waves so when we see this inner aspect of ourselves in outward projection we think we are seeing the face of a stranger. The nub of the difficulty is the disembodied aspect of the quantum ripple. Because the world we inhabit is limited by the dimensional context of space and time, it is difficult for our minds to grasp the dimensionless reality of the thoughts that enter them.
The best way to resolve one paradox is to invoke another so let me sketch out a further anti-intuitive aspect of quantum mechanics. Consider two subatomic particles, A and B, that are created from the same micro-event in such a way that A has a different 'spin' to B. Suppose further that these two particles now proceed to opposite 'ends' of the universe. They are so far apart that a signal travelling at the speed of light could not bridge the gap between them. The mind-bending scientific truth is that if one changes the spin on A, the spin / Page 78 / on B changes simultaneously. Since no signal can travel faster than light we are obviously dealing with a coupled system that cannot obey the spacetime logic we recognise. This paradoxical non-locality of quantum interactions means that even though human observers measure the space (or time) between the particles in terms of the millions of kilometres (or years) that (seem to) separate them, at a deeper level of reality they are still-somehow-together. So any point we call 'here' can simultaneously access all of space in exactly the same sense that, as the mystics assure us, any moment we call 'now' can simultaneously access all of time.
Mathematical theorist Rudy Rucker believes that mathematical truths occupy a 'mental space' which he dubs the Mindscape. 'A person who does mathematical research,' he opines, 'is an explorer of the Mindscape in much the same way that Armstrong, Livingstone or Cousteau are explorers of the physical features of our universe. '70 Other scientists seem to agree. Roger Penrose states:
When mathematicians communicate, this is made possible by each one having a direct route to truth [author emphasis], the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. 71"

 

I

ME

ENTANGLEMENTS

 

Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other ...en.wikipedia.org/wiki/Quantum_entanglement

Quantum entanglement

Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. But quantum entanglement does not enable the transmission of classical information faster than the speed of light (see discussion in next section below).

Quantum entanglement applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. At the same time, it prompts some of the more philosophically oriented discussions concerning quantum theory. The correlations predicted by quantum mechanics, and observed in experiment, reject the principle of local realism , which is that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement can be related to different interpretations of quantum mechanics.

 

 

Quantum entanglement just got a whole lot weirder, says Michael Brooks ... The team believe that quantum entanglement between the atoms is the only ..www.biophysica.com/quantum

Entanglement:  The weirdest link

New Scientist vol 181 issue 2440 - 27 March 2004, page 32

That spooky connection between tiny particles is appearing everywhere, and its consequences are even affecting the world that we experience. It seems to unravel the past, and may be what keeps us alive. Quantum entanglement just got a whole lot weirder, says Michael Brooks

ENTANGLEMENT. Erwin Schrödinger called this phenomenon the defining trait of quantum theory. Einstein famously dubbed it spukhafte Fernwirkungen: "spooky action at a distance". It is not hard to understand why. Set things up correctly, and you can instantaneously affect the physical properties of a particle on the other side of the universe simply by prodding its entangled twin.

This is no longer just a curiosity of the quantum world, visible only in excruciatingly delicate experiments. Physicists now believe that entanglement between particles exists everywhere, all the time, and have recently found shocking evidence that it affects the wider, "macroscopic" world that we inhabit.

It is a discovery that might have far-reaching consequences. Not only will it give us a better grip on technological applications, such as quantum computing and cryptography, and the teleportation of quantum states, it could also open up a whole new realm of reality, enabling us to retain and control quantum weirdness in our everyday world. And it's not just a strange kind of "remote control" over matter that is at stake. Entanglement could even be the key to understanding what gives rise to the phenomenon of life. It's enough to set Einstein spinning in his grave.

Entanglement has been an affront to our sensibilities for several decades now. Schrödinger discovered it through his newly formed quantum theory, when he examined the mathematical descriptions of two quantum particles that bump into one other. After the interaction, it is impossible to tease apart the two particles' characteristics. Once they are entangled, it makes no sense to talk about the properties of just one of them. All the information about the particles, such as their momentum and spin, lies only in their joint properties. So if something affects the quantum state of one particle, it will inevitably affect the quantum state of the other, no matter how far apart they are. It is this that gives entanglement the "spooky" character that Einstein found so distasteful.

Although it seems like something from the realm of fantasy, many physicists now use entanglement as a kind of resource for experiments and applications. Entangled pairs of quantum particles such as photons are routinely created and sent down microscopes or fired across vast distances. Their spooky properties are used to perform such feats as high-resolution imaging, quantum teleportation or quantum cryptography.

But, despite the growing use of entanglement as a technological tool, physicists are beginning to realise we have only just scratched the surface of its potential. "Are there some other forms of entanglement that we haven't yet discovered?" asks Benni Reznik, a theoretical physicist at Tel Aviv University in Israel. "I think there are."

Just how little we know about entanglement was made crystal clear last year by a collaboration led by Sayantani Ghosh at the University of Chicago (Nature, vol 425, p 48). The team analysed experiments done more than a decade ago with a sample of a magnetic salt containing holmium atoms, and compared them with theoretical predictions. What they found is extraordinary.

The holmium atoms within the salt behave like tiny magnets and respond to each others' magnetic fields by adjusting their relative orientation, just as a compass needle orients itself to align with the Earth's magnetic field. But the atoms change this settled orientation if they are placed in an external magnetic field. The degree to which they align with the field is known as the salt's "magnetic susceptibility".

Ghosh and his colleagues examined how the susceptibility of the salt varied with temperature. They expected it would decrease as the temperature rose, because the extra energy at higher temperatures disrupts the atoms' ability to maintain the optimum alignment. And it did. But at very low temperatures, the atoms were aligned to a greater degree than would be expected if they had normal quantum energy levels (Graphic omitted). The team believe that quantum entanglement between the atoms is the only explanation for this phenomenon.

It's a big shock: it shows that the quantum phenomenon of entanglement, whose power was thought to be confined to the infinitesimal world of subatomic particles, can produce effects that remain measurable on macroscopic scales.

Ghosh and his colleagues also showed that entanglement affects the salt's heat capacity, defined as the amount of heat needed to change the temperature of a kilogram of substance by 1 kelvin. Throw in some heat, and you can only determine exactly how far the salt's temperature will rise if you take entanglement between atoms into account.

According to Vlatko Vedral, a theoretical physicist at Imperial College in London, these discoveries are highly important. Vedral was one of the team that first predicted the effect, three years ago (Physical Review Letters, vol 87, p 017901). The fact that the prediction has been borne out by experiment catapults the mystery of entanglement into the list of big unanswered questions that scientists need to address, he says.

That's partly because physicists can no longer content themselves with using the quantum and classical energy level descriptions of a material if they want to determine and understand its properties. The effects of entanglement now have to be included as an integral part of any accurate calculation.

But the results also suggest that, if we knew where to look, we might find entanglement causing significant effects in other materials. "It's not just magnetic salts - this should be a more universal thing," Vedral says.

The best place to look first, he believes, might be the enigmatic phenomenon of high-temperature superconductivity. Vedral points out that superconductors contain pairs of electrons whose quantum descriptions, or wave functions, appear to be entangled. "The wave function describing the pair is not equal to the product of two wave functions," he says. "Mathematically, I can see there is entanglement."

So should entanglement be considered as a possible cause of high-temperature superconductivity? Might it show us how to make materials that are superconducting at room temperature? At this stage, it is too early to say: the effects of entanglement on Ghosh's magnetic salt only become noticeable below 1 kelvin. "That is almost absolute zero," Vedral admits. "What would be really interesting would be to find a material that exhibits the effects of entanglement at higher temperatures." Eventually, he thinks, we might well find such a material at room temperature. "I don't think it's going to be a very easy search, but I can't think of anything that would rule this out on the basis of fundamental theory. It doesn't look impossible to me."

While this might seem hopelessly optimistic at first glance, other recent discoveries about entanglement are suggesting otherwise. Entanglements at room temperature appear to be an everyday part of the universe. Reznik, for instance, has shown that all of empty space - what physicists refer to as the vacuum - is filled with pairs of particles that are entangled. "It's an unusual idea," says Reznik. "It was quite hard to get our first paper on this accepted." His paper was finally published last year in Foundations of Physics (vol 33, p 167).

Thomas Durt of Vrije University in Brussels also believes entanglement is everywhere. He has recently shown, from the basic equations that Schrödinger considered, that almost all quantum interactions produce entanglement, whatever the conditions. "When you see light coming from a faraway star, the photon is almost certainly entangled with the atoms of the star and the atoms encountered along the way," he says. And the constant interactions between electrons in the atoms that make up your body are no exception. According to Durt, we are a mass of entanglements.

Curiouser and curiouser

Of course, that is no guarantee we can use them. Reznik says he doesn't think you can take his vacuum entanglement and use it to perform feats such as teleportation. Indeed, he is not even sure how to demonstrate that this entanglement exists. Though the equations of quantum field theory show that it is present, he is still working out how to perform an experiment that makes vacuum entanglement more than a theoretical result.

These are all tantalising revelations, because they suggest that something priceless is within our grasp. But how do we reach it?

We certainly need to find a better handle on practical entanglement: at the moment, the only forms of it we have learned to use are somewhat constraining. The entangled photons used for cryptography and teleportation are produced by firing a photon into a "non-linear" crystal, such as beta barium borate. The optical properties of a non-linear crystal depend on its orientation, and a photon fired in at the correct angle will split into two entangled photons. But the entanglement between the photon pair is an artefact of the internal properties of the original photon - its path and polarisation (Link New Scientist, 30 October 1999, p 32. omitted). So entangled photons from a non-linear crystal effectively remain just one quantum system, rather than being the result of two distinct particles meeting and interacting. "It's a kind of entanglement, but not quite the same as between different quantum systems," Vedral says.

What physicists would dearly like, the resource that would open the way for the best experiments, is an unlimited source of pure two-particle entanglements. Despite the recent progress, this rich source of quantum magic has eluded them so far. So how do we take things forward? Schrödinger first discovered entanglement through analysing the mathematical descriptions of quantum theory, so perhaps mathematicians should be the pioneers. The trouble with this is that entanglement gives mathematicians a severe headache - especially when the entanglement is between anything more than two particles.

In theory, just bouncing particles off an entangled pair will establish another entanglement link that can then be put to work, but it's much easier said than done. Experimental physicists John Rarity and Paul Tapster were the first to entangle three photons, in their laboratory at the UK Defence Evaluation and Research Agency in Malvern, Worcestershire, five years ago. But no one has ever managed to work out how to describe the properties of such a system. For the most part, theorists can't even look at a given quantum state and tell if it is entangled - it is only possible in a few special cases. "Although I can define what it means to be entangled, that is, I can write down a state that's entangled and a state that's not, if you give me a state and ask whether it's entangled, then I have no efficient way of telling you that," says Vedral. In other words, he knows how to formulate the calculation, but it is so difficult that no computer can actually perform it.

But these problems may be nothing compared to the bombshell that Caslav Brukner of the University of Vienna has just dropped. As if our current understanding of entanglement between widely separated particles were not sketchy enough, Brukner, working with Vedral and two other Imperial College researchers, has uncovered a radical twist. They have shown that moments of time can become entangled too (link omitted).

They achieved this through a thought experiment that examines how quantum theory links successive measurements of a single quantum system. Measure a photon's polarisation, for example, and you will get a particular result. Do it again some time later, and you will get a second result. What Brukner and Vedral have found is a strange connection between the past and the future: the very act of measuring the photon polarisation a second time can affect how it was polarised earlier on. "It's really surprising," says Vedral.

This entanglement between moments in time is so bizarre that it could expose a hole in the very fabric of quantum theory, the researchers believe. The formulation does not allow messages to be sent back in time, but it still means that quantum mechanics seems to be bending the laws of cause and effect. On top of that, entanglement in time puts space and time on an equal footing in quantum theory, and that goes sharply against the grain.

Space and time have always been very different in quantum theory. A location in space is an "observable" - like momentum or spin, spatial coordinates are just another property any quantum particle can have. The passing of time, on the other hand, has always been part of the backdrop. An electron can have a particular value of spin, or momentum or location, but it cannot have a particular time.

But if time can become entangled, it should be considered as an observable, and there is no way to write that into quantum theory. "People have tried, but something in quantum mechanics always has to be violated if you want a proper time-observable," Vedral says. "So it could be that something in quantum mechanics has to be reformulated."

In other words, Brukner's result suggests that we might be missing something important in our understanding of how the world works. Maybe that shouldn't surprise us. After all, entanglement between two spatially separated objects already tells us that space doesn't really have the form that classical physics says it does: instantaneous cause and effect across cosmological distances is not something that any theory of the universe can cope with. And now Brukner's result seems to extend this "impossibility" to events separated in time as well.

It's not cause for despair, though. We know that relativity and quantum theory have to be meshed together if we are to create a "final" theory of how the universe works. It is too early to read much into Brukner's result, but maybe it is a clue about how to produce such a theory.

In the meantime, Vedral thinks he's identified an equally significant project to pursue. If, as Ghosh's result suggests, entanglement can produce macroscopic effects, is it such a stretch to reason that quantum entanglement might be the key to understanding life?

We know that quantum mechanics describes how atoms combine into molecules, and so underpins chemistry. And chemical processes underpin all biological processes, including the metabolic cycle and replication. So could entanglement support the emergent, macroscopic characteristic of chemistry that we call life? Reznik and Durt's revelations - that entanglements exist around us and inside us all the time - can only add to the intrigue. "I think it's a speculation worth making," Vedral says. "There may be some experiments in biology or biochemistry where we can see more of these effects, interpret some of the results in a different light. It would be a very exciting find."

Couple that with the ability to create materials that exploit our unfolding understanding of entanglement, and we might one day even gain the ability to use entanglement to create new forms of life. Now that is a spooky thought.

Michael Brooks

 

7
QUANTUM
107
26
8
12
ENTANGLEMENT
130
49
4

 

 

3
THE
33
15
6
6
SPOOKY
101
29
2
6
EFFECT
45
27
9

 

 

6
ELECTRO
78
33
6
8
MAGNETIC
72
36
9
12
ENTANGLEMENT
130
49
4
26
First Total
280
118
19
2+6
Add to Reduce
2+8+0
1+1+8
1+9
8
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
8
Essence of Number
1
1
1

 

 

-
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
5
-
+
=
15
1+5
=
6
-
6
-
6
-
-
-
14
-
-
14
-
-
-
-
-
14
-
+
=
42
4+2
=
6
-
6
-
6
-
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
-
5
-
2
1
-
7
3
5
4
5
-
2
+
=
34
3+4
=
7
-
7
-
7
-
-
5
-
20
1
-
7
12
5
13
5
-
20
+
=
88
8+8
=
16
1+6
7
-
7
-
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
-
5
14
20
1
14
7
12
5
13
5
14
20
+
=
130
1+3+0
=
4
-
4
-
4
-
-
5
5
2
1
5
7
3
5
4
5
5
2
+
=
49
4+9
=
13
1+3
4
-
4
-
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
5
-
-
5
-
-
5
-
5
5
-
=
-
5
occurs
x
6
=
30
3+0
3
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
23
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
22
-
-
12
-
49
-
22
2+3
1+2
5
5
-
-
5
-
-
5
-
5
5
-
-
-
2+2
-
-
1+2
-
4+9
-
2+2
5
3
E
N
T
A
N
G
L
E
M
E
N
T
-
-
4
-
-
3
-
13
-
4
-
-
5
5
2
1
5
7
3
5
4
5
5
2
-
-
-
-
-
-
-
1+3
-
-
5
3
E
N
T
A
N
G
L
E
M
E
N
T
-
-
4
-
-
3
-
4
-
4

 

 

12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
5
-
+
=
15
1+5
=
6
-
6
-
6
-
-
14
-
-
14
-
-
-
-
-
14
-
+
=
42
4+2
=
6
-
6
-
6
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
5
-
2
1
-
7
3
5
4
5
-
2
+
=
34
3+4
=
7
-
7
-
7
-
5
-
20
1
-
7
12
5
13
5
-
20
+
=
88
8+8
=
16
1+6
7
-
7
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
--
-
-
-
-
-
5
14
20
1
14
7
12
5
13
5
14
20
+
=
130
1+3+0
=
4
-
4
-
4
-
5
5
2
1
5
7
3
5
4
5
5
2
+
=
49
4+9
=
13
1+3
4
-
4
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
4
=
5
5
-
-
5
-
-
5
-
5
5
-
=
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
12
E
N
T
A
N
G
L
E
M
E
N
T
-
-
22
-
-
12
-
49
-
22
1+2
5
5
2
1
5
7
3
5
4
5
5
2
-
-
2+2
-
-
1+2
-
4+9
-
2+2
3
E
N
T
A
N
G
L
E
M
E
N
T
-
-
4
-
-
3
-
13
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+3
-
-
3
E
N
T
A
N
G
L
E
M
E
N
T
-
-
4
-
-
3
-
4
-
4

 

 

-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
5
-
1
+
=
16
1+6
=
7
-
7
-
7
-
-
-
14
-
-
14
-
-
-
-
-
14
-
19
+
=
61
6+1
=
7
-
7
-
7
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
5
-
2
1
-
7
3
5
4
5
-
2
-
+
=
34
3+4
=
7
-
7
-
7
-
-
5
-
20
1
-
7
12
5
13
5
-
20
-
+
=
88
8+8
=
16
1+6
7
-
7
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
5
14
20
1
14
7
12
5
13
5
14
20
19
+
=
149
1+4+9
=
14
1+4
5
-
5
-
-
5
5
2
1
5
7
3
5
4
5
5
2
1
+
=
50
5+0
=
5
-
5
-
5
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
5
-
-
5
-
-
5
-
5
5
-
-
=
-
5
occurs
x
6
=
30
3+0
3
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
23
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
22
-
-
13
-
50
-
23
2+3
1+3
5
5
-
-
5
-
-
5
-
5
5
-
-
-
-
2+2
-
-
1+3
-
5+0
-
2+3
5
4
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
4
-
-
4
-
5
-
5
-
-
5
5
2
1
5
7
3
5
4
5
5
2
1
-
-
-
-
-
-
-
-
-
-
5
4
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
4
-
-
4
-
5
-
5

 

 

-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
5
-
1
+
=
16
1+6
=
7
-
7
-
7
-
-
-
14
-
-
14
-
-
-
-
-
14
-
19
+
=
61
6+1
=
7
-
7
-
7
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
5
-
2
1
-
7
3
5
4
5
-
2
-
+
=
34
3+4
=
7
-
7
-
7
-
-
5
-
20
1
-
7
12
5
13
5
-
20
-
+
=
88
8+8
=
16
1+6
7
-
7
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
5
14
20
1
14
7
12
5
13
5
14
20
19
+
=
149
1+4+9
=
14
1+4
5
-
5
-
-
5
5
2
1
5
7
3
5
4
5
5
2
1
+
=
50
5+0
=
5
-
5
-
5
-
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
5
-
-
5
-
-
5
-
5
5
-
-
=
-
5
occurs
x
6
=
30
3+0
3
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
23
13
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
22
-
-
13
-
50
-
23
2+3
1+3
5
5
-
-
5
-
-
5
-
5
5
-
-
-
-
2+2
-
-
1+3
-
5+0
-
2+3
5
4
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
4
-
-
4
-
5
-
5
-
-
5
5
2
1
5
7
3
5
4
5
5
2
1
-
-
-
-
-
-
-
-
-
-
5
4
E
N
T
A
N
G
L
E
M
E
N
T
S
-
-
4
-
-
4
-
5
-
5

 

 

I

ME

YOU

ENTANGLEMENTS

MENTAL ANGLE ANGEL ANGLE MENTAL

ENTANGLE MENTALLY MENTALLY ENTANGLE

KARMAS THOUGHT ENTANGLEMENT THOUGHT KARMAS

0123456789 REAL REALITY REVEALED REALITY REAL 9876543210

0987654321 GODS REAL REALITY REAL GODS 1234567890

ONE GOD ONE I ONE ME ONE YOU ONE EVERYTHING GODS

INTO THE WITHIN OF THE WITHOUT THAT IS UNIVERSAL MIND OF DIVINE THOUGHT

ABCDEFGHIJKLMNOPQRSTUVWXYZ WELCOME HERO WELCOME ABCDEFGHIJKLMNOPQRSTUVWXYZ

 

 

THE CLOCKS ALL SAY THE HOURS OF HORUS HAVE ARRIVED UPON US

AND THE DICE ALL READ FIVE

LIVING FORM FROM OUT THE IN OF GODS LIVING FORM

 

 

-
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
9
5
-
-
-
-
-
9
5
-
9
-
-
-
+
=
42
4+2
=
6
=
6
-
6
-
-
-
14
-
-
-
-
-
9
14
-
-
-
-
-
9
14
-
9
-
-
-
+
=
69
6+9
=
15
1+5
6
-
6
-
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
5
9
2
1
-
-
2
7
-
7
9
-
-
3
-
7
3
5
+
=
66
6+6
=
12
1+2
3
-
3
-
-
21
-
3
5
18
20
1
-
-
20
25
-
16
18
-
-
3
-
16
12
5
+
=
183
1+8+3
=
12
1+2
3
-
3
-
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
21
14
3
5
18
20
1
9
14
20
25
-
16
18
9
14
3
9
16
12
5
+
=
252
2+5+2
=
9
-
9
-
9
-
-
3
5
3
5
9
2
1
9
5
2
7
-
7
9
9
5
3
9
7
3
5
+
=
108
1+0+8
=
9
-
9
-
9
-
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
occurs
x
2
=
4
-
4
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
-
-
-
-
7
-
-
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
-
9
occurs
x
5
=
45
4+5
9
18
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
27
-
-
20
-
108
-
27
1+8
2+0
-
-
-
-
9
-
-
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
-
2+7
-
-
2+0
-
1+0+8
-
2+7
9
2
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
9
-
-
2
-
9
-
9
-
-
3
5
3
5
9
2
1
9
5
2
7
-
7
9
9
5
3
9
7
3
5
-
-
-
-
-
-
-
-
-
-
9
2
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
9
-
-
2
-
9
-
9

 

 

20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
9
5
-
-
-
-
-
9
5
-
9
-
-
-
+
=
42
4+2
=
6
=
6
-
6
-
-
14
-
-
-
-
-
9
14
-
-
-
-
-
9
14
-
9
-
-
-
+
=
69
6+9
=
15
1+5
6
-
6
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
3
-
3
5
9
2
1
-
-
2
7
-
7
9
-
-
3
-
7
3
5
+
=
66
6+6
=
12
1+2
3
-
3
-
21
-
3
5
18
20
1
-
-
20
25
-
16
18
-
-
3
-
16
12
5
+
=
183
1+8+3
=
12
1+2
3
-
3
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
21
14
3
5
18
20
1
9
14
20
25
-
16
18
9
14
3
9
16
12
5
+
=
252
2+5+2
=
9
-
9
-
9
-
3
5
3
5
9
2
1
9
5
2
7
-
7
9
9
5
3
9
7
3
5
+
=
108
1+0+8
=
9
-
9
-
9
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
occurs
x
2
=
4
-
4
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
5
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
5
=
25
2+5
7
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
-
-
-
-
7
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
9
-
-
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
-
9
occurs
x
5
=
45
4+5
9
20
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
27
-
-
20
-
108
-
27
2+0
-
-
-
-
9
-
-
9
-
-
-
-
-
9
9
-
-
9
-
-
-
-
-
2+7
-
-
2+0
-
1+0+8
-
2+7
2
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
9
-
-
2
-
9
-
9
-
3
5
3
5
9
2
1
9
5
2
7
-
7
9
9
5
3
9
7
3
5
-
-
-
-
-
-
-
-
-
-
2
U
N
C
E
R
T
A
I
N
T
Y
-
P
R
I
N
C
I
P
L
E
-
-
9
-
-
2
-
9
-
9

 

 

5
SPACE
44
17
8
3
AND
19
10
1
4
TIME
47
20
2

 

 

-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
-
9
-
-
+
=
10
1+0
=
1
=
1
=
1
-
-
19
-
-
-
-
-
-
9
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
-
7
1
3
5
-
2
-
4
5
+
=
27
2+7
=
9
=
9
=
9
-
-
-
16
1
3
5
-
20
-
13
5
+
=
63
6+3
=
9
=
9
=
9
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
19
16
1
3
5
-
20
9
13
5
+
=
91
9+1
=
10
1+0
1
=
1
-
-
1
7
1
3
5
-
2
9
4
5
+
=
37
3+7
=
10
1+0
1
=
1
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
14
9
S
P
A
C
E
-
T
I
M
E
-
-
31
-
-
9
-
37
-
28
1+4
-
-
-
-
-
-
-
-
9
-
-
-
-
3+1
-
-
-
-
3+7
-
2+8
5
9
S
P
A
C
E
-
T
I
M
E
-
-
4
-
-
9
-
10
-
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+0
-
1+0
5
9
S
P
A
C
E
-
T
I
M
E
-
-
4
-
-
9
-
1
-
1

 

 

-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
-
9
-
-
+
=
10
1+0
=
1
=
1
=
1
-
-
19
-
-
-
-
-
-
9
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
-
7
1
3
5
-
2
-
4
5
+
=
27
2+7
=
9
=
9
=
9
-
-
-
16
1
3
5
-
20
-
13
5
+
=
63
6+3
=
9
=
9
=
9
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
--
-
-
-
-
-
-
19
16
1
3
5
-
20
9
13
5
+
=
91
9+1
=
10
1+0
1
=
1
-
-
1
7
1
3
5
-
2
9
4
5
+
=
37
3+7
=
10
1+0
1
=
1
-
9
S
P
A
C
E
-
T
I
M
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
14
9
S
P
A
C
E
-
T
I
M
E
-
-
31
-
-
9
-
37
-
28
1+4
-
-
-
-
-
-
-
-
9
-
-
-
-
3+1
-
-
-
-
3+7
-
2+8
5
9
S
P
A
C
E
-
T
I
M
E
-
-
4
-
-
9
-
10
-
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+0
-
1+0
5
9
S
P
A
C
E
-
T
I
M
E
-
-
4
-
-
9
-
1
-
1

 

 

6
ENERGY
74
38
2
8
ENERGIES
82
46
1

 

 

5
FORCE
47
29
2
6
FORCES
66
30
3

 

 

4
HEAT
34
16
7
5
LIGHT
56
29
2
9
Add to Reduce
90
45
9
-
Reduce to Deduce
9+0
4+5
-
9
Essence of Number
9
9
9

 

 

-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
9
-
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
15
-
9
-
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
2
-
4
-
3
-
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
-
1
20
-
13
-
3
-
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
20
15
13
9
3
-
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
-
1
2
6
4
9
3
-
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
-
--
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
-
1+2
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
8
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
-
--
1
2
6
4
9
3
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
8
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
6
-
9
-
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
15
-
9
-
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
1
2
-
4
-
3
-
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
1
20
-
13
-
3
-
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
1
20
15
13
9
3
-
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
1
2
6
4
9
3
-
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
--
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
1+2
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
--
1
2
6
4
9
3
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
9
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
15
-
9
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
2
-
4
-
3
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
-
1
20
-
13
-
3
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
20
15
13
9
3
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
-
1
2
6
4
9
3
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
-
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
-
1+2
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
8
3
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
-
--
1
2
6
4
9
3
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
8
3
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

S
=
1
-
9
SPLITTING
126
45
9
T
=
2
-
3
THE
33
15
6
A
=
1
-
4
ATOM
49
13
4
B
-
4
4
16
First Total
208
73
19
-
-
-
-
1+6
Add to Reduce
2+0+8
7+3
1+9
-
-
7
-
7
Second Total
10
10
10
8
-
-
-
-
Reduce to Deduce
1+0
1+0
1+0
-
-
7
-
7
Essence of Number
1
1
1

 

 

-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
38
-
1
-
-
9
-
-
9
5
-
-
-
8
-
-
-
-
6
-
+
=
38
3+8
=
11
1+1
2
=
2
83
-
19
-
-
9
-
-
9
14
-
-
-
8
-
-
-
-
15
-
+
=
74
8+3
=
11
1+1
2
=
2
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
25
-
-
7
3
-
2
2
-
-
7
-
2
-
5
-
1
2
-
4
+
=
35
3+5
=
8
=
8
=
8
79
-
-
16
12
-
20
20
-
-
7
-
20
-
5
-
1
20
-
13
+
=
134
1+3+4
=
8
=
8
=
8
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
208
-
19
16
12
9
20
20
9
14
7
-
20
8
5
-
1
20
15
13
+
=
208
2+0+8
=
10
1+0
1
=
1
73
-
1
7
3
9
2
2
9
5
7
-
2
8
5
-
1
2
6
4
+
=
73
7+3
=
10
1+0
1
=
1
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
2
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
9
-
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
45
-
-
16
-
73
-
46
4+5
1+6
-
-
-
-
2
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
4+5
-
-
1+6
-
7+3
-
4+6
9
7
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
9
-
-
7
-
10
-
10
-
-
1
7
3
9
2
2
9
5
7
-
2
8
5
-
1
2
6
3
-
-
-
-
-
-
-
1+0
-
1+0
6
7
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
6
-
-
7
-
1
-
1

 

 

NUCLEAR UNCLEAR

 

 

-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
38
-
1
-
-
9
-
-
9
5
-
-
8
-
-
-
6
-
+
=
38
3+8
=
11
1+1
2
=
2
83
-
19
-
-
9
-
-
9
14
-
-
8
-
-
-
15
-
+
=
74
8+3
=
11
1+1
2
=
2
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
25
-
-
7
3
-
2
2
-
-
7
2
-
5
1
2
-
4
+
=
35
3+5
=
8
=
8
=
8
79
-
-
16
12
-
20
20
-
-
7
20
-
5
1
20
-
13
+
=
134
1+3+4
=
8
=
8
=
8
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
208
-
19
16
12
9
20
20
9
14
7
20
8
5
1
20
15
13
+
=
208
2+0+8
=
10
1+0
1
=
1
73
-
1
7
3
9
2
2
9
5
7
2
8
5
1
2
6
4
+
=
73
7+3
=
10
1+0
1
=
1
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
2
2
-
-
-
2
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
9
-
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
45
-
-
16
-
73
-
46
4+5
1+6
-
-
-
-
2
2
-
-
-
2
-
-
-
2
-
-
-
-
4+5
-
-
1+6
-
7+3
-
4+6
9
7
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
9
-
-
7
-
10
-
10
-
-
1
7
3
9
2
2
9
5
7
2
8
5
1
2
6
3
-
-
-
-
-
-
-
1+0
-
1+0
6
7
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
6
-
-
7
-
1
-
1

 

 

-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
-
15
-
+
=
15
1+5
=
6
=
6
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
-
1
20
-
13
+
=
34
3+4
=
7
=
7
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
-
1
2
6
4
+
=
13
1+3
=
4
=
4
-
4
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
4
A
T
O
M
-
-
13
-
-
4
-
13
3+2
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
5
4
A
T
O
M
-
-
4
-
-
4
-
4
-
-
1
2
6
4
-
-
-
-
-
-
-
-
5
4
A
T
O
M
-
-
4
-
-
4
-
4

 

 

4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
15
-
+
=
15
1+5
=
6
=
6
4
A
T
O
M
-
-
-
-
-
--
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
1
20
-
13
+
=
34
3+4
=
7
=
7
4
A
T
O
M
-
-
-
-
-
--
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
1
2
6
4
+
=
13
1+3
=
4
=
4
4
A
T
O
M
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
6
-
-
-
6
occurs
x
1
=
6
4
A
T
O
M
-
-
13
-
-
4
-
13
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
4
A
T
O
M
-
-
4
-
-
4
-
4
-
1
2
6
4
-
-
-
-
-
-
-
-
4
A
T
O
M
-
-
4
-
-
4
-
4

 

 

-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
-
-
6
-
1
+
=
7
-
=
7
=
7
-
-
-
-
15
-
19
+
=
34
3+4
=
7
=
7
-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
1
2
-
4
-
+
=
7
-
=
7
=
7
-
-
1
20
-
13
-
+
=
34
3+4
=
7
=
7
-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
1
20
15
13
19
+
=
68
6+8
=
14
1+4
5
-
-
1
2
6
4
1
+
=
14
1+4
=
5
=
5
-
5
A
T
O
M
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
5
A
T
O
M
S
-
-
13
-
-
5
-
14
3+2
-
-
-
-
-
-
-
-
1+3
-
-
-
-
1+4
5
5
A
T
O
M
S
-
-
4
-
-
5
-
5
-
-
1
2
6
4
1
-
-
-
-
-
-
-
-
5
5
A
T
O
M
S
-
-
4
-
-
5
-
5

 

 
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
-
6
-
1
+
=
7
-
=
7
=
7
-
-
-
15
-
19
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
2
-
4
-
+
=
7
-
=
7
=
7
-
1
20
-
13
-
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
20
15
13
19
+
=
68
6+8
=
14
1+4
5
-
1
2
6
4
1
+
=
14
1+4
=
5
=
5
5
A
T
O
M
S
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
5
A
T
O
M
S
-
-
13
-
-
5
-
14
-
-
-
-
-
-
-
-
1+3
-
-
-
-
1+4
5
A
T
O
M
S
-
-
4
-
-
5
-
5
-
1
2
6
4
1
-
-
-
-
-
-
-
-
5
A
T
O
M
S
-
-
4
-
-
5
-
5

 

 

-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
-
8
-
-
-
-
15
-
+
=
23
2+3
=
5
=
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
2
-
5
-
1
2
-
4
+
=
14
1+4
=
5
=
5
-
-
20
-
5
-
1
20
-
13
+
=
59
5+9
=
14
1+4
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
20
8
5
-
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
-
2
8
5
-
1
2
6
4
+
=
28
2+8
=
10
1+0
1
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
3
-
-
-
-
-
-
--
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
--
--
--
--
--
--
--
-
-
7
SEVEN
7
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
19
7
T
H
E
-
A
T
O
M
-
-
26
-
-
7
-
28
1+9
-
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
10
7
T
H
E
-
A
T
O
M
-
-
8
-
-
7
-
10
1+0
-
2
8
5
-
1
2
6
4
-
-
-
-
-
-
-
1+0
1
7
T
H
E
-
A
T
O
M
-
-
8
-
-
4
-
1

 

 

7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
8
-
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
8
-
-
-
-
15
-
+
=
23
2+3
=
5
=
5
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
2
-
5
-
1
2
-
4
+
=
14
1+4
=
5
=
5
-
20
-
5
-
1
20
-
13
+
=
59
5+9
=
14
1+4
5
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
20
8
5
-
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
2
8
5
-
1
2
6
4
+
=
28
2+8
=
10
1+0
1
7
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
T
H
E
-
A
T
O
M
-
-
26
-
-
7
-
28
-
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
7
T
H
E
-
A
T
O
M
-
-
8
-
-
7
-
10
-
2
8
5
-
1
2
6
4
-
-
-
-
-
-
-
1+0
7
T
H
E
-
A
T
O
M
-
-
8
-
-
4
-
1

 

 

7
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
8
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
8
-
-
-
15
-
+
=
23
2+3
=
5
=
5
7
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
2
-
5
1
2
-
4
+
=
14
1+4
=
5
=
5
-
20
-
5
1
20
-
13
+
=
59
5+9
=
14
1+4
5
7
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
20
8
5
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
2
8
5
1
2
6
4
+
=
28
2+8
=
10
1+0
1
7
T
H
E
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
2
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
T
H
E
A
T
O
M
-
-
26
-
-
7
-
28
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
7
T
H
E
A
T
O
M
-
-
8
-
-
7
-
10
-
2
8
5
1
2
6
4
-
-
-
-
-
-
-
1+0
7
T
H
E
A
T
O
M
-
-
8
-
-
4
-
1

 

 

3
THE
33
15
6
6
ATOMIC
61
25
7
5
MINDS
59
23
5
1
I
9
9
9
15
Add to Reduce
162
72
27
1+5
Reduce to Deduce
1+6+2
7+2
2+7
8
Essence of Number
9
9
9

 

 

-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
6
-
9
-
-
-
9
5
-
1
-
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
-
8
-
-
-
-
15
-
9
-
-
-
9
14
-
19
-
9
+
=
83
8+3
=
11
1+1
2
=
2
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
1
2
-
4
-
3
-
4
-
-
4
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
-
20
-
5
-
1
20
-
13
-
3
-
13
-
-
4
-
-
-
+
=
79
7+9
=
16
1+6
7
=
7
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
1
20
15
13
9
3
-
13
9
14
4
19
-
9
+
=
162
1+6+2
=
9
=
9
=
9
-
-
2
8
5
-
1
2
6
4
9
3
-
4
9
5
4
1
-
9
+
=
63
6+3
=
9
=
9
=
9
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
-
``-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
4
-
-
-
4
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
7
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
38
-
-
15
-
63
-
36
-
1+5
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
7
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
11
-
-
6
-
9
-
9
-
-
2
8
5
-
1
2
6
3
9
3
-
3
9
5
4
1
-
9
-
-
1+1
-
-
-
-
-
-
-
7
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
2
-
-
6
-
9
-
9

 

 

5
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
6
-
9
-
-
-
9
5
-
1
-
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
8
-
-
-
-
15
-
9
-
-
-
9
14
-
19
-
9
+
=
83
8+3
=
11
1+1
2
=
2
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
1
2
-
4
-
3
-
4
-
-
4
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
20
-
5
-
1
20
-
13
-
3
-
13
-
-
4
-
-
-
+
=
79
7+9
=
16
1+6
7
=
7
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
1
20
15
13
9
3
-
13
9
14
4
19
-
9
+
=
162
1+6+2
=
9
=
9
=
9
-
2
8
5
-
1
2
6
4
9
3
-
4
9
5
4
1
-
9
+
=
63
6+3
=
9
=
9
=
9
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
``-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
4
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
38
-
-
15
-
63
-
36
1+5
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
11
-
-
6
-
9
-
9
-
2
8
5
-
1
2
6
3
9
3
-
3
9
5
4
1
-
9
-
-
1+1
-
-
-
-
-
-
-
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
2
-
-
6
-
9
-
9

 

 

5
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
6
-
9
-
-
9
5
-
1
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
8
-
-
-
15
-
9
-
-
9
14
-
19
9
+
=
83
8+3
=
11
1+1
2
=
2
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
2
-
5
1
2
-
4
-
3
4
-
-
4
-
-
+
=
25
2+5
=
7
=
7
=
7
-
20
-
5
1
20
-
13
-
3
13
-
-
4
-
-
+
=
79
7+9
=
16
1+6
7
=
7
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
20
8
5
1
20
15
13
9
3
13
9
14
4
19
9
+
=
162
1+6+2
=
9
=
9
=
9
-
2
8
5
1
2
6
4
9
3
4
9
5
4
1
9
+
=
63
6+3
=
9
=
9
=
9
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
2
=
2
=
2
``-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
38
-
-
15
-
63
-
36
1+5
-
-
-
-
-
-
-
9
-
-
9
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
6
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
11
-
-
6
-
9
-
9
-
2
8
5
1
2
6
3
9
3
3
9
5
4
1
9
-
-
1+1
-
-
-
-
-
-
-
6
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
2
-
-
6
-
9
-
9

 

 

A
=
1
-
4
ATOM
49
13
4
C
=
3
-
4
CELL
32
14
5
S
-
4
4
8
Add to Reduce
81
27
9
-
-
-
-
-
Reduce to Deduce
8+1
2+7
-
-
-
4
-
8
Essence of Number
9
9
9

 

 

-
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
15
-
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
-
1
2
-
4
-
3
5
3
3
+
=
21
2+1
=
3
=
3
=
3
-
-
1
20
-
13
-
3
5
12
12
+
=
66
6+6
=
12
1+2
3
=
3
-
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
-
1
20
15
13
--
3
5
12
12
+
=
81
8+1
=
9
=
9
=
9
-
-
1
2
6
4
-
3
5
3
3
+
=
27
2+7
=
9
=
9
=
9
-
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
--
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
--
--
--
--
--
--
-
-
-
--
--
7
SEVEN
7
-
-
-
-
-
8
-
--
--
--
--
--
--
-
-
-
--
--
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
--
--
9
NINE
9
-
-
-
-
-
24
8
A
T
O
M
-
C
E
L
L
-
-
21
-
-
8
-
27
-
27
2+4
-
-
-
-
-
-
3
-
3
3
-
-
2+1
-
-
-
-
2+7
-
2+7
6
8
A
T
O
M
-
C
E
L
L
-
-
3
-
-
8
-
9
-
9
--
-
1
2
6
4
-
3
5
3
3
-
-
-
-
-
--
-
-
-
-
6
8
A
T
O
M
-
C
E
L
L
-
-
3
-
-
8
-
9
-
9

 

 

8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
15
-
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
1
2
-
4
-
3
5
3
3
+
=
21
2+1
=
3
=
3
=
3
-
1
20
-
13
-
3
5
12
12
+
=
66
6+6
=
12
1+2
3
=
3
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
1
20
15
13
--
3
5
12
12
+
=
81
8+1
=
9
=
9
=
9
-
1
2
6
4
-
3
5
3
3
+
=
27
2+7
=
9
=
9
=
9
8
A
T
O
M
-
C
E
L
L
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
8
A
T
O
M
-
C
E
L
L
-
-
21
-
-
8
-
27
-
27
-
-
-
-
-
-
-
-
-
-
-
-
2+1
-
-
-
-
2+7
-
2+7
8
A
T
O
M
-
C
E
L
L
-
-
3
-
-
8
-
9
-
9
-
1
2
6
4
-
3
5
3
3
-
-
-
-
-
--
-
-
-
-
8
A
T
O
M
-
C
E
L
L
-
-
3
-
-
8
-
9
-
9

 

 

-
4
C
E
L
L
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
-
3
5
3
3
+
=
14
1+4
=
5
=
5
-
4
C
E
L
L
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
-
3
5
3
3
+
=
14
1+4
=
5
=
5
-
4
C
E
L
L
T
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-2-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
4
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
--
-
5
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
--
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
37
4
C
E
L
L
-
-
8
-
-
4
-
14
3+7
-
3
-
3
3
-
-
-
-
-
-
-
1+4
10
4
C
E
L
L
-
-
8
-
-
4
-
5
1+0
-
3
5
3
3
-
-
-
-
-
-
-
-
1
4
C
E
L
L
-
-
8
-
-
4
-
5

 

 

4
C
E
L
L
-
-
-
-
-
--
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
3
5
3
3
+
=
14
1+4
=
5
=
5
4
C
E
L
L
-
-
-
-
-
--
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
3
5
3
3
+
=
14
1+4
=
5
=
5
4
C
E
L
L
T
-
-
-
-
-
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
--
-
5
-
-
-
-
5
occurs
x
1
=
5
4
C
E
L
L
-
-
8
-
-
4
-
14
-
3
-
3
3
-
-
-
-
-
-
-
1+4
4
C
E
L
L
-
-
8
-
-
4
-
5
-
3
5
3
3
-
-
-
-
-
-
-
-
4
C
E
L
L
-
-
8
-
-
4
-
5

 

 

-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
-
`-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
3
5
3
3
-
+
=
14
1+4
=
5
=
5
-
`-
3
5
12
12
-
+
=
32
3+2
=
5
=
5
-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
19
+
=
51
5+1
=
6
=
6
-
-
3
5
3
3
1
+
=
15
1+5
=
6
=
6
-
5
C
E
L
L
S
T
-
-
-
-
-
-
-
--
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-2-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
4
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
--
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
--
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
36
5
C
E
L
L
S
-
-
9
-
-
5
-
15
3+6
-
3
-
3
3
-
-
-
-
-
-
-
-
1+5
9
5
C
E
L
L
S
-
-
9
-
-
5
-
6
-
-
3
5
3
3
1
-
-
-
-
-
-
-
-
9
5
C
E
L
L
S
-
-
9
-
-
5
-
6

 

 

5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
`-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
3
5
3
3
-
+
=
14
1+4
=
5
=
5
`-
3
5
12
12
-
+
=
32
3+2
=
5
=
5
5
C
E
L
L
S
-
-
-
-
-
--
-
-
`-
3
5
12
12
19
+
=
51
5+1
=
6
=
6
-
3
5
3
3
1
+
=
15
1+5
=
6
=
6
5
C
E
L
L
S
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
--
-
5
-
-
-
-
-
5
occurs
x
1
=
5
5
C
E
L
L
S
-
-
9
-
-
5
-
15
-
3
-
3
3
-
-
-
-
-
-
-
-
1+5
5
C
E
L
L
S
-
-
9
-
-
5
-
6
-
3
5
3
3
1
-
-
-
-
-
-
-
-
5
C
E
L
L
S
-
-
9
-
-
5
-
6

 

 

4
MIND
40
22
4
6
MATTER
77
23
5
10
Add to Reduce
117
45
9
1+0
Reduce to Deduce
1+1+7
4+5
-
1
Essence of Number
9
9
9

 

 

-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
+
=
31
3+1
=
4
=
4
=
4
-
-
13
-
-
4
-
13
1
20
20
5
18
+
=
94
9+4
=
13
1+3
4
=
4
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
+
=
117
1+2
=
9
=
9
=
9
-
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
--
--
--
--
--
--
--
--
--
--
--
-
-
3
-
-
-
-
-
-
-
-
-`
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
--
--
--
--
--
--
--
--
--
--
-
-
6
--
--
--
--
--
--
--
7
-
-
--
--
--
--
--
--
--
--
--
--
-
-
7
--
--
--
--
--
--
--
8
-
-
--
--
--
--
--
--
--
--
--
--
-
-
8
--
--
--
--
--
--
--
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
24
10
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
10
-
45
-
18
2+4
1+0
-
9
-
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
4+5
-
1+8
6
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9
-
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
6
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9

 

 

10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
9
14
-
-
-
-
-
-
-
-
+
=
23
2+3
=
5
=
5
=
5
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
+
=
31
3+1
=
4
=
4
=
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
94
9+4
=
13
1+3
4
=
4
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
+
=
117
1+2
=
9
=
9
=
9
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
10
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
10
-
45
-
18
1+0
-
9
-
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
4+5
-
1+8
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9

 

 

-
MIND MATTER
-
-
-
1
M
13
4
4
1
I
9
9
9
2
ND
18
9
9
2
MA
14
5
5
3
TTE
45
9
9
1
R
18
9
9
10
MATTER MIND
117
45
45
1+0
1+1+7
4+5
4+5
1
MIND MATTER
9
9
9

 

 

-
-
-
-
-
MATTER ANTI MATTER
-
-
-
M
=
4
-
3
MATTER
77
23
5
A
=
1
-
7
ANTI
44
17
8
M
=
4
--
5
MATTER
77
23
5
-
-
9
-
15
MATTER ANTI MATTER
198
63
18
-
-
-
-
1+5
-
1+9+8
6+3
1+8
--
--
9
--
6
MATTER ANTI MATTER
18
9
9
-
-
-
--
-
-
1+8
-
-
-
-
9
--
6
MATTER ANTI MATTER
9
9
9

 

 

-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
-
-
-
-
-
-
-
+
=
14
1+4
=
5
-
5
-
5
-
-
-
-
-
-
-
-
-
-
14
-
9
-
-
-
-
-
-
-
+
=
23
2+3
=
5
-
5
-
5
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
4
1
2
2
5
9
-
1
-
2
-
-
4
1
2
2
5
9
+
=
49
4+9
=
13
1+3
4
-
4
-
-
13
1
20
20
5
18
-
1
-
20
-
-
13
1
20
20
5
18
+
=
175
1+7+5
=
13
1+3
4
-
4
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
1
20
20
5
18
-
1
14
20
9
-
13
1
20
20
5
18
+
=
198
1+9+8
=
18
1+8
9
9
-
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
+
=
63
6+3
=
9
-
9
9
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
-
2
occurs
x
5
=
10
1+0
1
3
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
8
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
24
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
21
-
-
20
-
63
-
27
2+4
1+6
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
9
-
-
2+1
-
-
2+0
-
6+3
-
2+7
6
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
6
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
-
-
-
-
-
-
-
+
=
14
1+4
=
5
-
5
-
5
-
-
-
-
-
-
-
-
-
14
-
9
-
-
-
-
-
-
-
+
=
23
2+3
=
5
-
5
-
5
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
4
1
2
2
5
9
-
1
-
2
-
-
4
1
2
2
5
9
+
=
49
4+9
=
13
1+3
4
-
4
-
13
1
20
20
5
18
-
1
-
20
-
-
13
1
20
20
5
18
+
=
175
1+7+5
=
13
1+3
4
-
4
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
1
20
20
5
18
-
1
14
20
9
-
13
1
20
20
5
18
+
=
198
1+9+8
=
18
1+8
9
9
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
+
=
63
6+3
=
9
-
9
9
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
-
3
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
-
2
occurs
x
5
=
10
1+0
1
-
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
8
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
21
-
-
20
-
63
-
27
1+6
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
2+1
-
-
2+0
-
6+3
-
2+7
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

D
=
4
Q
4
DARK
34
16
7
M
=
4
Q
6
MATTER
77
23
5
M
=
4
Q
4
MIND
40
22
4
L
=
3
Q
5
LIGHT
56
29
2
-
-
15
-
19
First Total
207
90
18
-
-
1+5
-
1+9
Add to Reduce
1+1+1
9+0
1+8
-
-
6
Q
10
Second Total
9
9
9
-
-
-
-
1+0
Reduce to Deduce
-
-
-
-
-
6
Q
1
Essence of Number
9
9
9

 

 

-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
-
8
-
+
=
31
3+1
=
4
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
-
8
-
+
=
40
4+0
=
4
=
4
=
4
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
-
4
-
-
4
-
3
-
7
-
2
+
=
59
5+9
=
14
1+4
5
=
5
-
-
4
1
18
11
-
13
1
20
20
5
18
-
13
-
-
4
-
12
-
7
-
20
+
=
167
1+6+7
=
14
1+4
5
=
5
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
-
13
1
20
20
5
18
-
13
9
14
4
-
12
9
7
8
20
+
=
207
2+0+7
=
9
=
9
=
9
-
-
4
1
9
2
-
4
1
2
2
5
9
-
4
9
5
4
-
3
9
7
8
2
+
=
90
9+0
=
9
=
9
=
9
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
4
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
4
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
9
occurs
x
4
=
36
3+6
9
6
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
39
-
-
19
-
90
-
45
-
1+9
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
3+9
-
-
1+9
-
9+0
-
4+5
6
10
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
12
-
-
10
-
9
-
9
-
1+0
4
1
9
2
-
4
1
2
2
5
9
-
4
9
5
4
-
3
9
7
8
2
-
-
1+2
-
-
1+0
-
-
-
-
6
1
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
3
-
-
1
-
9
-
9

 

 

-
-
-
-
-
DARK MATTER
-
-
-
D
=
4
Q
4
DARK
34
16
7
M
=
4
Q
6
MATTER
77
23
5
-
-
8
-
10
DARK MATTER
111
39
12
-
-
-
-
1+0
-
1+1+1
3+9
1+2
-
-
8
Q
1
DARK MATTER
3
12
3
-
-
-
-
-
-
-
1+2
-
-
-
8
Q
1
DARK MATTER
3
3
3

 

 

-
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
-
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
-
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
-
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
-
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
-
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
-
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
2
2
-
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
1+6
8
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
1+0
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
3+6
9
24
10
D
A
R
K
-
M
A
T
T
E
R
-
-
21
-
-
10
-
39
-
30
2+4
1+0
-
-
9
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
3+9
-
3+0
6
1
D
A
R
K
-
M
A
T
T
E
R
-
-
3
-
-
1
-
12
-
3
-
-
4
1
9
2
-
4
1
2
2
5
9
-
-
-
-
-
-
-
1+2
-
-
6
1
D
A
R
K
-
M
A
T
T
E
R
-
-
3
-
-
1
-
3
-
3

 

 

10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
-
13
1
20
20
5
18
+
=
111
1+1+1
=
3
=
3
=
3
-
4
1
9
2
-
4
1
2
2
5
9
+
=
39
3+9
=
12
1+2
3
=
3
10
D
A
R
K
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
2
2
-
-
-
-
2
occurs
x
3
=
6
=
6
-
4
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
1+6
8
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
1+0
5
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
3+6
9
10
D
A
R
K
-
M
A
T
T
E
R
-
-
21
-
-
10
-
39
-
30
1+0
-
-
9
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
3+9
-
3+0
1
D
A
R
K
-
M
A
T
T
E
R
-
-
3
-
-
1
-
12
-
3
-
4
1
9
2
-
4
1
2
2
5
9
-
-
-
-
-
-
-
1+2
-
-
1
D
A
R
K
-
M
A
T
T
E
R
-
-
3
-
-
1
-
3
-
3

 

 

M
=
4
4
MIND
40
22
4
M
=
4
6
MATTER
77
23
5
M
=
4
6
MATTER
77
23
5
M
=
4
4
MIND
40
22
4
-
-
16
20
Add to Reduce
234
90
18
-
-
1+6
2+0
Reduce to Deduce
2+3+4
9+0
1+8
-
-
7
2
Essence of Number
9
9
9

 

 

-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
9
14
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
+
=
46
4+6
=
10
1+0
1
=
1
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
-
-
4
+
=
62
6+2
=
8
=
8
=
8
-
-
13
-
-
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
-
-
4
+
=
188
1+8+8
=
17
1+7
8
-
8
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
9
14
4
+
=
234
2+3+4
=
9
=
9
=
9
-
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
+
=
90
9+0
=
9
=
9
=
9
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
24
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
21
-
-
20
-
90
-
36
2+4
2+0
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
2+0
-
9+0
-
3+6
6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9
-
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
-
-
-
-
-
-
-
-
-
-
6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9

 

 

0
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
9
14
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
+
=
46
4+6
=
10
1+0
1
=
1
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
-
-
4
+
=
62
6+2
=
8
=
8
=
8
-
13
-
-
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
-
-
4
+
=
188
1+8+8
=
17
1+7
8
-
8
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
9
14
4
+
=
234
2+3+4
=
9
=
9
=
9
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
+
=
90
9+0
=
9
=
9
=
9
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
2
2
-
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
21
-
-
20
-
90
-
36
2+0
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
2+0
-
9+0
-
3+6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
-
-
-
-
-
-
-
-
-
-
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9

 

 

-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
46
4+6
=
10
1+0
1
=
1
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
4
1
2
2
5
9
-
4
-
-
4
-
4
-
-
4
-
4
1
2
2
5
9
+
=
62
6+2
=
8
=
8
=
8
-
-
13
1
20
20
5
18
-
13
-
-
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
188
1+8+8
=
17
1+7
8
=
8
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
1
20
20
5
18
-
13
9
14
4
-
13
9
14
4
-
13
1
20
20
5
18
+
=
234
2+3+4
=
9
=
9
=
9
-
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
+
=
90
9+0
=
9
=
9
=
9
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
24
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
20
-
90
-
36
2+4
2+0
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
2+1
-
-
2+0
-
9+0
-
3+6
6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
46
4+6
=
10
1+0
1
=
1
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
4
1
2
2
5
9
-
4
-
-
4
-
4
-
-
4
-
4
1
2
2
5
9
+
=
62
6+2
=
8
=
8
=
8
-
13
1
20
20
5
18
-
13
-
-
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
188
1+8+8
=
17
1+7
8
=
8
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
1
20
20
5
18
-
13
9
14
4
-
13
9
14
4
-
13
1
20
20
5
18
+
=
234
2+3+4
=
9
=
9
=
9
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
+
=
90
9+0
=
9
=
9
=
9
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
4
=
8
=
8
`
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
20
-
90
-
36
2+0
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
2+1
-
-
2+0
-
9+0
-
3+6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
--
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
9
5
3
1
2
-
4
-
2
7
+
=
33
3+3
=
6
=
6
=
6
-
-
18
5
12
1
20
-
22
-
20
25
+
=
123
1+2+3
=
6
=
6
=
6
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
18
5
12
1
20
9
22
9
20
25
+
=
141
1+4+1
=
6
=
6
=
6
-
-
9
5
3
1
2
9
4
9
2
7
+
=
51
5+1
=
6
=
6
=
6
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
--
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
3
-
-
-
--
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
7
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
14
10
R
E
L
A
T
I
V
I
T
Y
-
-
31
-``
-
10
-
51
-
33
1+4
1+0
9
-
-
-
-
9
-
9
-
-
-
-
3+1
-
-
1+0
-
5+1
-
3+3
5
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-``
-
1
-
6
-
6
-
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
5
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-
-
1
-
6
-
6

 

 

10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
9
5
3
1
2
-
4
-
2
7
+
=
33
3+3
=
6
=
6
=
6
-
18
5
12
1
20
-
22
-
20
25
+
=
123
1+2+3
=
6
=
6
=
6
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
18
5
12
1
20
9
22
9
20
25
+
=
141
1+4+1
=
6
=
6
=
6
-
9
5
3
1
2
9
4
9
2
7
+
=
51
5+1
=
6
=
6
=
6
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
3
-
-
-
--
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
--
-
-
7
-
-
7
occurs
x
1
=
7
-
7
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
10
R
E
L
A
T
I
V
I
T
Y
-
-
31
-``
-
10
-
51
-
33
1+0
9
-
-
-
-
9
-
9
-
-
-
-
3+1
-
-
1+0
-
5+1
-
3+3
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-``
-
1
-
6
-
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-
-
1
-
6
-
6

 

 

-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
8
-
6
-
-
-
6
-
-
-
-
-
-
-
9
-
9
-
-
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
14
-
-
-
-
-
-
8
-
15
-
-
-
15
-
-
-
-
-
-
-
9
-
9
-
-
+
=
70
7+0
=
7
=
7
=
7
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
5
9
1
3
-
2
-
5
-
9
7
-
-
6
-
9
5
3
1
2
-
4
-
2
7
+
=
92
9+2
=
11
1+1
2
=
2
-
-
7
5
-
5
18
1
12
-
20
-
5
-
18
25
-
-
6
-
18
5
12
1
20
-
22
-
20
25
+
=
245
2+4+5
=
11
1+1
2
=
1
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
7
5
14
5
18
1
12
-
20
8
5
15
18
25
-
15
6
-
18
5
12
1
20
9
22
9
20
25
+
+
351
1+8+2
=
9
-
9
=
9
-
-
7
5
5
5
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
+
+
135
1+3+5
=
9
-
9
=
2
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
3
=
6
=
6
3
``-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
5
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
5
=
45
=
9
45
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
45
-
-
25
-
135
-
54
4+5
2+5
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
4+5
-
-
2+5
-
1+3+5
-
5+4
9
7
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
7
-
9
-
9
-
-
7
5
5
5
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
9
7
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
7
-
9
-
9

 

 

G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
O
=
6
2
OF
21
12
3
R
=
9
10
RELATIVITY
141
51
6
-
-
24
25
Add to Reduce
315
135
18
-
-
2+4
2+5
Reduce to Deduce
3+1+5
1+3+5
1+8
-
-
6
7
Essence of Number
9
9
9

 

 

7
SPECIAL
65
29
2
6
THEORY
91
37
1
2
OF
21
12
3
10
RELATIVITY
141
51
6
25
First Total
318
129
12
2+5
Add to Reduce
3+1+8
1+2+9
1+2
7
Second Total
12
12
3
-
Reduce to Deduce
1+2
1+2
-
7
Essence of Number
3
3
3

 

 

G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
-
-
9
13
Add to Reduce
153
72
9
-
-
-
1+3
Reduce to Deduce
1+5+3
7+2
-
-
-
9
4
Essence of Number
9
9
9

 

 

T
=
2
3
THE
33
15
6
G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
O
=
6
2
OF
21
12
3
R
=
9
10
RELATIVITY
141
51
6
-
-
26
28
First Total
348
150
24
-
-
2+6
2+8
Add to Reduce
3+4+8
1+5+0
2+4
-
-
8
10
Second Total
15
6
6
-
-
-
1+0
Reduce to Deduce
1+5
-
-
-
-
8
1
Essence of Number
6
6
6

 

 

3
THE
33
15
6
7
SPECIAL
65
29
2
6
THEORY
91
37
1
2
OF
21
12
3
10
RELATIVITY
141
51
6
28
First Total
351
144
18
2+8
Add to Reduce
3+5+1
1+4+4
1+8
10
Second Total
9
9
9
1+0
Reduce to Deduce
-
-
-
1
Essence of Number
9
9
9

 

 

25
GENERAL THEORY OF RELATIVITY
315
135
9

 

 

28
THE SPECIAL THEORY OF RELATIVITY
351
144
9

 

 

-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
1
1
-
-
-
9
-
-
-
-
8
-
6
-
-
-
6
-
-
-
-
-
-
-
9
-
9
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
-
-
8
-
-
19
-
-
-
9
-
-
-
-
8
-
15
-
-
-
15
-
-
-
-
-
-
-
9
-
9
-
-
+
=
92
9+2
=
11
1+1
2
=
2
-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
7
5
3
-
1
3
-
2
-
5
-
9
7
-
-
6
-
9
5
3
1
2
-
4
-
2
7
+
=
88
8+8
=
16
1+6
7
=
7
-
-
20
-
5
-
-
16
5
3
-
1
12
-
20
-
5
-
18
25
-
-
6
-
18
5
12
1
20
-
22
-
20
25
+
=
259
2+5+9
=
16
1+6
7
=
7
-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
19
16
5
3
9
1
12
-
20
8
5
15
18
25
-
15
6
-
18
5
12
1
20
9
22
9
20
25
+
+
351
3+5+1
=
9
-
9
=
9
-
-
2
8
5
-
1
7
5
3
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
+
+
144
1+4+4
=
9
-
9
=
9
218
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
2
``-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
4
=
8
=
8
3
``-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
3
=
9
-
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
5
=
45
=
9
45
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
45
-
-
28
-
144
-
54
4+5
2+8
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
4+5
-
-
2+8
-
1+4+4
-
5+4
9
10
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
10
-
9
-
9
-
1+0
2
8
5
-
1
7
5
3
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
1+0
-
-
-
-
9
1
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
1
-
9
-
9

 

 

-
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
1
-
-
9
5
+
=
29
2+9
=
11
1+1
2
=
2
-
`-
-
-
-
-
-
-
-
-
9
14
19
-
-
9
14
+
=
65
6+5
=
11
1+1
2
=
2
-
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
-
-
1
3
2
5
9
2
-
5
-
-
-
2
5
-
-
+
=
25
2+5
=
7
=
7
=
7
-
`-
1
12
2
5
18
20
-
5
-
-
-
20
5
-
-
+
=
88
8+8
=
16
1+6
7
=
7
-
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
-
`-
1
12
2
5
18
20
-
5
9
14
19
20
5
9
14
+
=
153
1+5+3
=
9
=
9
=
9
-
-
1
3
2
5
9
2
-
5
9
5
1
2
5
9
5
+
=
63
6+3
=
9
=
9
=
9
-
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-T
-T
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
2
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-4
-`-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-`-
-
-
-
-
5
-
-
-
5
-
5
-
-
5
-
5
-
-
5
occurs
x
5
=
25
2+5
7
6
-`-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-`-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-`-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-`-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
-
9
occurs
x
3
=
27
2+7
9
25
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
20
-
1
14
-
63
-
27
2+5
1+4
-
-
-
5
-
-
-
5
-
5
-
-
5
-
5
-
-
2+0
-
-
1+4
-
6+3
-
2+7
7
5
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
2
-
1
5
-
9
-
9
-
-
1
3
2
5
9
2
-
5
9
5
1
2
5
9
5
-
-
-
-
-
--
-
-
-
-
7
5
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
2
-
1
5
-
9
-
9

 

 

14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
1
-
-
9
5
+
=
29
2+9
=
11
1+1
2
=
2
`-
-
-
-
-
-
-
-
-
9
14
19
-
-
9
14
+
=
65
6+5
=
11
1+1
2
=
2
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
-
1
3
2
5
9
2
-
5
-
-
-
2
5
-
-
+
=
25
2+5
=
7
=
7
=
7
`-
1
12
2
5
18
20
-
5
-
-
-
20
5
-
-
+
=
88
8+8
=
16
1+6
7
=
7
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
-
-
-
--
-
-
-
-
`-
1
12
2
5
18
20
-
5
9
14
19
20
5
9
14
+
=
153
1+5+3
=
9
=
9
=
9
-
1
3
2
5
9
2
-
5
9
5
1
2
5
9
5
+
=
63
6+3
=
9
=
9
=
9
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
2
-
-
2
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
3
=
6
=
6
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
5
-
-
-
5
-
5
-
-
5
-
5
-
-
5
occurs
x
5
=
25
2+5
7
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
-
9
occurs
x
3
=
27
2+7
9
14
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
20
-
1
14
-
63
-
27
1+4
-
-
-
5
-
-
-
5
-
5
-
-
5
-
5
-
-
2+0
-
-
1+4
-
6+3
-
2+7
5
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
2
-
1
5
-
9
-
9
-
1
3
2
5
9
2
-
5
9
5
1
2
5
9
5
-
-
-
-
-
--
-
-
-
-
5
A
L
B
E
R
T
-
E
I
N
S
T
E
I
N
-
-
2
-
1
5
-
9
-
9

 

 

DOES GOD PLAY DICE

THE NEW MATHEMATICS OF CHAOS

Ian Stewart 1989

Page 1

PROLOGUE

CLOCKWORK OR CHAOS?

"YOU BELIEVE IN A GOD WHO PLAYS DICE, AND I IN COMPLETE LAW AND ORDER."

Albert Einstein, Letter to Max Born

 

 

 

 

-
3
L
A
W
-
-
-
-
-
--
-
-
-
-
12
1
23
+
=
36
3+6
=
9
=
9
-
-
3
1
5
+
=
9
-
=
9
-
9
-
3
L
A
W
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
2
``-
-
-
-
-
-
2
-
-
-
-
-
``-
-
3
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
4
-
-
-
-
-
``-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
36
3
L
A
W
-
-
9
-
-
3
-
9
3+6
``-
-
-
-
-
-
-
-
-
-
-
-
9
3
L
A
W
-
-
9
-
-
3
-
9
-
-
3
1
5
-
-
-
-
-
-
-
-
9
3
L
A
W
-
-
9
-
-
3
-
9

 

 

3
L
A
W
-
-
-
-
-
--
-
-
-
12
1
23
+
=
36
3+6
=
9
=
9
-
3
1
5
+
=
9
-
=
9
-
9
3
L
A
W
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
-
3
-
-
-
-
3
occurs
x
1
=
3
-
-
-
5
-
-
5
occurs
x
1
=
5
3
L
A
W
-
-
9
-
-
3
-
9
``-
-
-
-
-
-
-
-
-
-
-
-
3
L
A
W
-
-
9
-
-
3
-
9
-
3
1
5
-
-
-
-
-
-
-
-
3
L
A
W
-
-
9
-
-
3
-
9

 

 

3
THE
33
15
6
3
LAW
36
9
9
2
OF
21
12
3
5
CAUSE
49
13
4
3
AND
19
10
1
6
EFFECT
45
27
9
22
First Total
203
86
32
2+2
Add to Reduce
2+0+3
8+6
3+2
4
Second Total
5
14
5
-
Reduce to Deduce
-
1+4
-
4
Essence of Number
5
5
5

 

 

21-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
-
6
-
-
-
-
-
1
-
-
-
5
-
-
-
-
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
8
-
-
-
-
-
-
15
-
-
-
-
-
19
-
-
-
14
-
-
-
-
-
-
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
1
5
-
-
6
-
3
1
3
-
5
-
1
-
4
-
5
6
6
5
3
2
+
=
66
6+6
=
12
1+2
3
=
3
-
-
20
-
5
-
12
1
23
-
-
6
-
3
1
21
-
5
-
1
-
4
-
5
6
6
5
3
20
+
=
147
1+4+7
=
12
1+2
3
=
3
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
12
1
23
-
15
6
-
3
1
21
19
5
-
1
14
4
-
5
6
6
5
3
20
+
=
203
2+0+3
=
5
=
5
=
5
-
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
+
=
86
8+6
=
14
1+4
5
=
5
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
4
=
4
=
4
-
``-
2
-
-
-`
-
-
-
-`
-
-
-`
-
-
-
-
-
-`
-
-
-
-`
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
-
-
-
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
16
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
29
-
-
22
-
86
-
32
1+6
2+2
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
2+9
-
-
2+2
-
8+6
-
3+2
7
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
11
-
-
4
-
14
-
5
-
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
-
-
1+1
-
-
-
-
1+4
-
-
7
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
2
-
-
4
-
5
-
5

 

 

22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
-
-
6
-
-
-
-
-
1
-
-
-
5
-
-
-
-
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
8
-
-
-
-
-
-
15
-
-
-
-
-
19
-
-
-
14
-
-
-
-
-
-
-
-
+
=
56
5+6
=
11
1+1
2
=
2
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
1
5
-
-
6
-
3
1
3
-
5
-
1
-
4
-
5
6
6
5
3
2
+
=
66
6+6
=
12
1+2
3
=
3
-
20
-
5
-
12
1
23
-
-
6
-
3
1
21
-
5
-
1
-
4
-
5
6
6
5
3
20
+
=
147
1+4+7
=
12
1+2
3
=
3
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
12
1
23
-
15
6
-
3
1
21
19
5
-
1
14
4
-
5
6
6
5
3
20
+
=
203
2+0+3
=
5
=
5
=
5
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
+
=
86
8+6
=
14
1+4
5
=
5
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
4
=
4
=
4
``-
2
-
-
-`
-
-
-
-`
-
-
-`
-
-
-
-
-
-`
-
-
-
-`
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
3
-
-
-
-
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
29
-
-
22
-
86
-
32
2+2
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
2+9
-
-
2+2
-
8+6
-
3+2
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
11
-
-
4
-
14
-
5
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
-
-
1+1
-
-
-
-
1+4
-
-
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
2
-
-
4
-
5
-
5

 

 

5
CAUSE
49
13
4
3
AND
19
10
1
6
EFFECT
45
27
9
14
Add to Reduce
113
50
14
1+4
Reduce to Deduce
1+1+3
5+0
1+4
5
Essence of Number
5
5
5

 

 

-
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
1
-
-
-
5
-
-
-
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
-
19
-
-
-
14
-
-
-
-
-
-
-
-
+
=
33
3+3
=
6
=
6
=
6
-
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
3
1
3
-
5
-
1
-
4
-
5
6
6
5
3
2
+
=
44
4+4
=
8
=
8
=
8
-
-
3
1
21
-
5
-
1
-
4
-
5
6
6
5
3
20
+
=
80
8+0
=
8
=
8
=
8
-
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
3
1
21
19
5
-
1
14
4
-
5
6
6
5
3
20
+
=
113
1+1+3
=
5
=
5
=
5
-
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
+
=
50
5+0
=
5
=
5
=
5
-
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
-
-
-`
-
-
-
-`
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
24
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
21
-
-
14
-
50
-
23
2+4
1+4
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
2+1
-
-
1+4
-
5+0
-
2+3
6
5
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
3
-
-
5
-
5
-
5
-
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
-
-
-
-
-
-
-
-
-
-
6
5
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
3
-
-
5
-
5
-
5

 

 
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
1
-
-
-
5
-
-
-
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
19
-
-
-
14
-
-
-
-
-
-
-
-
+
=
33
3+3
=
6
=
6
=
6
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
3
1
3
-
5
-
1
-
4
-
5
6
6
5
3
2
+
=
44
4+4
=
8
=
8
=
8
-
3
1
21
-
5
-
1
-
4
-
5
6
6
5
3
20
+
=
80
8+0
=
8
=
8
=
8
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
3
1
21
19
5
-
1
14
4
-
5
6
6
5
3
20
+
=
113
1+1+3
=
5
=
5
=
5
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
+
=
50
5+0
=
5
=
5
=
5
14
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
-
-`
-
-
-
-`
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
4
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
21
-
-
14
-
50
-
23
1+4
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
2+1
-
-
1+4
-
5+0
-
2+3
5
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
3
-
-
5
-
5
-
5
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
-
-
-
-
-
-
-
-
-
-
5
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
3
-
-
5
-
5
-
5

 

 

S
=
1
-
12
SCHRODINGERS
139
67
4
C
=
3
-
3
CAT
24
6
6
D
=
4
-
3
DID
17
17
8
C
=
3
-
10
CURIOUSITY
160
52
7
K
=
2
-
4
KILL
44
17
8
T
=
2
-
3
THE
33
15
6
C
=
3
-
3
CAT
24
6
6
-
-
18
4
38
First Total
441
180
45
-
-
1+8
-
3+8
Add to Reduce
4+4+1
1+8+0
4+5
-
-
9
-
11
Second Total
9
9
9
4
4
-
4
1+1
Reduce to Deduce
-
-
4
Q
-
9
5
2
Essence of Number
9
9
9

 

 

Daily Mail

Tuesday, June 1, 2010

Answers to Correspondents Compiled by Charles Legge

Page 59

QUESTION

I know the phrase 'curiosity killed the cat', but I have recently been told it continues 'satisfaction brought it back'. What is the origin of this?

ORIGINALLY 'care' killed the cat, not curiosity. That form of the expression is first found in Ben Jonson's play Every Man In His Humour, in 1598: 'Helterskelter, hang sorrow, care'll kill a cat, uptails all and a louse for the hangman.' In this sense, care meant 'worry or sorrow'. rather than the modern 'look after or provide for'. The play was first performed by the Lord Chamberlain's men, a troupe of actors of which Shakespeare was a member.

He obviously liked the line because he used it the following year in Much Ado About Nothing: 'What, courage man! What though care killed a cat, thou hast mettle in thee to kill care.'

The notion of curiosity has been frowned on, particularly by early theologians. St Augustine wrote in confessions (AD397) 'God fashioned hell for the inquisitive'', so the idea of curiosity killing the cat would seem logical.

Yet as late as 1898 the original form was still in use. Brewster's Dictionary Of Phrase And Fable had: 'Care killed the Cat. It is said that a cat has nine lives, but care would wear them all out.'

The. earliest known printed reference is in the O. Henry short story Schools And Schools from 1909: 'Curiosity can do more things than Kill a cat; and if emotions. well recognised as feminine, are inimical to feline life, then jealously would soon leave the world catless.'

The rejoinder 'satisfaction brought it back' is a curiosity in itself; the idea is obvious: if you are satisfied with your lot, you are unlikely to be curious about an alternative life.

When.the rejoinder was coined is uncertain, but it dates from the mid to late 20th century and seems to have been part of a ditty popularised
by a murderous Jack Nicholson in the movie The Shining (1980)

Curiosity killed the cat
Satisfaction brought it back
Safe and sound.
From hcad, to ground
From head.to ground
It was safe and sound.'

Jon Welham, Halifax.

 

 

THE GALACTIC CLUB

Intelligent life in outer space?

Ronald N. Bracewell 1974

Page 1

Chapter 1

ARE WE ALONE?


"Growing in size and complexity
Living things, masses of atoms, DNA, protein
Dancing a pattern ever more intricate.
Out of the cradle onto the dry land
Here it is standing
Atoms with consciousness
Matter with curiosity.
Stands at the sea
Wonders at wondering
I
A universe of atoms
An atom in the universe."

Richard P. Feynman

 

 

 

I

SAY

HAVE

I

MENTIONED DIVINE THOUGHT DIVINE CONSCIENCE

I

SAY

HAVE

I

MENTIONED

GODS

DIVINE LOVE DIVINE

HAVE

I

MENTIONED

THAT

?

I

HAVE

O

GOOD

 

 

THOSE PATENT PATIENT PATTERN MAKERS

 

 

Nature's Numbers
Ian Stewart 1995

Numerology is the easiest-and consequently the most dangerous-method for finding patterns. It is easy because anybody can do it and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. Here's a case in point. Kepler was fascinated with patterns in nature, and he devoted much of his life to looking for them in the behaviour of the planets. He devised a simple and tidy theory for the existence of precisely six planets (in his time only Mercury, Venus, Earth, Mars, Jupiter, and Saturn were known). He also discovered a very strange pattern relating the orbital period of a / planet- the time it takes to go once around the Sun-to its distance from the Sun. Recall that the square of a number is what you get when you multiply it by itself: for example, the square of 4 is 4 x 4 = 16. Similarly, the cube is what you get when you multiply it by itself twice: for example, the cube of 4 is 4 x 4 x 4 = 64. Kepler found that if you take the cube of the distance of any planet from the Sun and divide it by the square of its orbital period, you always get the same number. It was not an especially elegant number, but it was the same for all six planets.

Which of these numerological observations is the more significant? The verdict of posterity is that it is the second one, the complicated and rather arbitrary calculation with squares and cubes. This numerical pattern was one of the key steps towards Isaac Newton's theory of gravity, which has explained all sorts of puzzles about the motion of stars and planets. In contrast, Kepler's neat, tidy theory for the number of planets has been buried without trace. For a start it must have been wrong, because we now know of nine planets, not six. There could be even more, farther out from the Sun, and small enough to be undetectable But more important, we no longer expect to find a neat, tidy theory for the number of planets. We think that the Solar System condensed from a cloud of gas surrounding the Sun, and the number of planets presumably depended on the amount of matter in the gas cloud, how it was distributed, and how fast and in what directions it was moving. An equally plausible gas cloud could have given us eight planets, or eleven; the number is accidental, depending on the initial conditions of the gas cloud, rather than universal, reflecting a general law of nature"

Page 6

" The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which. For example, there are three stars, roughly equally spaced and in a straight line, in the belt of the constellation Orion. Is that a clue to a significant law of nature?
Here's a similar question. Io, Europa, and Ganymede are three of Jupiter's larger satellites. They orbit the planet in , respectively, 1.77, 3.55, and 7.16 days. Each of these numbers is almost exactly twice the previous one. Is that a significant pattern? Three stars in a row, in terms of orbital period. Which pattern if either, is an important clue..."
    "… In addition to numerical patterns there are geometric ones…"
    "… Until recently the main shapes that appealed to mathematicians were very simple ones: triangles, squares, pen / Page 7 /tagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on. All of these shapes can be found in nature, although some are far more common, or more evident, than others. The rainbow, for example, is a collection of circles, one for each colour. We don't normally see the entire circle just an arc; but rainbows seen from the air can be complete circles. You also see circles in the ripples on a pond, in the human eye, and on butterflies wings.
         Talking of ripples, the flow of fluids provides an inexhaustible supply of nature's patterns. There are waves of many different kinds-surging toward a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outward from an underwater earthquake…"
"…There are swirling spiral whirlpools and tiny vortices. And there is the apparently structureless, random frothing of turbulent flow, one of the great enigmas of mathematics and physics. There are similar patterns in the atmosphere, too, the most dramatic being the vast spiral of a hurricane…"
    "…There are also wave patterns on land. The most strikingly mathematical landscapes on Earth are to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Even when the wind blows steadily in a fixed direction, sand dunes form. The simplest pattern is that of transverse dunes, which-just like ocean waves-line up in parallel straight rows at right angles to the prevailing wind direction. Sometimes the rows themselves become wavy in which case they are called barchanoid ridges; sometimes they break up into / Page 8 / innumerable shield-shaped barchan dunes. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes-shaped like a U, with the rounded end pointing in the direction of the wind. These sometimes occur in clusters, and they resemble the teeth of a rake. If the wind direction is variable, other forms become possible. For example, clusters of sand shaped dunes can form, each having several irregular arms radiating from a central peak. They arrange themselves in a random pattern of spots.

Chapter 6

Page 81

"Nature's symmetries can be found on every scale, from the structure of subatomic particles to that of the entire universe. Many chemical molecules are symmetric. The methane molecule is a tetrahedron - a triangular-sided pyramid - with one carbon atom at its center and four hydrogen atoms at its corners Benzene has the sixfold symmetry of a regular hexagon. The fashionable molecule buckminsterfullerene is a truncated icosahedral cage of sixty carbon atoms. (An icosahedron is a regular solid with twenty triangular faces;
"truncated" means that the corners are cut off.) Its symmetry lends it a remarkable stability, which has opened up new possibilities for organic chemistry.
    On a slightly larger scale than molecules, we find symmetries in cellular structure; at the heart of cellular replication lies a tiny piece of mechanical engineering. Deep within each / Page 82  / living cell, there is a rather shapeless structure known as the centrosome, which sprouts long thin microtubules, basic components of the cell's internal "skeleton", like a diminutive sea urchin. Centrsomes were first discovered in 1887 and play an important role in organizing cell division. How-ever in one respect the structure of the centresome is astonishingly symmetric. Inside it has two structures, known as centrioles, positioned at right angles to each other. Each centriole is cylindrical, made from twenty-seven microtubules fused together along their lengths in threes, and arranged with perfect ninefold symmetry. The microtubules themselves also have an astonishingley regular symmetric form. They are hollow tubes, made from a perfect regular checkerboard pattern of units that contain two distinct proteins, alpha- and betatubulin. One day, perhaps, we will understand why nature chose these symmetric forms. But it is amazing to see symmetric structures at the core of a living cell. "

 

CELL = 3533 = CELL

3533 = CELL = 3533

CELL = 3533 = CELL

CELL = 5 = CELL

SEE EL EL SEE

C ELL ELL C

CIRCLE = 5 5 = CIRCLE

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

0
-
Z
=
8
-
4
ZERO
64
28
1
1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
58
13
4
3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
6
-
S
=
1
-
3
SIX
52
16
7
7
-
S
=
1
-
5
SEVEN
65
20
2
8
-
E
=
5
-
5
EIGHT
49
31
4
9
-
N
=
5
-
4
NINE
42
24
6
45
-
-
-
42
-
40
Add
522
225
45
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
9
-
-
-
6
-
4
Deduce
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
58
13
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
-
5
THREE
56
29
2
-
-
2
-
-
-
-
-
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
42
24
6
-
-
-
-
-
-
6
-
-
-
6
-
S
=
1
-
3
SIX
52
16
7
-
-
-
-
-
-
-
7
-
-
7
-
S
=
1
-
5
SEVEN
65
20
2
-
-
2
-
-
-
-
-
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
-
4
-
-
-
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
-
-
6
-
-
-
45
-
-
-
42
-
40
 Add
522
225
45
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
4+2
-
4+0
 Reduce
5+2+2
2+2+5
4+5
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
6
-
4
 Deduce
9
9
9
-
1
4
3
8
5
9
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
ZERO
-
ZERO
=
64
28
1
+
0
=
1
=
1
1
-
ONE
=
34
16
7
+
1
=
8
=
8
2
-
TWO
=
58
13
4
+
2
=
6
=
6
3
-
THREE
=
56
29
2
+
3
=
5
=
5
4
-
FOUR
=
60
24
6
+
4
=
10
1+0
1
5
-
FIVE
=
42
24
6
+
5
=
11
1+1
2
6
-
SIX
=
52
16
7
+
6
=
13
1+3
4
7
-
SEVEN
=
65
20
2
+
7
=
9
=
9
8
-
EIGHT
=
49
31
4
+
8
=
12
1+2
3
6
-
NINE
=
42
24
6
+
9
=
15
1+5
6
45
-
Add
-
522
225
45
-
45
-
90
-
45
4+5
-
Reduce
-
5+2+2
2+2+5
4+5
-
4+5
-
9+0
-
4+5
9
-
Deduce
-
9
9
9
-
9
-
9
-
9

 

 

1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
58
13
4
3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
6
-
S
=
1
-
3
SIX
52
16
7
7
-
S
=
1
-
5
SEVEN
65
20
2
8
-
E
=
5
-
5
EIGHT
49
31
4
9
-
N
=
5
-
4
NINE
42
24
6
45
-
-
-
34
-
36
Add
458
197
44
4+5
-
-
-
3+4
-
3+6
Reduce
4+5+8
1+9+7
4+4
9
-
-
-
7
4
9
Deduce
17
17
8
-
-
-
-
-
-
-
Produce
1+7
1+7
-
9
-
-
-
7
-
9
Essence
8
8
8

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
58
13
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
-
5
THREE
56
29
2
-
-
2
-
-
-
-
-
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
42
24
6
-
-
-
-
-
-
6
-
-
-
6
-
S
=
1
-
3
SIX
52
16
7
-
-
-
-
-
-
-
7
-
-
7
-
S
=
1
-
5
SEVEN
65
20
2
-
-
2
-
-
-
-
-
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
-
4
-
-
-
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
-
-
6
-
-
-
45
-
-
-
34
-
36
Add
458
197
44
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
3+4
-
3+6
Reduce
4+5+8
1+9+7
4+4
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
7
4
9
Deduce
17
17
8
-
1
4
3
8
5
18
14
8
9
-
-
-
-
-
-
-
Produce
1+7
1+7
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
7
-
9
Essence
8
8
8
-
1
4
3
8
5
9
5
8
9

 

 

0
-
4
ZERO
8
5
9
6
-
=
28
2+8
=
10
1+0
1
1
-
3
ONE
6
5
5
-
-
=
16
1+6
=
7
-
7
2
-
3
TWO
2
5
6
-
-
=
13
1+3
=
4
-
4
3
-
5
THREE
2
8
9
5
5
=
29
2+9
=
11
1+1
2
4
-
4
FOUR
6
6
3
9
-
=
24
2+4
=
6
-
6
5
-
4
FIVE
6
9
4
5
-
=
24
2+4
=
6
-
6
6
-
3
SIX
1
9
6
-
-
=
16
1+6
=
7
-
7
7
-
5
SEVEN
1
5
4
5
5
=
20
2+0
=
2
-
2
8
-
5
EIGHT
5
9
7
8
2
=
31
3+1
=
4
-
4
9
-
4
NINE
5
9
5
5
-
=
24
2+4
=
6
-
6
45
-
40
Add
42
70
58
43
12
-
225
-
-
63
-
45
4+5
-
4+0
-
4+2
7+0
5+8
4+3
1+2
-
2+2+5
-
-
6+3
-
4+5
9
-
4
Reduce
6
7
13
7
3
-
9
-
-
9
-
-
-
-
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
9
-
4
Deduce
6
7
4
7
3
-
9
-
-
9
-
9

 

 

O
=
6
3
ONE
34
16
7
T
=
2
3
TWO
58
13
4
T
=
2
5
THREE
56
29
2
F
=
6
4
FOUR
60
24
6
F
+
6
4
FIVE
42
24
6
S
=
1
3
SIX
52
16
7
S
=
1
5
SEVEN
65
20
2
E
=
5
5
EIGHT
49
31
4
N
=
5
4
NINE
42
24
6
-
-
39
36
-
458
197
44
-
-
3+9
3+6
-
4+5+8
1+9+7
4+4
-
-
12
9
-
17
17
8
-
-
1+2
-
-
-
-
-
-
-
3
9
-
8
8
8

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Page 32

5


To Sorcerers and Magicians number FIVE is the most powerful - five is the mark of the pentacle, a five pointed star drawn by extending the sides of a Pentagon. Five surely is in the possession of the occult. And the Pentagon is the geometric figure in which the golden ratio of classical art and architecture is found most.

 

 

THE

BALANCING

ONE TWO THREE FOUR

FIVE

NINE EIGHT SEVEN SIX

 

 

O
=
15
-
3
ONE
34
16
7
-
1
T
=
20
-
3
TWO
58
13
4
-
2
T
=
20
-
5
THREE
56
29
2
-
3
F
=
6
-
4
FOUR
60
24
6
-
4
-
-
61
-
15
Add
208
82
19
-
10
-
-
6+1
-
1+5
Reduce
2+0+8
8+2
1+9
-
1+0
-
-
7
-
6
Deduce
10
10
10
-
1
-
-
-
-
-
Produce
1+0
1+0
1+0
-
-
-
-
7
-
6
Essence
1
1
1
-
1

 

 

N
=
14
-
4
NINE
42
24
6
-
9
E
=
5
-
5
EIGHT
49
31
4
-
8
S
=
19
-
5
SEVEN
65
20
2
-
7
S
=
19
-
3
SIX
52
16
7
-
6
-
-
57
-
17
Add
208
91
19
-
30
-
-
5+7
-
1+7
Reduce
2+0+8
9+1
1+9
-
3+0
-
-
12
-
8
Deduce
10
10
10
-
3
-
-
1+2
-
-
Produce
1+0
1+0
1+0
-
-
-
-
3
-
8
Essence
1
1
1
-
3

 

 

4
FIVE
42
24
6

 

1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1

 

15
ONE TWO THREE FOUR
208
82
1
4
FIVE
42
24
6
17
NINE EIGHT SEVEN SIX
208
91
1

 

 

3
ONE
34
16
7
-
3
SIX
52
16
7
3
TWO
58
13
4
-
5
SEVEN
65
20
2
5
THREE
56
29
2
-
5
EIGHT
49
31
4
4
FOUR
60
24
6
-
4
NINE
42
24
6
15
Add
208
82
19
-
17
Add
208
91
19
1+5
Reduce
2+0+8
8+2
1+9
-
1+7
Reduce
2+0+8
9+1
1+9
6
Deduce
10
10
10
-
8
Deduce
10
10
10
-
Produce
1+0
1+0
1+0
-
-
Produce
1+0
1+0
1+0
6
Essence
1
1
1
-
8
Essence
1
1
1

 

ALWAYS BALANCING IS THAT FIVE THAT FIVE IS BALANCING ALWAYS

 

 

F
=
6
-
3
FOR
39
21
3
E
=
5
-
5
EVERY
75
30
3
A
=
1
-
6
ACTION
62
26
8
T
=
2
-
5
THERE
56
29
2
I
=
9
-
2
IS
28
19
1
A
=
1
-
2
AN
15
6
6
E
=
5
-
5
EQUAL
56
20
2
A
=
1
-
3
AND
19
10
1
O
=
6
-
8
OPPOSITE
115
43
7
R
=
9
-
8
REACTION
85
40
4
-
-
45
4
46
First Total
550
244
37
-
-
4+5
-
4+6
Add to Reduce
5+5+0
2+4+4
3+7
Q
-
9
-
10
Second Total
10
10
10
-
-
-
4
1+0
Reduce to Deduce
1+0
1+0
1+0
-
-
9
5
1
Essence of Number
1
1
1

 

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
9
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
57
First Total
995
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+7
Add to Reduce
9+9+5
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
12
Second Total
23
14
14
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+2
Reduce to Deduce
2+3
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

 
 
Top
  
 
Evokation
  
Previous Page
Index
 Next Page