Previous Page
   Next Page
 
Evokation
  
 
Index
  

 

A

MAZE

IN

ZAZAZA ENTER AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

 12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

WORK DAYS OF GOD

Herbert W Morris D.D.circa 1883

Page 22

"As all the words in the English language are composed out of the twenty-six letters of the alphabet,.."

 

 

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"

 

 

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

 

 

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

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

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

 

 

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"

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

I
=
9
-
1
I
9
9
9
-
-
S
=
1
-
3
SEE
29
11
2
-
2
I
=
9
-
2
IT
29
11
2
-
2
A
=
1
-
3
ALL
25
7
7
-
7
N
=
5
-
3
NOW
52
16
7
-
7
S
=
1
-
4
SAID
33
15
6
-
6
T
=
2
-
3
THE
33
15
6
-
6
B
=
2
-
5
BLIND
41
23
5
-
-
M
=
4
-
3
MAN
28
10
1
-
-
-
-
34
4
27
First
279
117
45
4
30
-
-
3+4
-
2+7
Add
2+7+9
1+1+7
4+5
-
3+0
-
-
7
-
9
Second
18
9
9
-
3
-
-
-
-
-
Reduce
1+8
-
-
-
-
-
-
7
-
9
Essence
9
9
9
-
3

 

 

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
-
-
41
4
53
First Total
616
238
49
-
-
4+1
-
5+3
Add to Reduce
6+1+6
2+3+8
4+9
-
-
5
-
8
Second Total
13
13
13
-
-
-
-
-
Reduce to Deduce
1+3
1+3
1+3
-
-
5
-
8
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
47
29
2
-
-
47
4
81
First Total
931
400
85
-
-
4+7
-
8+1
Add to Reduce
9+3+1
4+0+0
8+5
-
-
11
-
9
Second Total
13
4
13
-
-
1+5
-
-
Reduce to Deduce
1+3
-
1+3
-
-
2
-
9
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
-
-
13
-
12
Second Total
22
13
13
-
-
1+3
-
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

 

MATHEMATICS A LANGUAGE OF LETTERS AND NUMBERS

 

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
-
-
41
4
53
-
616
238
49
-
-
-
-
-
-
-
-
-
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
-
-
47
4
81
-
931
400
85
-
-
-
-
-
-
-
-
-
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
-
598
238
58
-
-
-
-
-
-
-
-
-
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
-
299
110
20
-
-
-
-
-
-
-
-
-
-
-
156
-
219
First Total
2444
986
212
-
-
1+5+6
-
2+1+9
Add to Reduce
2+4+4+4
9+8+6
2+1+2
-
-
12
-
12
Second Total
14
23
5
-
-
1+2
-
1+2
Reduce to Deduce
1+4
2+3
-
-
-
3
-
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
-
-
-
-
-
1
-
8
Essence of Number
9
9
9

 

 

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
-
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

 

 

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
-
-
41
4
53
-
616
238
49
-
-
-
-
-
-
-
-
-
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
-
-
47
4
81
-
931
400
85
-
-
-
-
-
-
-
-
-
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
-
598
238
58
-
-
-
-
-
-
-
-
-
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
-
299
110
20
-
-
-
-
-
-
-
-
-
-
-
156
-
219
First Total
2444
986
212
-
-
1+5+6
-
2+1+9
Add to Reduce
2+4+4+4
9+8+6
2+1+2
-
-
12
-
12
Second Total
14
23
5
-
-
1+2
-
1+2
Reduce to Deduce
1+4
2+3
-
-
-
3
-
3
Essence of Number
5
5
5

 

 

-
7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
8
9
1
+
=
18
1+8
=
9
-
9
-
-
-
-
-
-
8
9
19
+
=
36
3+6
=
9
=
9
-
7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
-
2
4
5
7
-
-
-
+
=
18
1+8
=
9
=
9
-
-
11
13
5
16
-
-
-
+
=
45
4+5
=
9
=
9
-
7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
-
11
13
5
16
8
9
19
+
=
81
8+1
=
9
=
9
-
-
2
4
5
7
8
9
1
+
=
36
3+6
=
9
=
9
-
7
K
M
E
P
H
I
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
``-
2
--
-
--
-
--
-
-
-
2
occurs
x
1
=
2
3
-
-
--
-
--
--
--
-
-
-
3
THREE
-
-
-
-
-
-
--
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
--
5
--
--
--
-
-
-
5
occurs
x
1
=
5
6
-
--
--
-
--
--
--
-
-
-
6
SIX
-
-
-
-
-
-
--
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
9
7
K
M
E
P
H
I
S
-
-
36
-
-
7
-
36
-
-
-
-
-
-
-
9
-
-
-
3+6
-
-
-
-
3+6
9
7
K
M
E
P
H
I
S
-
-
9
-
-
7
-
9
-
-
2
4
5
7
8
9
1
-
-
-
-
-
-
-
-
9
7
K
M
E
P
H
I
S
-
-
9
-
-
7
-
9

 

 

7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
-
-
-
-
8
9
1
+
=
18
1+8
=
9
-
9
-
-
-
-
-
8
9
19
+
=
36
3+6
=
9
=
9
7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
2
4
5
7
-
-
-
+
=
18
1+8
=
9
=
9
-
11
13
5
16
-
-
-
+
=
45
4+5
=
9
=
9
7
K
M
E
P
H
I
S
-
-
-
-
-
--
-
-
-
11
13
5
16
8
9
19
+
=
81
8+1
=
9
=
9
-
2
4
5
7
8
9
1
+
=
36
3+6
=
9
=
9
7
K
M
E
P
H
I
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
``-
2
--
-
--
-
--
-
-
-
2
occurs
x
1
=
2
-
--
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
-
--
5
--
--
--
-
-
-
5
occurs
x
1
=
5
-
--
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
7
K
M
E
P
H
I
S
-
-
36
-
-
7
-
36
-
-
-
-
-
-
9
-
-
-
3+6
-
-
-
-
3+6
7
K
M
E
P
H
I
S
-
-
9
-
-
7
-
9
-
2
4
5
7
8
9
1
-
-
-
-
-
-
-
-
7
K
M
E
P
H
I
S
-
-
9
-
-
7
-
9

 

 

-
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
-
-
8
9
-
-
6
-
-
-
-
8
9
-
+
=
40
4+0
=
4
-
4
-
4
-
`-
8
9
-
-
15
-
-
-
-
8
9
-
+
=
49
4+9
=
13
1+3
4
-
4
-
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
9
-
7
3
7
7
-
-
3
+
=
38
3+8
=
11
1+1
2
-
2
-
`-
-
-
5
18
-
7
12
25
16
-
-
3
+
=
83
8+3
=
11
1+1
2
-
2
-
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
-
`-
8
9
5
18
15
7
12
25
16
8
9
3
+
=
135
1+3+5
=
9
-
9
-
9
-
-
8
9
5
9
6
7
3
7
7
8
9
3
+
=
81
8+1
=
9
-
9
-
9
-
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
--
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
7
-
7
7
-
-
-
-
-
7
occurs
x
3
=
21
=
3
-
--
8
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
9
occurs
x
3
=
27
2+7
9
7
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
38
-
-
12
-
81
-
36
-
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
3+8
-
-
1+2
Q
8+1
-
3+6
7
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
11
-
-
3
-
9
-
9
-
-
8
9
5
9
6
7
3
7
7
8
9
3
-
-
1+1
-
-
-
-
-
-
-
7
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
2
-
-
3
-
9
-
9

 

 

12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
-
8
9
-
-
6
-
-
-
-
8
9
-
+
=
40
4+0
=
4
-
4
-
4
`-
8
9
-
-
15
-
-
-
-
8
9
-
+
=
49
4+9
=
13
1+3
4
-
4
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
5
9
-
7
3
7
7
-
-
3
+
=
38
3+8
=
11
1+1
2
-
2
`-
-
-
5
18
-
7
12
25
16
-
-
3
+
=
83
8+3
=
11
1+1
2
-
2
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
--
-
-
-
-
`-
8
9
5
18
15
7
12
25
16
8
9
3
+
=
135
1+3+5
=
9
-
9
-
9
-
8
9
5
9
6
7
3
7
7
8
9
3
+
=
81
8+1
=
9
-
9
-
9
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
-
-
-
5
-
-
-
-
-
-
-
-
--
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
7
-
7
7
-
-
-
-
-
7
occurs
x
3
=
21
=
3
--
8
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
9
occurs
x
3
=
27
2+7
9
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
38
-
-
12
-
81
-
36
-
-
9
-
9
-
-
-
-
-
-
9
-
-
-
3+8
-
-
1+2
Q
8+1
-
3+6
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
11
-
-
3
-
9
-
9
-
8
9
5
9
6
7
3
7
7
8
9
3
-
-
1+1
-
-
-
-
-
-
-
12
H
I
E
R
O
G
L
Y
P
H
I
C
-
-
2
-
-
3
-
9
-
9

 

 

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
-
Q
-
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

 

 

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

 

 

JUST SIX NUMBERS

Martin Rees

1
999

OUR COSMIC HABITAT I

PLANETS STARS AND LIFE

Page 24

"A proton is 1,836 times heavier than an electron, and the number 1,836 would have the same connotations to any 'intelligence' "
Page 24 / 25
"A manifestly artificial signal- even if it were as boring as lists of prime numbers, or the digits of 'pi' - would imply that 'intelli- gence' wasn't unique to the Earth and had evolved elsewhere. The nearest potential sites are so far away that signals would take many years in transit. For this reason alone, transmission would be primarily one-way. There would be time to send a measured response, but no scope for quick repartee!
Any remote beings who could communicate with us would have some concepts of mathematics and logic that paralleled our own. And they would also share a knowledge of the basic particles and forces that govern our universe. Their habitat may be very different (and the biosphere even more different) from ours here on Earth; but they, and their planet, would be made of atoms just like those on Earth. For them, as for us, the most important particles would be protons and electrons: one electron orbiting a proton makes a hydrogen atom, and electric currents and radio transmitters involve streams of electrons. A proton is 1,836 times heavier than an electron, and the number 1,836 would have the same connotations to any 'intelligence' able and motivated to transmit radio signals. All the basic forces and natural laws would be the same. Indeed, this uniformity - without which our universe would be a far more baffling place - seems to extend to the remotest galaxies that astronomers can study. (Later chapters in this book will, however, speculate about other 'universes', forever beyond range of our telescopes, where different laws may prevail.)
Clearly, alien beings wouldn't use metres, kilograms or seconds. But we could exchange information about the ratios of two masses (such as thc ratio of proton and electron masses) or of two lengths, which are 'pure numbers' that don't depend on what units are used: the statement that one rod is ten times as long as another is true (or false) whether we measure lengths / in feet or metres or some alien units"

 

"A proton is

1,836 times heavier than an electron, and the number 1,836

would have the same connotations to any 'intelligence'"

 

 

E
=
5
-
8
EIGHTEEN
73
46
1
T
=
2
-
9
THIRTYSIX
152
53
8
-
-
7
4
17
First Total
225
99
9
-
-
-
-
1+7
Add to Reduce
2+2+5
9+9
-
Q
-
7
-
8
Second Total
9
18
9
-
-
-
-
1+7
Reduce to Deduce
-
1+8
-
Q
-
7
-
8
Essence of Number
9
9
9

 

 

-
17
E
I
G
H
T
E
E
N
T
+
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
-
-
5
+
-
8
9
-
-
-
1
9
6
+
=
55
5+5
=
10
1+0
1
=
1
-
-
-
9
-
8
-
-
-
14
-
-
8
9
-
-
-
19
9
24
+
=
100
1+0+0
=
1
2+0
1
=
1
-
17
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
-
5
-
7
-
2
5
5
-
+
2
-
-
9
2
7
-
-
-
+
=
44
4+4
=
8
=
8
=
8
-
-
5
-
7
-
20
5
5
-
+
20
-
-
18
20
25
-
-
-
+
=
125
1+2+5
=
8
=
8
=
8
-
17
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
-
5
9
7
8
20
5
5
14
+
20
8
9
18
20
25
19
9
24
+
=
225
2+2+5
=
9
=
9
=
9
-
-
5
9
7
8
2
5
5
5
+
2
8
9
9
2
7
1
9
6
+
=
99
9+9
=
18
=
9
=
9
-
17
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
1
-
-
+
=
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
2
-
-
-
+
2
-
-
-
2
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
5
5
+
-
-
-
-
-
-
-
-
-
+
=
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
-
-
6
+
=
6
occurs
x
1
=
6
-
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
9
-
-
-
-
-
-
+
-
-
9
9
-
-
-
9
-
+
=
9
occurs
x
4
=
36
3+6
9
7
17
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
+
=
38
-
-
17
-
99
-
36
-
1+7
-
9
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
-
-
-
3+8
-
-
1+7
-
9+9
-
3+6
7
8
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
11
-
-
8
-
18
-
9
-
-
5
9
7
8
2
5
5
5
+
2
8
9
9
2
7
1
9
6
+
=
1+1
-
-
-
-
1+8
-
-
7
8
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
+
=
2
-
-
8
-
9
-
9

 

 

17
E
I
G
H
T
E
E
N
T
+
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
-
-
-
5
+
-
8
9
-
-
-
1
9
6
+
=
55
5+5
=
10
1+0
1
=
1
-
-
9
-
8
-
-
-
14
-
-
8
9
-
-
-
19
9
24
+
=
100
1+0+0
=
1
2+0
1
=
1
17
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
5
-
7
-
2
5
5
-
+
2
-
-
9
2
7
-
-
-
+
=
44
4+4
=
8
=
8
=
8
-
5
-
7
-
20
5
5
-
+
20
-
-
18
20
25
-
-
-
+
=
125
1+2+5
=
8
=
8
=
8
17
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
-
5
9
7
8
20
5
5
14
+
20
8
9
18
20
25
19
9
24
+
=
225
2+2+5
=
9
=
9
=
9
-
5
9
7
8
2
5
5
5
+
2
8
9
9
2
7
1
9
6
+
=
99
9+9
=
18
=
9
=
9
17
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
1
-
-
+
=
1
occurs
x
1
=
1
=
1
+
-
-
-
-
2
-
-
-
+
2
-
-
-
2
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
+
5
-
-
-
-
5
5
5
+
-
-
-
-
-
-
-
-
-
+
=
5
occurs
x
4
=
20
2+0
2
+
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
-
-
6
+
=
6
occurs
x
1
=
6
-
6
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
+
-
9
-
-
-
-
-
-
+
-
-
9
9
-
-
-
9
-
+
=
9
occurs
x
4
=
36
3+6
9
17
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
+
=
38
-
-
17
-
99
-
36
1+7
-
9
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
-
-
-
3+8
-
-
1+7
-
9+9
-
3+6
8
E
I
G
H
T
E
E
N
-
T
H
I
R
T
Y
S
I
X
-
-
11
-
-
8
-
18
-
9
-
5
9
7
8
2
5
5
5
+
2
8
9
9
2
7
1
9
6
+
=
1+1
-
-
-
-
1+8
-
-
8
E
I
G
H
T
E
E
N
+
T
H
I
R
T
Y
S
I
X
+
=
2
-
-
8
-
9
-
9

 

 

-
THE SOLAR SYSTEM
-
-
-
3
THE
33
15
6
5
SOLAR
65
29
2
6
SYSTEM
101
38
2
14
THE SOLAR SYSTEM
199
82
10
1+4
-
1+9+9
8+2
1+0
5
THE SOLAR SYSTEM
19
10
1
-
-
1+9
1+0
-
5
THE SOLAR SYSTEM
10
1
1
-
-
1+0
-
-
5
THE SOLAR SYSTEM
1
1
1

 

 

3
SUN
54
9
9
7
MERCURY
103
40
4
5
VENUS
81
18
9
5
EARTH
52
25
7
4
MOON
57
21
3
4
MARS
51
15
6
7
JUPITER
99
36
9
6
SATURN
93
21
3
6
URANUS
94
22
4
7
NEPTUNE
95
32
5
5
PLUTO
84
21
3
59
First Total
863
260
62
5+9
Add to Reduce
8+6+3
2+6
6+2
14
Second Total
17
8
8
1+4
Reduce to Deduce
1+7
-
-
5
Essence of Number
8
8
8

 

 

3
SUN
54
9
9
5
EARTH
52
25
7
4
MOON
57
21
3
12
First Total
163
55
19
1+2
Add to Reduce
1+6+3
5+5
6+2
3
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
3
Essence of Number
1
1
1

 

 

-
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

 

 

 

 

 

 

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 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.

 

 

PRINCIPIA MATHEMATICA
FREEDOM DEMOCRACY
FREEDOM DEMOCRACY, F,R, EEDOM, DE, M, OC, R, ACY
FREEDOM DEMOCRACY, F, R, EEDOM, DE, MOCRACY
THE, ZODIAC, OF, DENDERAH
THIRTEEN, ELEVEN
THIRTY THREE, THIRTY, THREE
NOSTRADAMUS
THE MYTH, THE, MYTH
LIVING, THE, MYTH
THE, MYTH, LIVES
AEON, AEONS
MIND, GRAVITY
DOPA, TRIBE
FIELD, OF, DREAMS
ELECTRO, MAGNETISM
KINETIC, KINETICS
KINETIC ENERGIES, KINETIC ENERGIES
FOUR, F, OUR
FOUR-FOR, FOUR, FOR
LUXOR
MATHEMATIKOI
ALGORITHM. ALGORITHMS
IRON MAGNETS
IRRESISTIBLE, FORCE
IMMORABLE, OBJECT
STARTING, FINISHING, ING
START, FINISH
GOD CONSCIOUS
CONSCIOUS
RELIGIONS

 

 

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

 

 

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

 

 

3
3
-
A+B+C
6
6
6
2
-
2
D+E
9
9
9
3
3
-
F+G+H
21
21
3
1
-
1
I
9
9
9
3
3
-
J+K+L
33
6
6
2
-
2
M+N
27
9
9
2
2
-
O+P
31
13
4
3
-
3
QRS
54
18
9
3
-
3
TUV
63
9
9
3
-
3
WXY
72
18
9
1
1
-
Z
26
8
8
26
12
14
First Total
351
126
81
2+6
1+2
1+4
Add to Reduce
3+5+1
1+2+6
8+1
8
3
5
Reduce to Deduce
9
9
9

 

 

3
A+B+C
6
6
6
-
D+E
-
-
-
3
F+G+H
21
21
3
-
I
-
-
-
3
J+K+L
33
6
6
-
M+N
-
-
-
2
O+P
31
13
4
-
QRS
-
-
-
-
TUV
-
-
-
-
WXY
-
-
-
1
Z
26
8
8
12
First Total
117
54
27
2+6
Add to Reduce
1+1+7
5+4
2+7
8
Reduce to Deduce
9
9
9

 

 

-
A+B+C
-
-
-
2
D+E
9
9
9
-
F+G+H
-
-
-
1
I
9
9
9
-
J+K+L
-
-
-
2
M+N
27
9
9
-
O+P
-
-
-
3
QRS
54
18
9
3
TUV
63
9
9
3
WXY
72
18
9
-
Z
-
-
-
14
First Total
234
72
54
1+4
Add to Reduce
2+3+4
7+2
5+4
5
Reduce to Deduce
9
9
9

 

 

4

ZERO
64
28
1
3

ONE
34
16
7
3

TWO
58
13
4
5

THREE
56
29
2
4

FOUR
60
24
6
4

FIVE
42
24
6
3

SIX
52
16
7
5

SEVEN
65
20
2
5

EIGHT
49
31
4
4

NINE
42
24
6
40
-
522
225
45
4+0
-
5+2+2
2+2+5
4+5
4
-
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
-
-
1+7
1+7
-
-
-
3
9
-
8
8
8

 

 

Z
=
8
4
ZERO
64
28
1
O
=
6
3
ONE
34
16
7
S
=
1
5
SEVEN
65
20
2
T
=
2
3
TWO
58
13
4
F
=
6
4
FOUR
60
24
6
S
=
1
3
SIX
52
16
7
T
=
2
5
THREE
56
29
2
F
=
6
4
FIVE
42
24
6
E
=
5
5
EIGHT
49
31
4
N
=
5
4
NINE
42
24
6
-
-
47
40
-
522
225
45
-
-
11
4+0
-
5+2+2
2+2+5
4+5
-
-
2
4
-
9
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
+
=
108
1+0+8
=
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
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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

 

 

3
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
+
=
108
1+0+8
=
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

 

 

-
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

 

 

-
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
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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
12
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
12
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
12
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
-
-
-
-
-
-
-
-
12
M
A
T
H
E
M
A
T
I
C
S
-
-
5
-
-
2
-
4

 

 

THE MAGIC MOUNTAIN


Thomas Mann 1875-1955

Page 417

"I preach mathematics."

 

 

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."

 

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

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
9
PRINCIPIA
95
59
5
-
-
-
-
-
-
-
-
-
-
M
=
4
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
6
-
-
-
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
6
7
-
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
I
=
9
8
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
7
6
-
-
-
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
6
-
-
-
H
=
8
13
1
H
8
8
8
-
-
-
-
-
-
6
-
8
-
E
=
5
14
1
E
5
5
5
-
-
-
-
-
5
6
-
-
-
M
=
4
15
1
M
13
4
4
-
-
-
-
4
7
6
-
-
-
A
=
1
16
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
T
=
2
17
1
T
20
2
2
-
-
2
-
-
-
6
-
-
-
I
=
9
18
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
C
=
3
19
1
C
3
3
3
-
-
-
3
-
-
6
7
-
-
A
=
1
20
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
-
-
99
-
20
-
189
99
99
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
99
4
9
PRINCIPIA
95
59
5
-
4
4
6
8
10
6
14
8
45
-
-
9+9
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
1+0
-
1+4
-
4+5
Q
-
18
-
20
First Total
189
99
9
-
4
4
6
8
1
6
5
8
9
-
-
1+8
-
2+0
Add to Reduce
1+8+9
9+9
-
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Second Total
18
18
9
-
4
4
6
8
1
6
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Essence of Number
9
9
9
-
4
4
6
8
1
6
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
9
PRINCIPIA
95
59
5
-
-
-
-
-
-
-
-
-
-
M
=
4
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
A
=
1
16
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
A
=
1
20
1
A
1
1
1
-
1
-
-
-
-
6
7
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
6
-
-
-
T
=
2
17
1
T
20
2
2
-
-
2
-
-
-
6
-
-
-
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
6
7
-
-
C
=
3
19
1
C
3
3
3
-
-
-
3
-
-
6
7
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
7
6
-
-
-
M
=
4
15
1
M
13
4
4
-
-
-
-
4
7
6
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
6
-
-
-
E
=
5
14
1
E
5
5
5
-
-
-
-
-
5
6
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
H
=
8
13
1
H
8
8
8
-
-
-
-
-
-
6
-
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
I
=
9
8
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
I
=
9
18
1
I
9
9
9
-
-
-
-
-
-
6
-
-
9
-
-
99
-
20
-
189
99
99
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
99
4
9
PRINCIPIA
95
59
5
-
4
4
6
8
10
6
14
8
45
-
-
9+9
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
1+0
-
1+4
-
4+5
Q
-
18
-
20
First Total
189
99
9
-
4
4
6
8
1
6
5
8
9
-
-
1+8
-
2+0
Add to Reduce
1+8+9
9+9
-
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Second Total
18
18
9
-
4
4
6
8
1
6
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Essence of Number
9
9
9
-
4
4
6
8
1
6
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
8
9
P
=
7
-
9
PRINCIPIA
95
59
5
-
-
-
-
-
-
-
-
-
M
=
4
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
8
9
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
7
-
-
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
7
-
-
A
=
1
16
1
A
1
1
1
-
1
-
-
-
-
7
-
-
A
=
1
20
1
A
1
1
1
-
1
-
-
-
-
7
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
T
=
2
17
1
T
20
2
2
-
-
2
-
-
-
-
-
-
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
7
-
-
C
=
3
19
1
C
3
3
3
-
-
-
3
-
-
7
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
7
-
-
-
M
=
4
15
1
M
13
4
4
-
-
-
-
4
7
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
-
-
-
E
=
5
14
1
E
5
5
5
-
-
-
-
-
5
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
7
8
-
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
7
8
-
H
=
8
13
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
I
=
9
8
1
I
9
9
9
-
-
-
-
-
-
-
-
9
I
=
9
18
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
99
-
20
-
189
99
99
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
99
4
9
PRINCIPIA
95
59
5
-
4
4
6
8
10
14
8
45
-
-
9+9
-
11
MATHEMATICA
94
40
4
-
-
-
-
-
1+0
1+4
-
4+5
Q
-
18
-
20
First Total
189
99
9
-
4
4
6
8
1
5
8
9
-
-
1+8
-
2+0
Add to Reduce
1+8+9
9+9
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Second Total
18
18
9
-
4
4
6
8
1
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
2
Essence of Number
9
9
9
-
4
4
6
8
1
5
8
9

 

 

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

 

 

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

 

 

-
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."

 

 

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

 

 

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

 

 

10
CURRICULUM
--
-
-
-
C
3
3
3
-
U
21
3
3
-
R
18
9
9
-
R
18
9
9
-
I
9
9
9
-
C
3
3
3
-
U
21
3
3
-
L
12
3
3
-
U
21
3
3
-
M
13
4
4
10
CURRICULUM
139
49
49
1+0
-
1+3+9
4+9
4+9
1
CURRICULUM
13
13
13
-
-
1+3
1+3
1+3
1
CURRICULUM
4
4
4

 

 

10
CURRICULUM
--
-
-
-
-
-
-
-
C
3
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
R
18
9
9
-
-
-
9
-
R
18
9
9
-
-
-
9
-
I
9
9
9
-
-
-
9
-
C
3
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
L
12
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
M
13
4
4
-
-
4
-
10
CURRICULUM
139
49
49
-
18
4
27
1+0
-
1+3+9
4+9
4+9
-
1+8
-
2+7
1
CURRICULUM
13
13
13
-
9
4
9
-
-
1+3
1+3
1+3
-
-
-
-
1
CURRICULUM
4
4
4
-
9
4
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. "

 

 

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
-
-
-
-
-
-
8
CALCULUS
2
2
2
9

 

 

11
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
=
1
-
-
-
-
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
11
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
21
-
+
=
73
7+3
=
10
1+0
1
=
1
11
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
21
19
+
=
92
9+2
=
11
1+0
2
=
2
-
-
3
1
3
3
3
3
3
1
+
=
20
2+0
=
2
=
2
=
2
11
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
-
3
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
=
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
FOUR
9
-
-
-
-
-
41
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
20
-
11
4+1
-
3
-
3
3
3
3
-
-
-
-
-
-
-
-
-
2+0
-
1+1
5
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2
-
-
3
1
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
5
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2

 

 

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
21
-
+
=
73
7+3
=
10
1+0
1
=
1
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
21
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
-
-
-
-
-
-
-
-
-
2+0
-
1+1
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2
-
3
1
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2

 

 

D
=
4
-
-
DICTIONARY
-
-
-
-
-
-
-
1
D
4
4
4
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
O
15
6
6
-
-
-
-
1
N
14
5
5
-
-
-
-
1
A
1
1
1
-
-
-
-
1
R
18
9
9
-
-
-
-
1
Y
25
7
7
D
=
4
Q
10
DICTIONARY
118
55
55
-
-
-
-
1+0
-
1+1+8
5+5
5+5
D
=
4
Q
1
DICTIONARY
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
D
=
4
Q
1
DICTIONARY
1
1
1

 

 

D
=
4
-
-
DICTIONARY
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
T
20
2
2
-
-
-
-
1
C
3
3
3
-
-
-
-
1
D
4
4
4
-
-
-
-
1
N
14
5
5
-
-
-
-
1
O
15
6
6
-
-
-
-
1
Y
25
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
1
R
18
9
9
D
=
4
Q
10
DICTIONARY
118
55
55
-
-
-
-
1+0
-
1+1+8
5+5
5+5
D
=
4
Q
1
DICTIONARY
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
D
=
4
Q
1
DICTIONARY
1
1
1

 

 

D
=
4
-
-
DICTIONARY
-
-
-
-
-
-
-
1
D
4
4
4
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C+T
23
5
5
-
-
-
-
1
I
9
9
9
-
-
-
-
1
O+N+A
30
12
3
-
-
-
-
1
R
18
9
9
-
-
-
-
1
Y
25
7
7
D
=
4
Q
10
DICTIONARY
118
55
46
-
-
-
-
1+0
-
1+1+8
5+5
4+6
D
=
4
Q
1
DICTIONARY
10
10
10
-
-
-
-
-
-
1+0
1+0
-
D
=
4
Q
1
DICTIONARY
1
1
1

 

 

T
=
2
Q
3
THE
33
15
6
E
=
5
Q
7
ENGLISH
74
38
2
D
=
4
Q
10
DICTIONARY
118
55
1
-
-
11
Q
20
Add to Reduce
225
108
9
Q
Q
1+1
-
2+0
Reduce to Deduce
1+1+8
1+0+8
-
-
-
2
Q
2
Essence of Number
9
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

 

 

www.merriam-webster.com/dictionary/algorithm

a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an ...

algorithm [ˈælgəˌrɪðəm]
n
1. (Mathematics) a logical arithmetical or computational procedure that if correctly applied ensures the solution of a problem Compare heuristic
2. (Mathematics) Logic Maths a recursive procedure whereby an infinite sequence of terms can be generated Also called algorism
[changed from algorism, through influence of Greek arithmos number]
algorithmic adj
aal·go·rithm (lg-rm)
n.
A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.
algorithmically adv

algorithm (lg-rthm)
A finite set of unambiguous instructions performed in a prescribed sequence to achieve a goal, especially a mathematical rule or procedure used to compute a desired result. Algorithms are the basis for most computer programming.

Noun 1. algorithm - a precise rule (or set of rules) specifying how to solve some problem
algorithmic program, algorithmic rule
formula, rule - (mathematics) a standard procedure for solving a class of mathematical problems; "he determined the upper bound with Descartes' rule of signs"; "he gave us a general formula for attacking polynomials"
sorting algorithm - an algorithm for sorting a list
stemming algorithm, stemmer - an algorithm for removing inflectional and derivational endings in order to reduce word forms to a common stem algorithm
any methodology for solving a certain kind of problem.
See also: Mathematics

 

 

Algorithm
From Wikipedia, the free encyclopedia

Flow chart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
In mathematics and computer science, an algorithm (i/ˈælɡərɪðəm/) is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.

More precisely, an algorithm is an effective method expressed as a finite list[1] of well-defined instructions[2] for calculating a function.[3] Starting from an initial state and initial input (perhaps empty),[4] the instructions describe a computation that, when executed, will proceed through a finite [5] number of well-defined successive states, eventually producing "output"[6] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[7]

Though al-Khwārizmī's algorism referred to the rules of performing arithmetic using Hindu-Arabic numerals and the systematic solution of linear and quadratic equations, a partial formalization of what would become the modern algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"[8] or "effective method";[9] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.[10]

Informal definition
For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.
While there is no generally accepted formal definition of "algorithm," an informal definition could be "a set of rules that precisely defines a sequence of operations."[11] For some people, a program is only an algorithm if it stops eventually; for others, a program is only an algorithm if it stops before a given number of calculation steps.[12]

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.

Boolos & Jeffrey (1974, 1999) offer an informal meaning of the word in the following quotation:

No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus an algorithm can be an algebraic equation such as y = m + n—two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in language understood by "the computer")[14] for a fast, efficient, "good"[15] process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities)[16] to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively"[17] produce, in a "reasonable" time,[18] output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.

[edit] Formalization

Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):

Minsky: "But we will also maintain, with Turing . . . that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. Although this may seem extreme, the arguments . . . in its favor are hard to refute".[19]

Gurevich: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".[20]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.

[edit] Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite state machine, state transition table and control table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Representations of algorithms can be classed into three accepted levels of Turing machine description:[21]
1 High-level description:
"...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head." 2 Implementation description:
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function." 3 Formal description:
Most detailed, "lowest level", gives the Turing machine's "state table". For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.
[edit] Implementation

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

[edit] Computer algorithms

Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test=true THEN GOTO step xxx) (a diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf Tausworthe 1977:100,114).
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the target machines to produce output from given input (perhaps null).

"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: ". . .we want good algorithms in some loosely defined aesthetic sense. One criterion . . . is the length of time taken to perform the algorithm . . .. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"[22] Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"[23]
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid).

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".[24]

Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example of using Euclid's algorithm will be shown below.

Computers (and computors), models of computation: A computer (or human "computor"[25]) is a restricted type of machine, a "discrete deterministic mechanical device"[26] that blindly follows its instructions.[27] Melzak's and Lambek's primitive models[28] reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters[29] (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.[30]

Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".[31] Minsky's machine proceeds sequentially through its five (or six depending on how one counts) instructions unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution)[32] operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1).[33] Rarely will a programmer have to write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.[34]

Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[35] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.[36]

This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[37] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" (division) instruction available rather than just subtraction (or worse: just Minsky's "decrement").

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code" a programmer can write structured programs using these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".[38] Tausworthe augments the three Böhm-Jacopini canonical structures:[39] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.[40] An additional benefit of a structured program will be one that lends itself to proofs of correctness using mathematical induction.[41]

Canonical flowchart symbols[42]: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only 4: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm-Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.

 

 

EVOLVE LOVE EVOLVE

LOVES SOLVE LOVES

EVOLVE LOVE EVOLVE

 

 

LOGARITHM


Logarithm - Wikipedia
en.wikipedia.org › wiki › Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed ...
Logarithmic scale · ?Natural logarithm · ?Common logarithm · ?Binary logarithm

 

Introduction to Logarithms - Math is Fun
www.mathsisfun.com › algebra › logarithms
Introduction to Logarithms. In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number?

?Working with Exponents · ?Logarithms Can Have Decimals · ?Exponents, Roots

 

-
-
-
-
9
ALGORITHM
103
49
49
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
LOGARITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
9
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
9
LOGARITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
9
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
10
ALGORITHM
103
49
49
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
LOGARITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

LOGARITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
10
ALGORITHM
103
49
49
-
1
2
3
4
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
LOGARITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

MINUS 5

 

LOGARITHM

ALGORITHM

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
4
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
4
4
4

 

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

ALGORITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

MINUS THE 5

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

Algorithm - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Algorithm

In mathematics and computer science, an algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and ...

 

A
=
1
-
9
ALGORITHM
103
49
4
A
=
1
-
10
ALGORITHMS
122
59
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
-
ALGORITHMS
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
L
12
3
3
-
-
-
-
1
G
7
7
7
-
-
-
-
1
O
15
6
6
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
H
8
8
8
-
-
-
-
1
M+S
32
14
5
A
=
1
-
10
ALGORITHMS
122
59
50
-
-
-
-
1+0
-
1+2+2
5+9
5+0
A
=
1
-
1
ALGORITHMS
5
14
5
-
-
-
-
-
-
-
1+4
-
A
=
1
-
1
ALGORITHMS
5
5
5

 

 

-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
9
-
8
-
1
+
=
24
2+4
=
6
=
6
-
-
-
-
-
15
-
9
-
8
-
19
+
=
51
5+1
=
6
=
6
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
3
7
-
9
-
2
-
4
-
+
=
26
2+6
=
8
=
8
-
-
1
12
7
-
18
-
20
-
13
-
+
=
71
7+1
=
8
=
8
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
12
7
15
18
9
20
8
13
19
+
=
122
1+2+2
=
5
1+0
5
-
-
1
3
7
6
9
9
2
8
4
1
+
=
50
5+0
=
5
1+0
5
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
9
-
-
-
-
-
-
9
occurs
x
1
=
9
5
10
A
L
G
O
R
I
T
H
M
S
-
-
27
-
-
10
-
41
-
1+0
-
-
-
-
9
9
-
-
-
-
-
-
2+7
-
-
1+0
-
4+1
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5
-
-
1
3
7
6
9
9
2
8
4
1
-
-
-
-
-
-
-
-
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
T
=
2
-
1
T
20
2
2
L
=
3
-
1
L
12
3
3
M
=
4
-
1
M
13
4
4
5
-
5
-
-
5
-
-
5
O
=
6
-
1
O
15
6
6
G
=
7
-
1
G
7
7
7
H
=
8
-
1
H
8
8
8
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Preface to the New Edition

Page 5
Twelve years ago a little boy entered my imagination as he hopped across the centuries and played with numbers. I began to see how the simple architecture of our decimal system could be constructed in secret ways — not a building project this time but an abstract one. On the surface of our arithmetic countless combinations of numbers take part in tedious and exacting calculations but underneath it all there is pattern, governed by a repeating code of integers. The Sigma Code reduces numbers to a single digit and the illusion of the many is seen to be but the reflection of a few. This is not a book on maths: this is a book for anyone who can carry out simple sums in their heads, and who won't be short-changed knowingly.
When Number 9 first came out I received mail from many who played with numbers. They chased patterns; some had special numbers and even mystical systems. I was tempted to write about numerology but resisted. I wanted to write about the intricacy of what the.. numbers actually do and leave the reader to wonder about the larger irrational that seems to hover around such constructions.
If I were writing this book today the numbers would have featured in a wider context of structuring nature's patterns, and also playing the role of animator in algorithms that create unique architectural forms and shapes. I would also include my previous research into other base systems. But this book was a first step which came from a child-like urge, like playing with building blocks, to build out of our numbers — just the simple 1, 2, 3, up to number 9.

 

RESEARCH R E SEARCH ER RESEARCH

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

NUMBER = 534259 = 1 = 534259 NUMBER

NUMBER = 234559 NUMBER

NUMBER = 534259 = 1 = 534259 NUMBER

 

 

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

SBUMNER = 1234559 = SBUMNER

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

THE

BALANCING

ONE TWO THREE FOUR

FIVE

NINE EIGHT SEVEN SIX

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

1234 5 6789

 

 

prime number - Whatis Techtarget
https://whatis.techtarget.com/definition/prime-number
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

 


Prime number - Wikipedia
https://en.wikipedia.org/wiki/Prime_number

The first 25 prime numbers (all the prime numbers less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in the OEIS).
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several proofs of the theorem.

 

Prime numbers
www-groups.dcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html
Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to ...
Prime numbers

Number theory index History Topics Index

Version for printing

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

 

PRIME NUMBERS

 

P
=
2
-
5
PRIME
61
34
7
N
=
2
-
7
NUMBERS
92
38
2
-
-
17
-
12
Add to Reduce
153
72
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
8
-
3
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
34
-
11
-
61
34
34
-
-
-
-
-
-
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
10
-
92
38
29
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

6 +8 = 14 1+4 = 5

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
7
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
7
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
7
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
7
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
7
9

 

 

I = 9 9 = I
ME = 9 9 = ME
BRAIN + BODY = 9 9 = BODY + BRAIN
LIGHT + DARK = 9 9 = DARK + LIGHT
ENERGY + MASS = 9 9 = MASS +ENERGY
MIND + MATTER = 9 9 = MATTER + MIND
MAGNETIC + FIELD = 9 9 = FIELD + MAGNETIC
POSITIVE + NEGATIVE = 9 9 = NEGATIVE + POSITIVE
973 OM AZAZAZAZAZAZAZAZAZZAZAZAZAZAZAZAZAZAOM 973

 

 

-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
-
-
-
-
4
FOUR
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
18
7
A
L
G
E
B
R
A
-
-
27
-
-
7
-
28
1+8
-
-
-
-
-
-
9
-
-
-
2+7
-
-
-
-
2+8
9
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
10
-
-
1
3
7
5
2
9
1
-
-
-
-
-
-
-
1+0
9
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
1

 

 

7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
7
A
L
G
E
B
R
A
-
-
27
-
-
7
-
28
-
-
-
-
-
-
9
-
-
-
2+7
-
-
-
-
2+8
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
10
-
1
3
7
5
2
9
1
-
-
-
-
-
-
-
1+0
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
1

 

 

-
-
-
-
9
CORRECTED
-
-
-
-
1
2
3
4
5
6
7
8
9
C
=
3
-
1
C
3
3
3
-
1
-
3
-
-
-
7
8
-
O
=
6
-
1
O
15
6
6
-
1
-
-
-
-
6
7
8
-
R
=
9
-
1
R
18
9
9
-
1
-
-
-
-
-
7
8
9
R
=
9
-
1
R
18
9
9
-
1
-
-
-
-
-
7
8
9
E
=
5
-
1
E
5
5
5
-
1
-
-
-
5
-
7
8
-
C
=
3
-
1
C
3
3
3
-
1
-
3
-
-
-
7
8
-
T
=
2
-
1
T
20
2
2
-
1
2
-
-
-
-
7
8
-
E
=
5
-
1
E
5
5
5
-
1
-
-
-
5
-
7
8
-
D
=
4
-
1
D
4
4
4
-
1
-
-
4
-
-
7
8
-
-
-
46
-
9
CORRECTED
91
46
46
-
1
2
6
4
10
6
7
8
18
-
-
4+6
-
-
-
9+1
4+6
4+6
-
-
-
-
-
1+0
-
-
-
1+8
-
-
10
-
9
CORRECTED
10
10
10
-
1
2
6
4
1
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
9
CORRECTED
1
1
1
-
1
2
6
4
1
6
7
8
9

 

 

-
-
-
-
9
CORRECTED
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
T
20
2
2
-
1
2
-
-
-
-
7
8
-
C
=
3
-
1
C
3
3
3
-
1
-
3
-
-
-
7
8
-
C
=
3
-
1
C
3
3
3
-
1
-
3
-
-
-
7
8
-
D
=
4
-
1
D
4
4
4
-
1
-
-
4
-
-
7
8
-
E
=
5
-
1
E
5
5
5
-
1
-
-
-
5
-
7
8
-
E
=
5
-
1
E
5
5
5
-
1
-
-
-
5
-
7
8
-
O
=
6
-
1
O
15
6
6
-
1
-
-
-
-
6
7
8
-
R
=
9
-
1
R
18
9
9
-
1
-
-
-
-
-
7
8
9
R
=
9
-
1
R
18
9
9
-
1
-
-
-
-
-
7
8
9
-
-
46
-
9
CORRECTED
91
46
46
-
1
2
6
4
10
6
7
8
18
-
-
4+6
-
-
-
9+1
4+6
4+6
-
-
-
-
-
1+0
-
-
-
1+8
-
-
10
-
9
CORRECTED
10
10
10
-
1
2
6
4
1
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
9
CORRECTED
1
1
1
-
1
2
6
4
1
6
7
8
9

 

 

9
CORRECTED
-
-
-
-
1
2
3
4
5
6
7
8
9
1
C+O
18
9
9
-
1
2
3
4
-
6
7
8
9
1
R
18
9
9
-
1
2
3
4
-
6
7
8
9
1
R
18
9
9
-
1
2
3
4
-
6
7
8
9
1
E
5
5
5
-
1
2
3
4
5
6
7
8
-
1
C+T
23
5
5
-
1
2
3
4
5
6
7
8
-
1
E+D
9
9
9
-
1
2
3
4
-
6
7
8
9
9
CORRECTED
91
46
46
-
1
2
3
4
10
6
7
8
36
-
-
9+1
4+6
4+6
-
-
-
-
-
1+0
-
-
-
3+6
9
CORRECTED
10
10
10
-
1
2
3
4
1
6
7
8
9
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
9
CORRECTED
1
1
1
-
1
2
3
4
1
6
7
8
9

 

 

9
CORRECTED
-
-
-
-
5
9
1
C+O
18
9
9
-
-
9
1
R
18
9
9
-
-
9
1
R
18
9
9
-
-
9
1
E
5
5
5
-
5
-
1
C+T
23
5
5
-
5
-
1
E+D
9
9
9
-
-
9
9
CORRECTED
91
46
46
-
10
36
-
-
9+1
4+6
4+6
-
1+0
3+6
9
CORRECTED
10
10
10
-
1
9
-
-
1+0
1+0
1+0
-
-
-
9
CORRECTED
1
1
1
-
1
9

 

 

ALGEBRA

 

-
-
-
-
7
ALGEBRA
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
7
ALGEBRA
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
L
=
3
2
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
G
=
7
3
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
B
=
2
5
1
B
2
2
2
-
-
2
-
4
-
6
-
8
-
R
=
9
6
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
7
ALGEBRA
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
B
=
2
5
1
B
2
2
2
-
-
2
-
4
-
6
-
8
-
L
=
3
2
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
G
=
7
3
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
R
=
9
6
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
7
ALGEBRA
-
-
-
-
1
2
3
5
7
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
B
=
2
5
1
B
2
2
2
-
-
2
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
5
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
7
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
9
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
5
7
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
5
7
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
5
7
9

 

 

-
-
-
-
9
EQUATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
1
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
Q
=
7
2
1
Q
17
8
8
-
-
-
-
-
-
-
-
8
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
O
=
3
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
7
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
S
=
1
9
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
28
-
9
EQUATIONS
121
49
40
-
2
2
3
4
10
6
7
8
9
-
-
2+8
-
-
-
1+2+1
4+9
4+0
-
-
-
-
-
1+0
-
-
-
-
-
-
10
-
9
EQUATIONS
4
13
4
-
2
2
3
4
1
6
7
8
9
-
-
1+0
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
9
EQUATIONS
4
4
4
-
2
2
3
4
1
6
7
8
9

 

 

-
-
-
-
9
EQUATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
1
1
E
5
5
5
-
-
-
-
4
5
-
7
-
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
4
-
-
7
8
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
-
7
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
4
-
-
7
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
4
-
-
7
-
-
I
=
9
6
1
I
9
9
9
-
-
-
-
4
-
-
7
-
9
O
=
6
7
1
O
15
6
6
-
-
-
-
4
-
6
7
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
4
5
-
7
-
-
S
=
1
9
1
S
19
10
1
-
1
-
-
4
-
-
7
-
-
-
-
40
-
9
EQUATIONS
121
49
40
-
2
2
3
4
10
6
7
8
9
-
-
4+0
-
-
-
1+2+1
4+9
4+0
-
-
-
-
-
1+0
-
-
-
-
-
-
4
-
9
EQUATIONS
4
13
4
-
2
2
3
4
1
6
7
8
9
-
-
-
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
9
EQUATIONS
4
4
4
-
2
2
3
4
1
6
7
8
9

 

 

-
-
-
-
9
EQUATIONS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
4
1
A
1
1
1
-
1
-
-
4
-
-
7
-
-
S
=
1
9
1
S
19
10
1
-
1
-
-
4
-
-
7
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
4
-
-
7
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
-
7
-
-
E
=
5
1
1
E
5
5
5
-
-
-
-
4
5
-
7
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
4
5
-
7
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
4
-
6
7
-
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
4
-
-
7
8
-
I
=
9
6
1
I
9
9
9
-
-
-
-
4
-
-
7
-
9
-
-
40
-
9
EQUATIONS
121
49
40
-
2
2
3
4
10
6
7
8
9
-
-
4+0
-
-
-
1+2+1
4+9
4+0
-
-
-
-
-
1+0
-
-
-
-
-
-
4
-
9
EQUATIONS
4
13
4
-
2
2
3
4
1
6
7
8
9
-
-
-
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
9
EQUATIONS
4
4
4
-
2
2
3
4
1
6
7
8
9

 

 

-
-
-
-
9
EQUATIONS
-
-
-
-
1
2
3
5
6
8
9
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
S
=
1
9
1
S
19
10
1
-
1
-
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
-
E
=
5
1
1
E
5
5
5
-
-
-
-
5
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
5
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
6
-
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
-
-
8
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
9
-
-
40
-
9
EQUATIONS
121
49
40
-
2
2
3
10
6
8
9
-
-
4+0
-
-
-
1+2+1
4+9
4+0
-
-
-
-
1+0
-
-
-
-
-
4
-
9
EQUATIONS
4
13
4
-
2
2
3
1
6
8
9
-
-
-
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
9
EQUATIONS
4
4
4
-
2
2
3
1
6
8
9

 

 

T
=
2
-
1
THE
33
15
6
E
=
5
-
1
EQUATION
102
39
3
-
-
4
-
9
Add to Reduce
135
54
9
-
-
-
-
-
Reduce to Deduce
1+3+5
5+4
-
-
-
-
-
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

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
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
2
3
4
5
6
7
8
9
K
-
2
1
1
K
11
2
2
-
-
2
-
4
-
6
-
8
-
I
-
9
2
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
N
-
5
3
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
T
-
2
5
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
I
-
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
C
-
3
7
1
C
3
3
3
-
-
-
3
4
-
6
-
8
-
-
-
46
-
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ENERGIES
-
-
-
-
-
-
-
-
-
-
-
-
-
E
-
5
8
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
N
-
5
9
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
10
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
R
-
9
11
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
G
-
7
12
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
I
-
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
E
-
5
14
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
S
-
1
15
1
S
19
10
1
-
1
-
-
4
-
6
-
8
-
-
-
46
-
8
ENERGIES
82
55
46
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
8+1
K
=
2
Q
7
KINETIC
71
35
35
-
1
4
3
4
3
6
7
8
9
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
4
3
4
3
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
4
3
4
3
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
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
2
3
4
5
6
7
8
9
K
-
2
1
1
K
11
2
2
-
-
2
-
4
-
6
-
8
-
I
-
9
2
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
N
-
5
3
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
T
-
2
5
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
I
-
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
C
-
3
7
1
C
3
3
3
-
-
-
3
4
-
6
-
8
-
E
-
5
8
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
N
-
5
9
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
10
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
R
-
9
11
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
G
-
7
12
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
I
-
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
E
-
5
14
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
S
-
1
15
1
S
19
10
1
-
1
-
-
4
-
6
-
8
-
-
-
-
-
-
ENERGIES
-
-
-
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
8+1
K
=
2
Q
7
KINETIC
71
35
35
-
1
4
3
4
3
6
7
8
9
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
4
3
4
3
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
4
3
4
3
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
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
2
3
4
5
6
7
8
9
S
-
1
15
1
S
19
10
1
-
1
-
-
4
-
6
-
8
-
K
-
2
1
1
K
11
2
2
-
-
2
-
4
-
6
-
8
-
T
-
2
5
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
C
-
3
7
1
C
3
3
3
-
-
-
3
4
-
6
-
8
-
N
-
5
3
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
8
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
N
-
5
9
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
10
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
E
-
5
14
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
G
-
7
12
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
I
-
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
I
-
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
R
-
9
11
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
I
-
9
2
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
-
ENERGIES
-
-
-
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
8+1
K
=
2
Q
7
KINETIC
71
35
35
-
1
4
3
4
3
6
7
8
9
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
4
3
4
3
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
4
3
4
3
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
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
2
3
5
7
9
S
-
1
15
1
S
19
10
1
-
1
-
-
-
-
-
K
-
2
1
1
K
11
2
2
-
-
2
-
-
-
-
T
-
2
5
1
T
20
2
2
-
-
2
-
-
-
-
C
-
3
7
1
C
3
3
3
-
-
-
3
-
-
-
N
-
5
3
1
N
14
5
5
-
-
-
-
5
-
-
E
-
5
4
1
E
5
5
5
-
-
-
-
5
-
-
E
-
5
8
1
E
5
5
5
-
-
-
-
5
-
-
N
-
5
9
1
N
14
5
5
-
-
-
-
5
-
-
E
-
5
10
1
E
5
5
5
-
-
-
-
5
-
-
E
-
5
14
1
E
5
5
5
-
-
-
-
5
-
-
G
-
7
12
1
G
7
7
7
-
-
-
-
-
7
-
I
-
9
6
1
I
9
9
9
-
-
-
-
-
-
9
I
-
9
13
1
I
9
9
9
-
-
-
-
-
-
9
R
-
9
11
1
R
18
9
9
-
-
-
-
-
-
9
I
-
9
2
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
-
ENERGIES
-
-
-
-
1
4
3
30
7
36
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
8+1
K
=
2
Q
7
KINETIC
71
35
35
-
1
4
3
3
7
9
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
4
3
3
7
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
4
3
3
7
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

 

 

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

 

 

-
-
-
-
-
C
-
-
-
-
-
-
-
-
-
-
R
-
-
-
-
-
-
-
-
-
-
U
-
-
-
-
-
-
-
-
-
-
C
-
-
-
-
-
-
-
-
-
-
I
-
-
-
-
-
C
R
U
C
I
F
I
X
I
O
N
-
-
-
-
-
I
-
-
-
-
-
-
-
-
-
-
X
-
-
-
-
-
-
-
-
-
-
I
-
-
-
-
-
-
-
-
-
-
O
-
-
-
-
-
-
-
-
-
-
N
-
-
-
-
-

 

 

-
-
-
-
C
-
-
-
-
-
-
-
-
R
-
-
-
-
-
-
-
-
U
-
-
-
-
-
-
-
-
C
-
-
-
-
C
R
U
C
I
F
I
E
D
-
-
-
-
F
-
-
-
-
-
-
-
-
I
-
-
-
-
-
-
-
-
E
-
-
-
-
-
-
-
-
D
-
-
-
-

 

 

-
-
-
-
A
-
-
-
-
-
-
-
-
T
-
-
-
-
-
-
-
-
O
-
-
-
-
-
-
-
-
N
-
-
-
-
A
T
O
N
E
M
E
N
T
-
-
-
-
M
-
-
-
-
-
-
-
-
E
-
-
-
-
-
-
-
-
N
-
-
-
-
-
-
-
-
T
-
-
-
-

 

 

`-
16
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
6
-
-
-
-
-
+
=
12
1+2
=
3
=
3
=
3
-
-
-
-
-
-
-
15
-
-
-
-
-
15
-
-
-
-
-
+
=
30
3+0
=
3
=
3
=
3
`-
16
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
9
5
5
4
-
4
-
4
5
4
-
3
9
1
3
7
+
=
69
6+9
=
15
1+5
6
=
6
-
-
6
18
5
5
4
-
13
-
4
5
13
-
3
18
1
3
25
+
=
123
1+2+3
=
6
=
6
=
6
`-
16
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
18
5
5
4
15
13
-
4
5
13
15
3
18
1
3
25
+
=
153
1+5+3
=
9
=
9
=
9
-
-
6
9
5
5
4
6
4
-
4
5
4
6
3
9
1
3
7
+
=
81
8+1
=
9
=
9
=
9
-
16
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
2
--
--
--
--
--
--
--
-
-
  
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
1
-
-
-
4
-
4
-
4
-
4
-
-
-
-
-
-
-
-
4
occurs
x
4
=
16
1+6
7
-
-
  
-
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
6
-
-
-
-
6
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
  
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
8
--
--
--
--
--
--
--
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
10
16
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
35
-
-
16
-
81
-
45
1+0
1+6
  
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+5
-
-
1+6
-
8+1
-
4+5
1
7
F
R
E
E
D
O
M
-
D
E
M
O
C
R
A
C
Y
-
-
8
-
-
7
-
9
-
9

 

 

16
FREEDOM DEMOCRACY
-
-
-
-
F
6
6
6
-
R
18
9
9
-
E+E+D+O+M
42
24
6
-
D+E
9
9
9
-
M
13
4
4
-
O+C
18
9
9
-
R
18
9
9
-
A+C+Y
29
11
2
16
FREEDOM DEMOCRACY
-
-
-

 

 

16
FREEDOM DEMOCRACY
-
-
-
-
F
6
6
6
-
R
18
9
9
-
E+E+D+O+M
42
24
6
-
D+E
9
9
9
-
M+O+C+R+A+C+Y
78
33
6
16
FREEDOM DEMOCRACY
153
81
36
1+6
-
1+5+3
8+1
3+6
7
FREEDOM DEMOCRACY
9
9
9

 

 

3
THE
33
15
6
6
ZODIAC
58
31
4
2
OF
21
12
3
8
DENDERAH
59
41
5
19
First Total
171
99
18
1+9
Add to Reduce
1+7+1
9+9
1+8
10
Second Total
9
18
9
1+0
Reduce to Deduce
-
1+8
-
1
Essence of Number
9
9
9

 

 

8
THIRTEEN
99
45
9
6
ELEVEN
63
27
9

 

 

-
THIRTY THREE
-
-
-
6
THIRTY
100
37
1
5
THREE
56
29
2
11
THIRTY THREE
156
66
3
1+1
-
1+5+6
6+6
-
2
THIRTY THREE
12
12
3
-
-
1+2
1+2
-
2
THIRTY THREE
3
3
3

 

 

-t-
THE MYTH
-
-
-
3
THE
33
15
6
4
MYTH
66
21
3
7
THE MYTH
99
36
9
-
-
9+9
3+6
-
7
THE MYTH
18
9
9
-
-
1+8
-
-
7
THE MYTH
9
9
9

 

 

6
LIVING
73
37
1
3
THE
33
15
6
4
MYTH
66
21
3
13
First Total
172
73
10
1+3
Add to Reduce
1+7+2
7+3
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

3
THE
33
15
6
4
MYTH
66
21
3
5
LIVES
67
22
4
12
First Total
166
58
13
1+2
Add to Reduce
1+6+6
5+8
1+3
3
Second Total
13
13
4
-
Reduce to Deduce
1+3
1+3
-
3
Essence of Number
4
4
4

 

 

4
AEON
35
17
8
5
AEONS
54
18
9

 

 

4
MIND
40
22
4
7
GRAVITY
102
39
3
11
Add to Reduce
142
61
7
1+1
Reduce to Deduce
1+4+2
6+1
-
2
Essence of Number
7
7
7

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
-
13
GRAVITATIONAL
149
59
5
F
=
6
-
5
FIELD
36
27
9
-
-
15
-
21
First Total
218
101
20
-
-
1+5
-
2+1
Add to Reduce
2+1+8
1+0+1
2+0
-
-
6
-
3
Second Total
11
2
2
-
-
-
-
-
Reduce to Deduce
1+1
-
-
-
-
6
-
3
Essence of Number
2
2
2

 

 

T
=
2
-
3
THE
33
15
6
F
=
6
-
5
FORCE
47
29
2
O
=
6
-
2
OF
21
12
3
G
=
7
-
7
GRAVITY
102
39
3
-
-
21
-
17
First Total
203
95
14
-
-
2+1
-
1+7
Add to Reduce
2+0+3
9+5
1+4
-
-
3
-
8
Second Total
5
14
5
-
-
-
-
-
Reduce to Deduce
-
1+4
-
-
-
3
-
8
Essence of Number
5
5
5

 

 

-
7
G
R
A
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
-
-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
-
7
G
R
A
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
9
1
4
-
2
7
+
=
30
3+0
=
3
=
3
=
3
-
-
7
18
1
22
-
20
25
+
=
93
9+3
=
12
1+2
3
=
3
-
7
G
R
A
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
18
1
22
9
20
25
+
=
102
1+0+2
=
3
=
3
=
3
-
-
7
9
1
4
9
2
7
+
=
39
3+9
=
12
1+2
3
=
3
-
7
G
R
A
V
I
T
Y
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
--
-
-`-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
SIX
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
22
7
G
R
A
V
I
T
Y
-
-
23
-
1
7
-
39
-
21
2+2
-
-
9
-
-
9
-
-
-
-
2+3
-
-
-
-
3+9
-
2+1
4
7
G
R
A
V
I
T
Y
-
-
5
-
1
7
-
12
-
3
-
-
7
9
1
4
9
2
7
-
-
-
-
-
-
-
1+2
-
-
4
7
G
R
A
V
I
T
Y
-
-
5
-
1
7
-
3
-
3

 

 

7
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
`-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
`-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
7
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
7
9
1
4
-
2
7
+
=
30
3+0
=
3
=
3
=
3
`-
7
18
1
22
-
20
25
+
=
93
9+3
=
12
1+2
3
=
3
7
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
`-
7
18
1
22
9
20
25
+
=
102
1+0+2
=
3
=
3
=
3
-
7
9
1
4
9
2
7
+
=
39
3+9
=
12
1+2
3
=
3
7
G
R
A
V
I
T
Y
-T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-`-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-`-
7
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
9
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
7
G
R
A
V
I
T
Y
-
-
23
-
1
7
-
39
-
21
-
-
9
-
-
9
-
-
-
-
2+3
-
-
-
-
3+9
-
2+1
7
G
R
A
V
I
T
Y
-
-
5
-
1
7
-
12
-
3
-
7
9
1
4
9
2
7
-
-
-
-
-
-
-
1+2
-
-
7
G
R
A
V
I
T
Y
-
-
5
-
1
7
-
3
-
3

 

 

GODS RA VITALITY GRAVITY VITALITY RA GODS

 

G
=
7
-
7
GRAVITY
102
39
3
G
=
7
-
4
GODS
45
18
9
R
=
9
-
2
RA
19
10
1
V
=
4
-
8
VITALITY
118
37
1
-
-
27
-
21
First Total
284
104
14
-
-
2+7
-
2+1
Add to Reduce
2+8+4
9+5
1+4
-
-
9
-
3
Second Total
14
14
5
-
-
-
-
-
Reduce to Deduce
1+4
1+4
-
-
-
9
-
3
Essence of Number
5
5
5

 

 

4
DOPA
36
18
9
5
TRIBE
54
27
9
9
Add to Reduce
90
45
18
-
Reduce to Deduce
9+0
4+5
 1+8
9
Essence of Number
9
9
9

 

 

5
FIELD
36
27
9
2
OF
21
12
3
6
DREAMS
60
24
6

 

 

7
ELECTRO
78
33
6
9
MAGNETISM
101
38
2

 

 

4
FOUR
-
-
-
-
F
6
6
6
-
O+U+R
54
18
9
4
FOUR
60
24
15
-
-
6+0
2+4
1+5
4
FOUR
6
6
6

 

 

--
FOUR-FOR
-
-
-
4
FOUR
60
24
6
3
FOR
39
21
3
7
FOUR FOR
99
45
9
-
-
9+9
4+5
-
7
FOUR FOR
18
9
9
-
-
1+8
-
-
7
FOUR FOR
9
9
9

 

 

-
5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
6
-
+
=
12
1+2
=
3
=
3
=
3
-
-
-
-
24
15
-
+
=
39
3+9
=
12
1+2
3
=
3
-
5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
-
3
3
-
-
9
+
=
15
1+5
=
6
=
6
=
6
-
-
12
21
-
-
18
+
=
51
5+1
=
6
=
6
=
6
-
5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
-
12
21
24
15
18
+
=
90
9+0
=
9
=
9
=
9
-
-
3
3
6
6
9
+
=
27
2+7
=
9
=
9
=
9
-
5
L
U
X
O
R
-
-
-
-
-
-
-
-
-
-
1
-
-
--
--
--
--
--
--
1
--
--
--
--
--
--
--
2
-
-
--
--
--
--
--
--
2
--
--
--
--
--
--
--
-
-
3
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
--
-
-
-
-
-
4
--
--
--
--
--
--
--
5
-
-
--
-
-
-
-
-
5
--
--
--
--
--
--
--
-
--
-
6
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
--
-
-
-
-
-
7
--
--
--
--
--
--
--
8
-
-
--
-
-
-
-
-
8
--
--
--
--
--
--
--
-
-
--
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
27
5
L
U
X
O
R
-
-
18
-
-
5
-
27
-
18
2+7
-
--
-
-
-
9
-
-
1+8
-
-
-
-
2+7
-
1+8
9
5
L
U
X
O
R
-
-
9
-
-
5
-
9
-
9

 

 

5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
-
-
6
6
-
+
=
12
1+2
=
3
=
3
=
3
-
-
-
24
15
-
+
=
39
3+9
=
12
1+2
3
=
3
5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
3
3
-
-
9
+
=
15
1+5
=
6
=
6
=
6
-
12
21
-
-
18
+
=
51
5+1
=
6
=
6
=
6
5
L
U
X
O
R
-
-
-
-
-
--
-
-
-
-
-
12
21
24
15
18
+
=
90
9+0
=
9
=
9
=
9
-
3
3
6
6
9
+
=
27
2+7
=
9
=
9
=
9
5
L
U
X
O
R
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
6
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
5
L
U
X
O
R
-
-
18
-
-
5
-
27
-
18
-
-
-
-
-
9
-
-
1+8
-
-
-
-
2+7
-
1+8
5
L
U
X
O
R
-
-
9
-
-
5
-
9
-
9

 

 

-
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
-
-
-
--
-
-
-
-
-
-
9
-
6
5
-
-
-
-
5
-
-
1
+
=
26
2+6
=
8
=
8
=
8
-
-
9
-
15
14
-
-
-
-
14
-
-
19
+
=
71
7+1
=
8
=
8
=
8
-
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
-
-
-
--
-
-
-
-
-
-
9
-
6
5
-
4
1
7
-
5
2
-
+
=
28
2+8
=
10
1+0
1
=
1
-`
-
9
-
15
14
-
13
1
7
-
5
20
-
+
=
64
6+4
=
10
1+0
1
=
1
-
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
-
-
-
--
-
-
-
-
-
-
9
18
15
14
-
13
1
7
14
5
20
19
+
=
135
1+3+5
=
9
=
9
=
9
-
-
9
9
6
5
-
4
1
7
5
5
2
1
+
=
54
5+4
=
9
=
9
=
9
-
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
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
=
6
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
11
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
34
-
-
11
-
54
-
36
1+1
-
9
9
-
-
-
-
-
-
-
-
-
-
-
-
3+4
-
-
1+1
-
5+4
-
3+6
11
11
I
R
O
N
-
M
A
G
N
E
T
S
-
-
7
-
-
2
-
9
-
9

 

 

12
IRRESISTIBLE
137
56
2
5
FORCE
47
29
2
17
First Total
184
85
4
1+7
Add to Reduce
1+8+4
8+5
-
8
Second Total
13
13
4
-
Reduce to Deduce
1+3
1+3
-
8
Essence of Number
4
4
4

 

 

9
IMMOVABLE
92
38
2
6
OBJECT
55
19
1
15
First Total
147
57
3
1+5
Add to Reduce
1+4+7
5+7
-
6
Second Total
12
12
3
-
Reduce to Deduce
1+2
1+2
-
6
Essence of Number
3
3
3

 

 

8
STARTING
108
36
9
9
FINISHING
131
59
5
3
ING
30
21
3

 

 

5
START
78
15
6
6
FINISH
65
38
2

 

 

-
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
--
-
-
6
-
-
-
6
5
1
-
9
6
-
1
+
=
34
3+4
=
7
-
-
=
7
-
`-
-
15
-
-
-
15
14
19
-
9
15
-
19
+
=
106
1+0+6
=
7
-
-
=
7
-
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
-
-
7
-
4
-
3
-
-
-
3
-
-
3
-
+
=
20
2+0
=
2
-
-
=
2
-
`-
7
-
4
-
3
-
-
-
3
-
-
21
-
+
=
38
3+8
=
11
1+1
-
=
2
-
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
-
-
-
-
-
-
-
-
-
`-
7
15
4
-
3
15
14
19
3
9
15
21
19
+
=
144
1+4+4
=
9
=
9
=
9
-
-
7
6
4
-
3
6
5
1
3
9
6
3
1
+
=
54
5+4
=
9
=
9
=
9
-
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
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
-
-
-
-
6
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
35
-
-
12
-
54
-
45
1+0
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+5
-
-
1+2
-
5+4
-
4+5
1
12
G
O
D
-
C
O
N
S
C
I
O
U
S
-
-
8
-
-
3
-
9
-
9

 

 

-
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
--
-
-
6
5
1
-
9
6
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
`-
-
15
14
19
-
9
15
-
19
+
=
64
6+4
=
10
1+0
1
=
1
-
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
-
-
3
-
-
-
3
-
-
3
-
+
=
9
-
=
9
-
-
=
9
-
`-
3
-
-
-
3
-
-
21
-
+
=
27
2+7
=
9
-
-
=
9
-
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
-
`-
3
15
14
19
3
9
15
21
19
+
=
118
1+1+8
=
10
1+0
1
=
1
-
-
3
6
5
1
3
9
6
3
1
+
=
37
3+7
=
10
1+0
1
=
1
-
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
6
-
-
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
21
9
C
O
N
S
C
I
O
U
S
-
-
24
-
-
9
-
37
-
28
2+1
-
-
-
-
-
-
9
-
-
-
-
-
2+4
-
-
-
-
3+7
-
2+8
3
9
C
O
N
S
C
I
O
U
S
-
-
6
-
-
9
-
10
-
10
-
-
3
6
5
1
3
9
6
3
1
-
-
-
-
-
-
-
1+0
-
1+0
3
9
C
O
N
S
C
I
O
U
S
-
-
6
-
-
9
-
1
-
1

 

 

9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
-
-
6
5
1
-
9
6
-
1
+
=
28
2+8
=
10
1+0
1
=
1
`-
-
15
14
19
-
9
15
-
19
+
=
64
6+4
=
10
1+0
1
=
1
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
-
3
-
-
-
3
-
-
3
-
+
=
9
-
=
9
-
-
=
9
`-
3
-
-
-
3
-
-
21
-
+
=
27
2+7
=
9
-
-
=
9
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
--
-
-
-
-
`-
3
15
14
19
3
9
15
21
19
+
=
118
1+1+8
=
10
1+0
1
=
1
-
3
6
5
1
3
9
6
3
1
+
=
37
3+7
=
10
1+0
1
=
1
9
C
O
N
S
C
I
O
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
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
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
9
C
O
N
S
C
I
O
U
S
-
-
24
-
-
9
-
37
-
28
-
-
-
-
-
-
9
-
-
-
-
-
2+4
-
-
-
-
3+7
-
2+8
9
C
O
N
S
C
I
O
U
S
-
-
6
-
-
9
-
10
-
10
-
3
6
5
1
3
9
6
3
1
-
-
-
-
-
-
-
1+0
-
1+0
9
C
O
N
S
C
I
O
U
S
-
-
6
-
-
9
-
1
-
1

 

 

-
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
-
`-
-
-
-
9
-
9
15
14
19
+
=
66
6+6
=
12
1+2
3
-
3
-
-
-
-
-
9
-
9
6
5
1
+
=
30
3+0
=
3
-
3
-
3
--
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
9
5
3
-
7
-
-
-
-
+
=
24
2+4
=
6
-
6
-
6
-
`-
18
5
12
-
7
-
-
-
-
+
=
42
4+2
=
6
-
6
-
6
--
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
--
`-
18
5
12
9
7
9
15
14
19
+
=
108
1+0+8
=
9
-
9
-
9
-
-
9
5
3
9
7
9
6
5
1
+
=
54
5+4
=
9
-
9
-
9
-
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
2
--
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
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-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
9
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
14
9
R
E
L
I
G
I
O
N
S
-
-
31
-
-
9
-
54
-
27
1+4
-
9
-
-
9
-
9
-
-
-
-
-
3+1
-
-
-
-
5+4
-
2+7
5
9
R
E
L
I
G
I
O
N
S
-
-
4
-
-
9
-
9
-
9

 

 

9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
`-
-
-
-
9
-
9
15
14
19
+
=
66
6+6
=
12
1+2
3
-
3
-
-
-
-
9
-
9
6
5
1
+
=
30
3+0
=
3
-
3
-
3
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
-
9
5
3
-
7
-
-
-
-
+
=
24
2+4
=
6
-
6
-
6
`-
18
5
12
-
7
-
-
-
-
+
=
42
4+2
=
6
-
6
-
6
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
--
-
-
-
-
`-
18
5
12
9
7
9
15
14
19
+
=
108
1+0+8
=
9
-
9
-
9
-
9
5
3
9
7
9
6
5
1
+
=
54
5+4
=
9
-
9
-
9
9
R
E
L
I
G
I
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
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
9
R
E
L
I
G
I
O
N
S
-
-
31
-
-
9
-
54
-
27
-
9
-
-
9
-
9
-
-
-
-
-
3+1
-
-
-
-
5+4
-
2+7
9
R
E
L
I
G
I
O
N
S
-
-
4
-
-
9
-
9
-
9

 

 

 

 

0
-
Z
=
8
-
4
ZERO
64
28
1
1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
56
29
2
3
-
T
=
2
-
5
THREE
52
16
7
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
65
20
2
6
-
S
=
1
-
3
SIX
42
24
6
7
-
S
=
1
-
5
SEVEN
58
13
4
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
56
29
2
-
-
2
-
-
-
-
-
-
-
3
-
T
=
2
-
5
THREE
52
16
7
-
-
-
-
-
-
-
7
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
65
20
2
-
-
2
-
-
-
-
-
-
-
6
-
S
=
1
-
3
SIX
42
24
6
-
-
-
-
-
-
6
-
-
-
7
-
S
=
1
-
5
SEVEN
58
13
4
-
-
-
-
4
-
-
-
-
-
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
2
4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
7
2
-
T
=
2
-
3
TWO
56
29
2
-
-
2
-
-
-
3
-
T
=
2
-
5
THREE
52
16
7
-
-
-
-
-
7
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
6
-
5
-
F
=
6
-
4
FIVE
65
20
2
-
-
2
-
-
-
6
-
S
=
1
-
3
SIX
42
24
6
-
-
-
-
6
-
7
-
S
=
1
-
5
SEVEN
58
13
4
-
-
-
4
-
-
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
8
18
14
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
8
9
5

 

 

N
U
M
B
E
R
S
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ZERO
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
Z
=
8
-
-
Z
26
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
O
=
5
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
0
-
-
-
27
-
4
ZERO
64
28
2+8
=
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
1
-
-
-
16
-
3
ONE
34
16
1+6
=
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
5
-
W
23
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
2
-
-
-
13
-
3
TWO
58
13
1+3
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
3
-
-
-
29
-
5
THREE
56
29
2+9
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
-
-
U
21
3
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
4
-
-
-
24
-
4
FOUR
60
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
24
-
4
FIVE
42
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
X
=
6
-
-
X
24
6
6
-
-
-
-
-
-
-
-
6
-
-
-
6
-
-
-
16
-
3
SIX
52
16
1+6
=
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
7
-
-
-
20
-
5
SEVEN
65
20
2+0
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
-
-
G
7
7
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
8
-
-
-
31
-
5
EIGHT
49
31
3+1
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
9
-
-
-
24
-
4
NINE
42
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
45
-
-
-
-
70
40
Add
522
225
45
-
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
-
7+0
4+0
Reduce
5+2+2
2+2+5
4+5
-
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
-
7
4
Deduce
9
9
9
-
9
-
2
6
3
8
7
6
7
6
9

 

 

N
U
M
B
E
R
S
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
-
-
-
-
4
ZERO
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
Z
=
8
-
-
Z
26
8
8
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
O
=
5
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
1
-
-
-
-
-
3
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
2
-
-
-
-
-
3
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
5
-
W
23
5
5
-
-
-
-
-
5
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
3
-
-
-
-
-
5
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
4
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
-
-
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
5
-
-
-
-
-
4
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
6
-
-
-
-
-
3
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
X
=
6
-
-
X
24
6
6
-
-
-
-
-
-
6
-
-
-
7
-
-
-
-
-
5
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
8
-
-
-
-
-
5
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
-
-
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
8
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
9
-
-
-
-
-
4
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
45
-
-
-
-
70
40
ADD
522
225
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
-
7+0
4+0
REDUCE
5+2+2
2+2+5
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
-
7
4
DEDUCE
9
9
9
-
2
6
3
8
7
6
7
6
9

 

 

N
U
M
B
E
R
S
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ZERO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Z
=
8
-
1
Z
26
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
0
-
O
=
6
-
4
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
-
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
1
-
E
=
5
-
3
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
-
3
W
23
5
5
-
-
-
-
-
-
-
5
-
-
-
-
2
-
O
=
6
-
3
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
-
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
3
-
E
=
5
-
5
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
-
-
U
21
3
3
-
-
-
-
-
3
-
-
-
-
-
-
4
-
R
=
9
-
4
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
5
-
E
=
5
-
4
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
6
-
X
=
6
-
3
X
24
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
-
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
7
-
N
=
5
-
7
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
-
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
-
-
G
7
7
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
8
-
T
=
2
-
5
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
N
=
5
-
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
-
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
9
-
E
=
5
-
4
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
45
-
-
-
42
-
40
Add
522
225
45
-
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
-
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
6
-
4
Deduce
9
9
9
-
9
-
2
6
3
8
7
6
7
6
9

 

 

ZERO ONE TWO THREEFOUR 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 THREEFOUR FIVE SIX SEVEN EIGHT NINE

 

LETTERS TRANSPOSED INTO NUMBERS REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
-
-
-
-
-
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

 

LETTERS TRANSPOSED INTO NUMBERS REARRANGED IN NUMERICAL ORDER

 

ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

 

1
occurs
x
2
=
2
=
2
2
occurs
x
3
=
6
=
6
3
occurs
x
1
=
3
=
3
4
occurs
x
2
=
8
=
8
5
occurs
x
14
=
70
7+0
7
6
occurs
x
7
=
42
4+2
6
7
occurs
x
1
=
7
=
7
8
occurs
x
3
=
24
2+4
6
9
occurs
x
7
=
63
6+3
9
45
-
-
40
-
225
-
54
4+5
-
-
4+0
-
2+2+5
-
5+4
9
-
-
4
-
9
-
9

 

 

Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
8
-
-
6
-
6
5
-
-
-
-
6
-
-
8
-
-
-
-
-
6
-
-
-
-
9
-
-
-
1
9
6
-
1
-
-
-
5
-
-
9
-
8
-
-
5
9
5
-
112
26
-
-
15
-
15
14
-
-
-
-
15
-
-
8
-
-
-
-
-
15
-
-
-
-
9
-
-
-
19
9
24
-
19
-
-
-
14
-
-
9
-
8
-
-
14
9
14
-
256
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
-
5
9
-
-
-
-
5
-
2
5
-
-
2
-
9
5
5
-
6
-
3
9
-
6
-
4
5
-
-
-
-
-
-
5
4
5
-
-
5
-
7
-
2
-
-
-
-
5
113
-
5
18
-
-
-
-
5
-
20
23
-
-
20
-
18
5
5
-
6
-
21
18
-
6
-
22
5
-
-
-
-
-
-
5
22
5
-
-
5
-
7
-
20
-
-
-
-
5
266
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
26
5
18
15
-
15
14
5
-
20
23
15
-
20
8
18
5
5
-
6
15
21
18
-
6
9
22
5
-
19
9
24
-
19
5
22
5
14
-
5
9
7
8
20
-
14
9
14
5
522
8
5
9
6
-
6
5
5
-
2
5
6
-
2
8
9
5
5
-
6
6
3
9
-
6
9
4
5
-
1
9
6
-
1
5
4
5
5
-
5
9
7
8
2
-
5
9
5
5
225
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
5
-
-
-
--
5
5
-
-
5
-
-
-
-
-
5
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
5
-
-
-
-
-
5
-
5
5
5
-
-
-
6
-
6
-
-
-
-
--
6
-
-
-
-
-
-
-
6
6
-
-
-
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
9
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
45
-
5
-
-
-
--
5
5
-
-
5
-
-
-
-
-
5
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
5
-
-
5
5
-
-
-
-
5
4+5
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
9
8
5
9
6
-
6
5
5
-
2
5
6
-
2
8
9
5
5
-
6
6
3
9
-
6
9
4
5
-
1
9
6
-
1
5
4
5
5
-
5
9
7
8
2
-
5
9
5
5
-
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
9

 

 

1
occurs
x
2
=
2
=
2
2
occurs
x
3
=
6
=
6
3
occurs
x
1
=
3
=
3
4
occurs
x
2
=
8
=
8
5
occurs
x
14
=
70
7+0
7
6
occurs
x
7
=
42
4+2
6
7
occurs
x
1
=
7
=
7
8
occurs
x
3
=
24
2+4
6
9
occurs
x
7
=
63
6+3
9
45
-
-
40
-
225
-
54
4+5
-
-
4+0
-
2+2+5
-
5+4
9
-
-
4
-
9
-
9

 

 

Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
-
8
-
-
6
6
5
-
-
-
6
-
8
-
-
-
-
6
-
-
-
9
-
-
1
9
6
1
-
-
-
5
-
9
-
8
-
5
9
5
-
112
26
-
-
15
15
14
-
-
-
15
-
8
-
-
-
-
15
-
-
-
9
-
-
19
9
24
19
-
-
-
14
-
9
-
8
-
14
9
14
-
256
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
-
-
5
9
-
-
-
5
2
5
-
2
-
9
5
5
6
-
3
9
6
-
4
5
-
-
-
-
5
4
5
-
5
-
7
-
2
-
-
-
5
113
-
5
18
-
-
-
5
20
23
-
20
-
18
5
5
6
-
21
18
6
-
22
5
-
-
-
-
5
22
5
-
5
-
7
-
20
-
-
-
5
266
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
-
26
5
18
15
15
14
5
20
23
15
20
8
18
5
5
6
15
21
18
6
9
22
5
19
9
24
19
5
22
5
14
5
9
7
8
20
14
9
14
5
522
8
5
9
6
6
5
5
2
5
6
2
8
9
5
5
6
6
3
9
6
9
4
5
1
9
6
1
5
4
5
5
5
9
7
8
2
5
9
5
5
225
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
-
5
-
-
--
5
5
-
5
-
-
-
-
5
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
5
5
5
-
-
-
-
5
-
5
5
5
-
-
-
6
6
-
-
-
--
6
-
-
-
-
-
6
6
-
-
6
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
8
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
9
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
9
-
-
9
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
45
-
5
-
-
--
5
5
-
5
-
-
-
-
5
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
5
5
5
-
-
5
5
-
-
-
5
4+5
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
9
8
5
9
6
6
5
5
2
5
6
2
8
9
5
5
6
6
3
9
6
9
4
5
1
9
6
1
5
4
5
5
5
9
7
8
2
5
9
5
5
-
Z
E
R
O
O
N
E
T
W
O
T
H
R
E
E
F
O
U
R
F
I
V
E
S
I
X
S
E
V
E
N
E
I
G
H
T
N
I
N
E
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
-
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
56
29
2
T
=
2
-
5
THREE
52
16
7
F
=
6
-
4
FOUR
60
24
6
F
=
6
-
4
FIVE
65
20
2
S
=
1
-
3
SIX
42
24
6
S
=
1
-
5
SEVEN
58
13
4
E
=
5
-
5
EIGHT
49
31
4
N
=
5
-
4
NINE
42
24
6
-
-
34
-
36
-
458
197
44
-
-
3+4
-
3+6
-
4+5+8
1+9+7
4+4
-
-
7
4
9
-
17
17
8
-
-
-
-
-
-
1+7
1+7
-
-
-
7
-
9
-
8
8
8

 

 

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

 

 

B
=
2
-
7
BREATHE
59
32
5
O
=
6
-
2
ON
29
11
2
M
=
4
-
2
ME
18
9
9
B
=
2
-
6
BREATH
54
27
9
O
=
6
-
2
OF
21
12
3
G
=
7
-
3
GOD
26
17
8
B
-
27
-
22
Add to Reduce
207
180
36
-
-
2+7
-
2+2
Reduce to Deduce
2+0+7
1+8+0
3+6
-
-
9
-
4
Essence of Number
9
9
9

 

 

-
-
-
-
9
NINE
-
-
-
-
-
-
1
-
N
14
5
5
-
-
-
1
-
I
9
9
9
-
-
-
2
-
N+E
19
10
1
N
=
5
4
4
NINE
42
24
15
-
-
-
-
-
-
4+2
2+4
1+5
N
=
5
3
4
NINE
6
6
6

 

 

-
-
-
-
0
ZERO
-
-
-
-
-
-
2
-
Z+E
31
13
4
-
-
-
1
-
R
18
9
9
-
-
-
1
-
O
15
6
6
Z
=
8
4
4
ZERO
64
28
19
-
-
-
-
-
-
9+1
1+9
1+9
Z
=
8
4
-
ZERO
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
Z
=
8
4
-
ZERO
10
10
10

 

 

-
-
-
-
1
ONE
-
-
-
-
-
-
1
-
O
15
6
6
-
-
-
1
-
N
14
5
5
-
-
-
1
-
E
5
5
5
O
=
6
3
4
ONE
34
16
16
-
-
-
-
-
-
3+4
1+6
1+6
O
=
6
3
4
ONE
7
7
7

 

 

-
-
-
-
2
TWO
-
-
-
-
-
-
1
-
T
20
2
2
-
-
-
1
-
W
23
5
5
-
-
-
1
-
O
15
6
6
T
=
2
3
4
TWO
58
13
13
-
-
-
-
-
-
5+8
1+3
1+3
T
=
2
3
4
TWO
13
4
4
-
-
-
-
-
-
1+3
-
-
T
=
2
3
4
TWO
4
4
4

 

 

-
-
-
-
3
THREE
-
-
-
-
-
-
1
-
T
20
2
2
-
-
-
1
-
H
8
8
8
-
-
-
1
-
R
18
9
9
-
-
-
2
-
E+E
10
10
1
T
=
2
5
4
THREE
56
29
20
-
-
-
-
-
-
5+6
2+9
2+0
T
=
2
5
4
THREE
11
11
2
-
-
-
-
-
-
1+1
1+1
-
T
=
2
5
4
THREE
2
2
2

 

 

-
-
-
-
4
FOUR
-
-
-
-
-
-
1
-
F
6
6
6
-
-
-
2
-
O+U
36
9
9
-
-
-
1
-
R
18
9
9
F
=
6
4
4
FOUR
60
24
24
-
-
-
-
-
-
6+0
2+4
2+4
F
-
6
4
4
FOUR
6
6
6

 

 

-
-
-
-
5
FIVE
-
-
-
-
-
-
1
-
F
6
6
6
-
-
-
1
-
I
9
9
9
-
-
-
2
-
V+E
27
9
9
F
=
6
4
4
FIVE
60
24
24
-
-
-
-
-
-
6+0
2+4
2+4
F
=
6
4
4
FIVE
6
6
6

 

 

-
-
-
-
6
SIX
-
-
-
-
-
-
1
-
S
19
10
1
-
-
-
1
-
I
9
9
9
-
-
-
1
-
X
24
6
6
S
=
1
3
4
SIX
52
25
16
-
-
-
-
-
-
5+2
2+5
1+6
S
=
1
3
4
SIX
7
7
7

 

 

-
-
-
-
7
SEVEN
-
-
-
-
-
-
1
-
S
19
10
1
-
-
-
2
-
E+V
27
9
9
-
-
-
2
-
E+N
19
10
1
S
=
1
5
4
SEVEN
65
29
11
-
-
-
-
-
-
6+5
2+9
1+1
S
=
2
5
4
SEVEN
11
11
2
-
-
-
-
-
-
1+1
1+1
-
S
=
2
5
4
SEVEN
2
2
2

 

 

-
-
-
-
8
EIGHT
-
-
-
-
-
-
1
-
E
5
5
5
-
-
-
1
-
I
9
9
9
-
-
-
3
-
G+H+T
35
17
8
E
=
5
5
4
EIGHT
49
31
4
-
-
-
-
-
-
4+9
3+1
-
E
=
5
5
4
EIGHT
13
4
4
-
-
-
-
-
-
1+3
-
-
E
=
5
5
4
EIGHT
4
4
4

 

 

-
-
-
-
9
NINE
-
-
-
-
-
-
1
-
N
14
5
5
-
-
-
1
-
I
9
9
9
-
-
-
2
-
N+E
19
10
1
N
=
5
4
4
NINE
42
24
15
-
-
-
-
-
-
4+2
2+4
1+5
N
=
5
3
4
NINE
6
6
6

 

 

-
-
-
10
-
TEN
--
-
-
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
T
=
2
10
3
TEN
39
12
12
-
-
-
-
-
-
3+9
1+2
1+2
T
=
2
10
3
TEN
12
3
3
-
-
-
-
-
-
1+2
-
-
T
=
2
10
3
TEN
3
3
3

 

 

-
-
-
11
-
ELEVEN
--
-
-
-
-
-
-
3
E+L+E
22
13
4
-
-
-
-
2
V+E
27
9
9
-
-
-
-
1
N
14
5
5
E
=
5
11
6
ELEVEN
63
27
18
-
-
-
-
-
-
6+3
2+7
1+8
E
=
5
11
6
ELEVEN
9
9
9

 

 

-
-
-
11
-
ELEVEN
--
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
L
12
3
3
-
-
-
-
1
E
5
5
5
-
-
-
-
1
V
22
4
4
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
E
=
5
11
6
ELEVEN
63
27
27
-
-
-
-
-
-
6+3
2+7
2+7
E
=
5
11
6
ELEVEN
9
9
9

 

 

-
-
-
12
-
TWELVE
--
-
-
-
-
-
-
4
T+W+E+L
60
15
6
-
-
-
-
2
V+E
27
9
9
T
=
2
12
6
TWELVE
87
24
15
-
-
-
-
-
-
8+7
2+4
1+5
T
=
2
12
6
TWELVE
15
6
6
6
-
-
-
-
-
1+5
-
-
T
=
2
12
6
TWELVE
6
6
6

 

 

-
-
-
12
-
TWELVE
--
-
-
-
-
-
-
1
T
20
2
2
-
-
-
-
1
W
23
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
L
12
3
3
-
-
-
-
1
V
22
4
4
-
-
-
-
1
E
5
5
5
T
=
2
12
6
TWELVE
87
24
24
-
-
-
-
-
-
8+7
2+4
2+4
T
=
2
12
6
TWELVE
15
6
6
6
-
-
-
-
-
1+5
-
-
T
=
2
12
6
TWELVE
6
6
6

 

 

-
-
-
-
TEEN
--
-
-
-
-
-
1
T
20
2
2
-
-
-
1
E
5
5
5
-
-
-
1
E
5
5
5
-
-
-
1
N
14
5
5
T
=
2
4
TEEN
44
17
17
-
-
-
-
-
4+4
1+7
1+7
T
=
2
4
TEEN
8
8
8

 

 

-
-
-
13
-
THIRTEEN
--
-
-
-
-
-
-
2
T+H
28
10
1
-
-
-
-
1
I
9
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
4
T+E+E+N
44
17
8
T
=
2
13
8
THIRTEEN
99
45
27
-
-
-
-
-
-
9+9
4+5
2+7
T
=
2
13
8
THIRTEEN
18
9
9
6
-
-
-
-
-
1+8
-
-
T
=
2
13
8
THIRTEEN
9
9
9

 

 

-
-
-
13
-
THIRTEEN
--
-
-
-
-
-
-
1
T
20
2
2
-
-
-
-
1
H
8
8
8
-
-
-
-
1
I
9
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
T
=
2
13
8
THIRTEEN
99
45
27
-
-
-
-
-
-
9+9
4+5
2+7
T
=
2
13
8
THIRTEEN
18
9
9
6
-
-
-
-
-
1+8
-
-
T
=
2
13
8
THIRTEEN
9
9
9

 

 

-
-
-
14
-
FOURTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
2
O+U
36
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
4
T+E+E+N
44
17
8
F
=
6
14
8
FOURTEEN
104
41
32
-
-
-
-
-
-
1+0+4
4+1
3+2
F
=
6
14
8
FOURTEEN
5
5
5

 

 

-
-
-
14
-
FOURTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
1
O
15
6
6
-
-
-
-
1
U
21
3
3
-
-
-
-
1
R
18
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
F
=
6
14
8
FOURTEEN
104
41
32
-
-
-
-
-
-
1+0+4
4+1
3+2
F
=
6
14
8
FOURTEEN
5
5
5

 

 

-
-
-
15
-
FIFTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
1
I
9
9
9
-
-
-
-
5
F+T+E+E+N
50
23
5
F
=
6
15
7
FIFTEEN
65
38
20
-
-
-
-
-
-
6+5
3+8
2+0
F
=
6
15
7
FIFTEEN
11
11
2
6
-
-
-
-
-
1+1
1+1
-
F
=
6
15
7
FIFTEEN
2
2
2

 

 

-
-
-
15
-
FIFTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
1
I
9
9
9
-
-
-
-
1
F
6
6
6
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
F
=
6
15
7
FIFTEEN
65
38
20
-
-
-
-
-
-
6+5
3+8
2+0
F
=
6
15
7
FIFTEEN
11
11
2
6
-
-
-
-
-
1+1
1+1
-
F
=
6
15
7
FIFTEEN
2
2
2

 

 

-
-
-
16
-
SIXTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
1
I
9
9
9
-
-
-
-
4
X+T+E+E
54
18
9
-
-
-
-
1
N
14
5
5
S
=
1
16
7
SIXTEEN
96
42
24
-
-
-
-
-
-
9+6
4+2
2+4
S
=
1
16
7
SIXTEEN
15
6
6
6
-
-
-
-
-
1+5
-
-
S
=
1
16
7
SIXTEEN
6
6
6

 

 

-
-
-
16
-
SIXTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
1
I
9
9
9
-
-
-
-
1
X
24
6
6
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
S
=
1
16
7
SIXTEEN
96
42
33
-
-
-
-
-
-
9+6
4+2
3+3
S
=
1
16
7
SIXTEEN
15
6
6
6
-
-
-
-
-
1+5
-
-
S
=
1
16
7
SIXTEEN
6
6
6

 

 

-
-
-
17
-
SEVENTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
2
E+V
27
9
9
-
-
-
-
6
E+N+T+E+E+N
63
27
9
S
=
1
17
9
SEVENTEEN
109
46
19
-
-
-
-
-
-
1+0+9
4+6
1+9
S
=
1
17
9
SEVENTEEN
10
10
10
6
-
-
-
-
-
1+0
1+0
1+0
S
=
1
17
9
SEVENTEEN
1
1
1

 

 

-
-
-
17
-
SEVENTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
1
E
5
5
5
-
-
-
-
1
V
22
4
4
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
S
=
1
17
9
SEVENTEEN
109
46
19
-
-
-
-
-
-
1+0+9
4+6
1+9
S
=
1
17
9
SEVENTEEN
10
10
10
6
-
-
-
-
-
1+0
1+0
1+0
S
=
1
17
9
SEVENTEEN
1
1
1

 

 

-
-
-
18
-
EIGHTEEN
--
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
I
9
9
9
-
-
-
-
5
G+H+T+E+E
45
27
9
-
-
-
-
1
N
14
5
5
E
=
5
18
8
EIGHTEEN
73
46
28
-
-
-
-
-
-
7+3
4+6
2+8
E
=
5
18
8
EIGHTEEN
10
10
10
6
-
-
-
-
-
1+0
1+0
1+0
E
=
5
18
8
EIGHTEEN
1
1
1

 

 

-
NINETEEN
171
81
9
-
9
1
N
14
5
5
-
5
1
I
9
9
9
-
9
1
N
14
5
5
-
5
1
E
5
5
5
-
5
1
T
20
2
2
-
9
1
E
5
5
5
-
5
1
E
5
5
5
-
5
1
N
14
5
5
-
5
8
NINETEEN
86
41
41
-
30
-
-
8+6
4+1
4+1
-
3+0
8
NINETEEN
14
5
5
-
3
-
9
1+4
-
-
-
-
8
NINETEEN
5
5
5
-
3

 

 

N
5
-
8
NINETEEN
-
-
-
-
-
-
1
N
14
5
5
-
-
-
1
I
9
9
9
-
-
-
1
N
14
5
5
-
-
-
1
E
5
5
5
-
-
-
1
T
20
2
2
-
-
-
1
E
5
5
5
-
-
-
1
E
5
5
5
-
-
-
1
N
14
5
5
N
5
-
8
NINETEEN
86
41
41
-
-
-
-
-
8+6
4+1
4+1
N
5
-
8
NINETEEN
14
41
41
-
-
-
-
-
1+4
4+1
4+1
N
5
-
8
NINETEEN
5
5
5

 

 

-
TEEN
--
-
-
1
T
20
2
2
3
E+E+N
24
15
6
4
TEEN
44
17
8
-
-
4+4
1+7
-
4
TEEN
8
8
8

 

 

-
-
-
-
TEEN
--
-
-
-
-
-
1
T
20
2
2
-
-
-
1
E
5
5
5
-
-
-
1
E
5
5
5
-
-
-
1
N
14
5
5
T
=
2
4
TEEN
44
17
17
-
-
-
-
-
4+4
1+7
1+7
T
=
2
4
TEEN
8
8
8

 

TEN ELEVEN TWELVE THIRTEEN FOURTEEN FIFTEEN SIXTEEN SEVENTEEN EIGHTEEN NINETEEN

T55 5L5V55 T55LV5 THIRT555 FOURT555 FIFT555 SIXT555 S5V55T555 5IGHT555 5I55T555

TEN ELEVEN TWELVE THIRTEEN FOURTEEN FIFTEEN SIXTEEN SEVENTEEN EIGHTEEN NINETEEN

5 X 37 X 5

LOOK AT THE 5FIVES LOOK AT THE 5FIVES LOOK AT THE 5FIVES THE 5FIVES THE 5FIVES

 

 

ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

O55 T5O THR55 FOUR FIV5 SIX S5V55 5IGHT 5I5

ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE

5 X 13 X 5

TEN ELEVEN TWELVE THIRTEEN FOURTEEN FIFTEEN SIXTEEN SEVENTEEN EIGHTEEN NINETEEN

5 X 37 X 5

 

ZERO 5 X 1 X 5 ZERO

 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

5 X 13 X 5

5 X 37 X 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

5 X 50 X 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

TWENTY 5 X 3 X 5 TWENTY

 

 

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
-
10
MULTIPLIED
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

 

 

 

-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
35
6
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
8
6
A
T
U
M
-
-
1
-
-
4
-
2

 

 

6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
6
A
T
U
M
-
-
10
-
-
4
-
10
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
6
A
T
U
M
-
-
1
-
-
4
-
2

 

AUTUMN ATUM AUTUMN

 

-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
-
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
-
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
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
30
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
3+0
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
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
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
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
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

AUTUMN ATUM AUTUMN

QUANTUM ATUM QUANTUM

 

 

-
QUANTUM
-
-
-
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
3
.....QU...N.............
52
16
7
7
QUANTUM
107
26
8

 

 

-
QUANTUM
-
-
-
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 GODS 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

 

 

PROBLEMS - PROBLEMS

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

LETTERS RE ARRANGED NUMERICALLY

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
9

 

LETTERS RE ARRANGED NUMERICALLY

 

SOLVE PROBLEMS SOLVE

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
35
-
-
-
98
44
35
-
-
-
-
-
-
-
-
-
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
37
-
-
-
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

LETTERS RE ARRANGED NUMERICALLY

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
9

 

 

EVOLVE LOVE EVOLVE

LOVE EVOLVE LOVE

EVOLVE REVOLVE EVOLVE

REVOLVE EVOLVE REVOLVE

SOLVE LOVES SOLVE

 

 

H
=
8
-
4
HOLY
60
24
6
C
=
3
-
10
CONSCIENCE
90
45
9
T
=
2
-
3
THE
33
15
6
I
=
9
-
5
INNER
60
33
6
V
=
4
-
5
VOICE
54
27
9
-
-
26
-
27
First Total
297
144
27
-
-
2+6
-
2+7
Add to Reduce
2+9+7
1+4+4
2+7
Q
-
8
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
8
5
9
Essence of Number
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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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

 

..

 
-
-
11
ATOMIC POWER
163
64
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
25
-
-
-
61
25
25
-
-
-
-
-
-
-
-
-
-
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
8
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
W
=
5
9
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
32
-
-
-
77
32
32
-
1
2
3
4
10
12
7
8
12
-
-
-
-
11
ATOMIC POWER
-
-
-
-
-
-
-
-
1+0
1+2
-
-
1+2
A
=
1
-
6
ATOMIC
61
25
7
-
1
2
3
4
1
3
7
8
9
P
=
7
-
5
POWER
77
32
5
-
-
-
-
-
-
-
-
-
-
-
-
8
-
11
ATOMIC POWER
138
57
12
-
1
2
3
4
1
3
7
8
9
-
-
-
-
1+1
-
1+3+8
5+7
1+2
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
ATOMIC POWER
12
12
3
-
1
2
3
4
1
3
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
SUN EARTH MOON
3
3
3
-
1
2
3
4
1
3
7
8
9

 

 
-
-
11
ATOMIC POWER
163
64
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
8
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
O
=
6
8
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
W
=
5
9
1
W
23
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
1
2
3
4
10
12
7
8
18
-
-
-
-
11
ATOMIC POWER
-
-
-
-
-
-
-
-
1+0
1+2
-
-
1+8
A
=
1
-
6
ATOMIC
61
25
7
-
1
2
3
4
1
3
7
8
9
P
=
7
-
5
POWER
77
32
5
-
-
-
-
-
-
-
-
-
-
-
-
8
-
11
ATOMIC POWER
138
57
12
-
1
2
3
4
1
3
7
8
9
-
-
-
-
1+1
-
1+3+8
5+7
1+2
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
ATOMIC POWER
12
12
3
-
1
2
3
4
1
3
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
ATOMIC POWER
3
3
3
-
1
2
3
4
1
3
7
8
9

 

 
-
-
11
ATOMIC POWER
163
64
19
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
W
=
5
9
1
W
23
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
O
=
6
8
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
P
=
7
7
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
1
2
3
4
10
12
7
8
18
-
-
-
-
11
ATOMIC POWER
-
-
-
-
-
-
-
-
1+0
1+2
-
-
1+8
A
=
1
-
6
ATOMIC
61
25
7
-
1
2
3
4
1
3
7
8
9
P
=
7
-
5
POWER
77
32
5
-
-
-
-
-
-
-
-
-
-
-
-
8
-
11
ATOMIC POWER
138
57
12
-
1
2
3
4
1
3
7
8
9
-
-
-
-
1+1
-
1+3+8
5+7
1+2
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
ATOMIC POWER
12
12
3
-
1
2
3
4
1
3
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
2
ATOMIC POWER
3
3
3
-
1
2
3
4
1
3
7
8
9

 

 

ATOMIC ENERGY

 

-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ATOMIC
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
25
-
-
ATOMIC
61
25
25
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ENERGY
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
11
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
Y
=
7
12
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
38
-
-
ENERGY
74
38
38
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
15
6
14
8
18
-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
1
2
3
4
6
6
5
8
9
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
6
6
5
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC ENERGY
9
9
9
-
1
2
3
4
6
6
5
8
9

 

 

-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
11
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
Y
=
7
12
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
15
6
14
8
18
-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
1
2
3
4
6
6
5
8
9
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
6
6
5
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC ENERGY
9
9
9
-
1
2
3
4
6
6
5
8
9

 

 

-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
8
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
G
=
7
11
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
Y
=
7
12
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
15
6
14
8
18
-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
1
2
3
4
6
6
5
8
9
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
6
6
5
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC ENERGY
9
9
9
-
1
2
3
4
6
6
5
8
9

 

 

-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
11
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
Y
=
7
12
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
4
15
6
14
8
18
-
-
-
-
-
ATOMIC ENERGY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
1
2
3
4
6
6
5
8
9
E
=
5
Q
6
ENERGY
74
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC ENERGY
135
63
9
-
1
2
3
4
6
6
5
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC ENERGY
9
9
9
-
1
2
3
4
6
6
5
8
9

 

LETTERS REARRANGED NUMERICALLY

 

ATOMIC NUMBERS

 

-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
13
ATOMIC NUMBERS
153
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ATOMIC
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
25
-
-
ATOMIC
61
25
25
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
7
NUMBERS
92
38
29
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
4
6
8
10
6
7
8
18
-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
-
-
-
-
1+0
-
-
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
2
4
6
8
1
6
7
8
9
E
=
5
Q
6
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC NUMBERS
153
63
9
-
2
4
6
88
1
6
7
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC NUMBERS
9
9
9
-
2
4
6
8
1
6
7
8
9

 

 

-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
13
ATOMIC NUMBERS
153
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ATOMIC
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
7
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
7
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
7
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
7
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
7
8
9
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
7
8
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
7
8
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
7
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
7
8
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
7
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
7
8
-
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
7
8
9
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
7
8
-
-
-
-
-
-
-
-
-
-
-
2
4
6
8
10
6
7
8
18
-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
-
-
-
-
1+0
-
-
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
2
4
6
8
1
6
7
8
9
E
=
5
Q
6
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC NUMBERS
153
63
9
-
2
4
6
88
1
6
7
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC NUMBERS
9
9
9
-
2
4
6
8
1
6
7
8
9

 

LETTERS REARRANGED NUMERICALLY

 

-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
13
ATOMIC NUMBERS
153
63
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ATOMIC
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
7
8
-
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
7
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
7
8
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
7
8
-
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
7
8
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
7
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
7
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
7
8
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
7
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
7
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
7
8
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
7
8
9
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
7
8
9
-
-
-
-
-
-
-
-
-
-
2
4
6
8
10
6
7
8
18
-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
-
-
-
-
1+0
-
-
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
2
4
6
8
1
6
7
8
9
E
=
5
Q
6
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC NUMBERS
153
63
9
-
2
4
6
88
1
6
7
8
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC NUMBERS
9
9
9
-
2
4
6
8
1
6
7
8
9

 

 

-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
1
2
3
4
5
6
9
K
=
2
Q
6
ATOMIC
61
25
7
-
-
-
-
-
-
-
-
E
=
5
Q
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
Q
Q
7
Q
13
ATOMIC NUMBERS
153
63
9
-
1
2
3
4
5
6
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ATOMIC
-
-
-
-
1
2
3
4
5
6
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
-
C
=
3
6
1
C
3
3
3
-
-
-
3
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
I
=
9
5
1
I
18
9
9
-
-
-
-
-
-
-
9
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
2
4
6
8
10
6
18
-
-
-
-
-
ATOMIC NUMBERS
-
-
-
-
-
-
-
-
1+0
-
1+8
A
=
1
Q
6
ATOMIC
61
25
7
-
2
4
6
8
1
6
9
E
=
5
Q
6
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
Q
Q
6
Q
12
ATOMIC NUMBERS
153
63
9
-
2
4
6
88
1
6
9
-
-
-
-
1+5
-
1+3+5
6+3
-
-
-
-
-
-
-
-
-
Q
Q
6
Q
6
ATOMIC NUMBERS
9
9
9
-
2
4
6
8
1
6
9

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
-
3
GOD
26
26
8
P
=
7
-
8
PARTICLE
84
39
3
-
-
16
-
14
Add to Reduce
143
80
17
-
-
1+6
-
1+4
Reduce to Deduce
1+4+3
8+0
1+7
-
-
7
-
5
Essence of Number
8
8
8

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
-
3
GOD
26
26
8
P
=
7
-
9
PARTICLES
103
40
4
-
-
16
-
15
Add to Reduce
162
81
18
-
-
1+6
-
1+5
Reduce to Deduce
1+6+2
8+1
1+8
-
-
7
-
6
Essence of Number
9
9
9

 

 

Higgs boson - Wikipedia
https://en.wikipedia.org/wiki/Higgs_boson

The Higgs boson is an elementary particle in the Standard Model of particle physics. First suspected to exist in the 1960s, it is the quantum excitation of the Higgs field, a fundamental field of crucial importance to particle physics theory. Unlike other known fields such as the electromagnetic field, it has a non-zero constant ...

Higgs boson
Subatomic particle

The Higgs boson is an elementary particle in the Standard Model of particle physics. First suspected to exist in the 1960s, it is the quantum excitation of the Higgs field, a fundamental field of crucial importance to particle physics theory. Wikipedia

Composition: Elementary particle

Classification: Boson

Symbol: H°

Mass: 125.09±0.21 (stat.)±0.11 (syst.) GeV/c² (CMS+ATLAS)

Electric charge: 0 e

Discovered: Large Hadron Collider (2011–2013)

Mean lifetime: 1.56×10-22 s (predicted)

 

T
=
2
-
3
THE
33
15
6
H
=
8
-
5
HIGGS
50
41
5
B
=
2
-
5
BOSON
65
29
2
-
-
12
-
13
First Total
148
85
13
-
-
1+2
-
1+3
Add to Reduce
1+4+8
8+5
1+3
-
-
3
-
4
Second Total
13
13
4
-
-
-
-
-
Reduce to Deduce
1+3
1+3
-
-
-
3
-
4
Essence of Number
4
4
4

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
THE
-
-
-
-
-
-
3
4
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
15
-
3
THE
33
15
15
-
-
-
3
4
-
-
-
-
-
-
-
-
-
-
HIGGS
-
-
-
-
-
-
3
4
-
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
-
-
32
-
5
HIGGS
50
41
32
-
-
-
3
4
-
-
-
-
-
-
-
-
-
-
BOSON
-
-
-
-
-
-
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
20
-
5
BOSON
65
29
20
-
-
-
3
4
-
-
-
-
-
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

LETTERS RE-ARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
5
-
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
5
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
-
6
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
-
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
8
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
9
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
1
3
5
7
9

 

 

H
=
8
-
5
HIGGS
50
32
5
B
=
2
-
5
BOSON
65
20
2
P
=
7
-
8
PARTICLE
84
39
3
-
-
17
-
18
First Total
199
91
10
-
-
1+7
-
1+8
Add to Reduce
1+9+9
9+1
1+0
-
-
8
-
18
Second Total
10
10
1
-
-
-
-
1+8
Reduce to Deduce
1+0
1+0
-
-
-
8
-
9
Essence of Number
1
1
1

 

 

BREATHE ON ME BREATH OF GOD

 

 

HOLY BIBLE

Scofield References

Page 7

GENESIS

C 2 V 7

And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
OXYGEN
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
X
=
6
-
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
Y
=
7
-
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
36
-
6
OXYGEN
90
36
36
-
1
2
3
4
10
12
14
8
9
-
-
3+6
-
-
-
9+0
3+6
3+6
-
-
-
-
-
1+0
1+2
1+4
-
-
-
-
9
-
6
OXYGEN
9
9
9
-
1
2
3
4
1
3
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
5
6
7
-
-
-
-
-
OXYGEN
-
-
-
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
6
-
X
=
6
-
1
X
24
6
6
-
-
6
-
Y
=
7
-
1
Y
25
7
7
-
-
-
7
G
=
7
-
1
G
7
7
7
-
-
-
7
E
=
5
-
1
E
5
5
5
-
5
-
-
N
=
5
-
1
N
14
5
5
-
5
-
-
-
-
36
-
6
OXYGEN
90
36
36
-
10
12
14
-
-
3+6
-
-
-
9+0
3+6
3+6
-
1+0
1+2
1+4
-
-
9
-
6
OXYGEN
9
9
9
-
1
3
5

 

 

-
-
-
-
-
-
-
-
-
-
5
6
7
-
-
-
-
-
OXYGEN
-
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
5
-
-
N
=
5
-
1
N
14
5
5
-
5
-
-
O
=
6
-
1
O
15
6
6
-
-
6
-
X
=
6
-
1
X
24
6
6
-
-
6
-
Y
=
7
-
1
Y
25
7
7
-
-
-
7
G
=
7
-
1
G
7
7
7
-
-
-
7
-
-
36
-
6
OXYGEN
90
36
36
-
10
12
14
-
-
3+6
-
-
-
9+0
3+6
3+6
-
1+0
1+2
1+4
-
-
9
-
6
OXYGEN
9
9
9
-
1
3
5

 

 

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

 

 

 

THE SCULPTURE OF VIBRATIONS 1971

 

......
 
Top
  
 
Evokation
  
Previous Page
Index
 Next Page