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Evokation
 
 
Index
 

 

A

MAZE

IN

ZAZAZA ENTERS AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

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

 

 

A 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

 

BEYOND THE VEIL ANOTHER VEIL ANOTHER VEIL BEYOND

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

....

 

 

 

-
ALL IS NUMBER
-
-
-
3
ALL
25
7
7
2
IS
28
10
1
6
NUMBER
73
28
1
11
ALL IS NUMBER
126
45
9
1+1
-
1+2+6
4+5
-
2
ALL IS NUMBER
9
9
9

 

 

-
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
-
5
9
-
-
-
1
-
-
-
-
-
5
-
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
14
9
-
-
-
19
-
-
-
-
-
14
-
-
-
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
4
5
9
-
1
3
-
3
1
-
7
3
1
7
5
+
=
52
5+2
=
7
=
7
=
7
-
-
21
-
-
22
5
18
-
1
12
-
12
1
-
7
21
1
7
5
+
=
133
1+3+3
=
7
=
7
=
7
-
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
21
14
9
22
5
18
19
1
12
-
12
1
14
7
21
1
7
5
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
3
5
9
4
5
9
1
1
3
-
3
1
5
7
3
1
7
5
+
=
72
7+2
=
9
=
9
=
9
-
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
1
1
-
-
-`
1
-
-
-
1
-
-
-
-
1
occurs
x
4
=
4
=
4
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
3
-
3
-
-
-
3
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
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
16
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
29
-
-
17
-
72
-
27
1+6
1+7
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+9
-
-
1+7
-
7+2
-
2+7
7
8
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
11
-
-
8
-
9
-
9
-
-
3
5
9
4
5
9
1
1
3
-
3
1
5
7
3
1
7
5
-
-
1+1
-
-
-
-
-
-
-
7
8
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
2
-
-
8
-
9
-
9

 

 

17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
5
9
-
-
-
1
-
-
-
-
-
5
-
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
14
9
-
-
-
19
-
-
-
-
-
14
-
-
-
-
-
+
=
56
5+6
=
11
1+1
2
=
2
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
3
-
-
4
5
9
-
1
3
-
3
1
-
7
3
1
7
5
+
=
52
5+2
=
7
=
7
=
7
-
21
-
-
22
5
18
-
1
12
-
12
1
-
7
21
1
7
5
+
=
133
1+3+3
=
7
=
7
=
7
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
21
14
9
22
5
18
19
1
12
-
12
1
14
7
21
1
7
5
+
=
189
1+8+9
=
18
1+8
9
=
9
-
3
5
9
4
5
9
1
1
3
-
3
1
5
7
3
1
7
5
+
=
72
7+2
=
9
=
9
=
9
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
1
1
-
-
-`
1
-
-
-
1
-
-
-
-
1
occurs
x
4
=
4
=
4
-
3
-
-
-
-
-
-
-
3
-
3
-
-
-
3
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
17
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
29
-
-
17
-
72
-
27
1+7
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+9
-
-
1+7
-
7+2
-
2+7
8
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
11
-
-
8
-
9
-
9
-
3
5
9
4
5
9
1
1
3
-
3
1
5
7
3
1
7
5
-
-
1+1
-
-
-
-
-
-
-
8
U
N
I
V
E
R
S
A
L
-
L
A
N
G
U
A
G
E
-
-
2
-
-
8
-
9
-
9

 

 

-
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
1
-
5
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
9
19
-
14
-
-
-
-
-
+
=
42
4+2
=
6
=
6
=
6
-
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
3
3
-
-
-
-
-
3
4
2
5
9
+
=
30
3+0
=
3
=
3
=
3
-
-
1
12
12
-
-
-
-
-
21
13
2
5
18
+
=
84
8+4
=
12
1+2
3
=
3
-
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
12
12
-
9
19
-
14
21
13
2
5
18
+
=
126
1+2+6
=
9
=
9
=
9
-
-
1
3
3
-
9
1
-
5
3
4
2
5
9
+
=
45
4+5
=
9
=
9
=
9
-
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
3
-
-
-
-
-`
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
21
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
24
-
-
11
-
45
-
27
2+1
1+1
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
2+4
-
-
1+1
-
4+5
-
2+7
3
2
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
6
-
-
2
-
9
-
9

 

 

11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
1
-
5
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
9
19
-
14
-
-
-
-
-
+
=
42
4+2
=
6
=
6
=
6
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
1
3
3
-
-
-
-
-
3
4
2
5
9
+
=
30
3+0
=
3
=
3
=
3
-
1
12
12
-
-
-
-
-
21
13
2
5
18
+
=
84
8+4
=
12
1+2
3
=
3
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
1
12
12
-
9
19
-
14
21
13
2
5
18
+
=
126
1+2+6
=
9
=
9
=
9
-
1
3
3
-
9
1
-
5
3
4
2
5
9
+
=
45
4+5
=
9
=
9
=
9
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
3
-
-
-
-
-`
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
11
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
24
-
-
11
-
45
-
27
1+1
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
2+4
-
-
1+1
-
4+5
-
2+7
2
A
L
L
-
I
S
-
N
U
M
B
E
R
-
-
6
-
-
2
-
9
-
9

 

 

-
ALL IS NUMBER
-
-
-
3
ALL
25
7
7
2
IS
28
10
1
6
NUMBER
73
28
1
11
ALL IS NUMBER
126
45
9
1+1
-
1+2+6
4+5
-
2
ALL IS NUMBER
9
9
9

 

 

THE CITY OF REVELATION

John Michell 1972

Gnostic Numbers 

Page 118

"Exactly how they came by their science of numbers is not certain, but they appear to have made the discovery that the numerical code of the Hebrew cabala and those of other mystical systems throughout the world were all degenerate versions of the same once universal system of knowledge that returns within the reach of human perception at certain intervals in time. As the revealed books of the Old Testament were written in a code to be interpreted by reference to number, so were the revelations of the gnostic prophets expressed in words and phrases formed on a system of proportion, which gave life and power to the Christian myth, while allowing initiates to gain a further understanding of the balance of forces that produce the world of phenomena."  

Page 121 / How it was ever supposed that the Hebrew alphabet of twenty- two letters, together with various geometrical symbols might serve to represent the entire moving pattern of the universe is not now easy to understand; but, since all ancient philosophy, religion, magic, the arts and sciences were based on the concept of a correspondence between numbers and cosmic law, it is impossible to appreciate the history of the past without some actual experience of the fundamental truth behind this approach to cosmology. Plato gives a remarkable account in Cratylos of the origin of language and letters. The philosopher is asked whether there is any particular significance in names, for surely they are simply a matter of convention and one is more or less as good as another. After all, foreigners call things by different names and appear to manage just as well as the Greeks in this respect. The answer given is that despite appearances the matter is by no means so simple. Words are the tools of expression, and the making of these, as of any other tools, is the task of a skilled craftsman, in this case the lawgiver. Language has grown corrupt over the ages, and names have deviated from their original perfect forms, which are those used by the gods. But all names were originally formed on certain principles, through knowledge of which it is possible to discover the archetypal meaning of words in current use. 'So perhaps the man who knows about names considers their value and is not confused if some letter is added, transposed or subtracted, or even if the force of the name is expressed in quite different letters.' This is Plato's clearest reference to the mystical science of the cabala, in which letters, words and whole phrases may be substituted for others of the same numerical value. The force of a name is to be found in its number, and can be expressed through any combination of letters, provided the sum of the letters amounts to the appropriate number by gematria.

 

GNOSIS GODS SON IS

 

LETTERS AND NUMBERS AND NUMBERS AND LETTERS

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
N
=
7
-
7
NUMBERS
92
56
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
N
=
5
1
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
2
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
3
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
7
1
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
56
-
7
NUMBERS
92
56
56
-
1
2
3
4
10
6
7
8
9
-
-
5+6
-
-
-
9+2
5+6
5+6
-
-
-
-
-
1+0
-
-
-
-
-
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
-
-
-
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
N
=
7
-
7
NUMBERS
92
56
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
7
1
S
19
1
1
-
1
-
-
-
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
2
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
3
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
N
=
5
1
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
56
-
7
NUMBERS
92
56
56
-
1
2
3
4
10
6
7
8
9
-
-
5+6
-
-
-
9+2
5+6
5+6
-
-
-
-
-
1+0
-
-
-
-
-
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
-
-
-
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
N
=
7
-
7
NUMBERS
92
56
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
S
=
1
7
1
S
19
1
1
-
1
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
U
=
3
2
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
3
1
M
13
4
4
-
-
-
-
4
-
-
N
=
5
1
1
N
14
5
5
-
-
-
-
-
5
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
56
-
7
NUMBERS
92
56
56
-
1
2
3
4
10
9
-
-
5+6
-
-
-
9+2
5+6
5+6
-
-
-
-
-
1+0
-
-
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
-
-
-
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

THE

LANGUAGE OF NUMBERS THE NUMBERS OF LANGUAGE

 

 

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.

 

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

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 typew
riters 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.' "

 

I = 9 9 = I

R = 9 9 = R

 

OF

T9ME AND STA9S

A9thu9 C. Cla9ke,1972

Page 15

THE N9NE B9LL9ON NAMES OF GOD

'Th9s 9s a sl9ghtly unusual 9equest,'sa9d D9 Wagne9, w9th what he hoped was commendable 9est9a9nt.' As fa9 as 9 know, 9t's the f99st t9me anyone's been asked to supply a T9betan monaste9y with an Automat9c Sequence Compute9. 9 don't w9sh to be 9nqu9s9t9ve, but 9 should ha9dly have thought that you9- ah - establ9shment had much use for such a mach9ne.Could you expla9n just what you 9ntend to do w9th 9t?'

'Gladly,' 9epl9ed the lama, 9eadjust9ng h9s s9lk 9obes and ca9efully putting away the sl9de 9ule he had been us9ng fo9 cu99ency conve9s9ons. 'You9 Ma9k V Compute9 can ca99y out any 9out9ne mathemat9cal ope9at9on 9nvolv9ng up to ten d9g9ts. Howeve9, for ou9 wo9k we a9e 9nte9ested 9n lette9s, not numbe9s. As we w9sh you to mod9fy the output c9rcu9ts,the mach9ne w9ll be p99nt9ng wo9ds not columns of f9gu9es.'

'9 dont qu9te unde9stand…'

'Th9s 9s a p9oject on wh9ch we have been work9ng fo9 the last th9ee centu99es - s9nce the lamase9y was founded, 9n fact.9t 9s somewhat al9en to you9 way of thought, so9 hope you w9ll l9sten with an open m9nd wh9le 9 expla9n 9t

'Natu9ally.'

'9t 9s 9eally qu9te s9mple.We have been comp9l9ng a l9st wh9ch shall conta9n all the poss9ble names of God'

'9 beg you9 pa9don?' / Page16 / 'We have 9eason to bel9eve' cont9nued the lama 9mpe9tu9bably, ' that all such names can be w99tten with not mo9e than n9ne lette9s 9n an alphabet we have dev9sed,'

'And you have been do9ng th9s for three centu99es?

'Yes: we expected9t would take us about f9fteen thousand yea9s to complete the task.'

'Oh, Dr Wagne9 looked a l9ttle dazed. 'Now9 see why you wanted to h99e one of ou9 mach9nes. But what exactly9s the pu9pose of th9s p9oject ?

'The lama hes9tated fo9 a f9act9on of a second, and Wagne9 wonde9ed9f he had offended h9m.9f so the9e was no t9ace of annoyance9n the 9eply.

'Call9t 99tual, 9f you l9ke, but 9t's a fundamental pa9t of ou9 bel9ef. All the many names of the Sup9eme Be9ng - God , Jehova , Allah , and so on - they a9e only man made labels. The9e 9s a ph9losoph9cal p9oblem of some d9ff9culty he9e, wh9ch9 do not p9opose to d9scuss, but somewhe9e among all the poss9ble comb9nat9ons of lette9s that can occu9 a9e what one may call the 9eal names of God. By systemat9c pe9mutat9on of lette9s, we have been t9y9ng to l9st them all'

9 see. You've been sta9t9ng at AAAAAAA… and wo9k-9ng up to ZZZZZZZZ …'

'Exactly - though we use a spec9al alphabet of ou9 own. Mod9fy9ng the elect9omat9c typew99te9s to deal w9th th9s 9s of cou9se t99v9al. A 9athe9 mo9e 9nte9est9ng p9oblem 9s that of dev9s9ng su9table c99cu9ts to el9m9nate 9 9d9culous comb9nat9ons. Fo9 example, no lette9 must occu9 mo9e than th9ee t9mes 9n sucess9on.'

'Th9ee? Su9ely you mean two.'

'Th9ee 9s co99ect; 9 am af9a9d 9t would take too long to expla9n why , even 9f you unde9stood ou9 language.'/ Page 17 / '9'm su9e 9t would,' sa9d Wagne9 hast9ly. 'Go on.'

'Luck9ly, 9t w9ll be a s9mple matte9 to adapt you9 Automat9c Sequence Compute9 fo9 th9s wo9k, s9nce once 9t has been p9og9ammed p9ope9ly 9t w9ll pe9mute each lette9 9n tu9n and p99nt the 9esult. What would have taken us f9fteen thousand years 9t w9ll be able to do 9n a hund9ed days.'

'Dr Wagne9 was sca9cely consc9ous of the fa9nt sounds f9om the Manhatten st9eets fa9 below. He was 9n a d9ffe9ent wo9ld, a wo9ld of natu9al, not man-made mounta9ns. H9gh up 9n the99 9emote ae99es these monks had been pat9ently at wo9k gene9at9on afte9 gene9at9on, comp9l9ng the99 l9sts of mean9ngless wo9ds. Was the9e any l9m9ts to the foll9es of mank9nd ? St9ll, he must g9ve no h9nt of h9s 9nne9 thoughts. The custome9 was always 99ght…"

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"

 

 

-
-
-
-
-
ABACUS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
4
5
6
7
8
9
B
=
2
2
1
B
2
2
2
-
-
2
-
4
5
6
7
8
9
A
=
1
3
1
A
1
1
1
-
1
-
-
4
5
6
7
8
9
C
=
3
4
1
C
3
3
3
-
-
-
3
4
5
6
7
8
9
U
=
3
5
1
U
21
3
3
-
-
-
3
4
5
6
7
8
8
S
=
1
6
1
S
19
10
1
-
1
-
-
4
5
6
7
8
9
-
-
11
-
6
ABACUS
47
20
11
-
3
2
6
4
5
6
7
8
9
-
-
1+1
-
-
-
4+7
2+0
1+1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
6
ABACUS
11
2
2
-
3
2
6
4
5
6
7
8
9
-
-
1+1
-
-
-
1+1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
6
ABACUS
2
2
2
-
3
2
6
4
5
6
7
8
9

 

 

-
-
-
-
-
ABACUS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
4
5
6
7
8
9
A
=
1
3
1
A
1
1
1
-
1
-
-
4
5
6
7
8
9
S
=
1
6
1
S
19
10
1
-
1
-
-
4
5
6
7
8
9
B
=
2
2
1
B
2
2
2
-
-
2
-
4
5
6
7
8
9
C
=
3
4
1
C
3
3
3
-
-
-
3
4
5
6
7
8
9
U
=
3
5
1
U
21
3
3
-
-
-
3
4
5
6
7
8
8
-
-
11
-
6
ABACUS
47
20
11
-
3
2
6
4
5
6
7
8
9
-
-
1+1
-
-
-
4+7
2+0
1+1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
6
ABACUS
11
2
2
-
3
2
6
4
5
6
7
8
9
-
-
1+1
-
-
-
1+1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
6
ABACUS
2
2
2
-
3
2
6
4
5
6
7
8
9

 

 

Abacus - Wikipedia
https://en.wikipedia.org/wiki/Abacus

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system. The exact origin of the abacus is still unknown.

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system. The exact origin of the abacus is still unknown. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal.
Abaci come in different designs. Some designs, like the bead frame consisting of beads divided into tens, are used mainly to teach arithmetic, although they remain popular in the post-Soviet states as a tool. Other designs, such as the Japanese soroban, have been used for practical calculations even involving several digits. For any particular abacus design, there usually are numerous different methods to perform a certain type of calculation, which may include basic operations like addition and multiplication, or even more complex ones, such as calculating square roots. Some of these methods may work with non-natural numbers (numbers such as 1.5 and ?3/4).
Although today many use calculators and computers instead of abaci to calculate, abaci still remain in common use in some countries. Merchants, traders and clerks in some parts of Eastern Europe, Russia, China and Africa use abaci, and they are still used to teach arithmetic to children.[1] Some people who are unable to use a calculator because of visual impairment may use an abacus.

 

A
=
1
-
1
A
1
1
1
C
=
3
-
8
COUNTING
103
40
4
T
=
2
-
5
TABLE
40
13
4
-
-
6
-
14
Add to Reduce
144
54
9
-
-
-
-
1+4
Reduce to Deduce
1+4+4
5+4
1+6
-
-
6
-
5
Essence of Number
9
9
9

 

 

G
=
7
4
GOOD
41
23
5
A
=
1
3
AND
19
10
1
E
=
4
4
EVIL
48
21
3
-
-
12
11
Add to Reduce
108
54
9
-
-
1+2
1+1
Reduce to Deduce
1+0+8
5+4
-
-
-
3
2
Essence of Number
9
9
9

 

 

-
-
-
-
-
RIGHT AND WRONG
-
-
-
R
=
8
-
5
RIGHT
62
35
8
A
=
4
-
3
AND
19
10
1
W
=
5
-
5
WRONG
77
32
5
-
-
17
-
13
RIGHT AND WRONG
158
77
14
-
-
1+7
-
1+3
-
1+5+8
7+7
1+4
6
-
8
-
4
RIGHT AND WRONG
14
14
5
-
-
-
-
-
-
1+4
1+4
-
6
-
8
-
4
RIGHT AND WRONG
5
5
5

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
RIGHT AND WRONG
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
R
=
9
1
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
I
=
9
2
1
I
9
9
9
-
-
-
3
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
3
-
-
-
-
-
-
-
-
35
-
5
-
62
35
35
-
-
-
-
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
3
-
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
3
4
-
-
-
-
-
-
-
10
-
3
-
19
10
10
-
-
-
-
-
-
-
-
-
-
W
=
5
9
1
W
23
5
5
-
-
-
3
-
5
-
-
-
-
R
=
9
10
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
O
=
6
11
1
O
15
6
6
-
-
-
3
-
-
6
-
-
-
N
=
5
12
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
G
=
7
13
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
-
-
32
-
5
-
77
32
32
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
RIGHT AND WRONG
158
77
14
-
1
2
3
4
15
6
14
8
27
-
-
-
-
1+3
-
1+5+8
7+7
1+4
-
-
-
-
-
1+5
-
1+4
-
2+7
-
-
-
-
4
RIGHT AND WRONG
14
14
5
-
1
2
3
4
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
RIGHT AND WRONG
5
5
5
-
1
2
3
4
6
6
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
RIGHT AND WRONG
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
R
=
9
1
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
I
=
9
2
1
I
9
9
9
-
-
-
3
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
3
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
3
-
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
3
4
-
-
-
-
-
W
=
5
9
1
W
23
5
5
-
-
-
3
-
5
-
-
-
-
R
=
9
10
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
O
=
6
11
1
O
15
6
6
-
-
-
3
-
-
6
-
-
-
N
=
5
12
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
G
=
7
13
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
RIGHT AND WRONG
158
77
77
-
1
2
3
4
15
6
14
8
27
-
-
-
-
1+3
-
1+5+8
7+7
7+7
-
-
-
-
-
1+5
-
1+4
-
2+7
-
-
-
-
4
RIGHT AND WRONG
14
14
14
-
1
2
3
4
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
RIGHT AND WRONG
5
5
5
-
1
2
3
4
6
6
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
RIGHT AND WRONG
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
3
-
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
3
-
-
-
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
3
4
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
W
=
5
9
1
W
23
5
5
-
-
-
3
-
5
-
-
-
-
N
=
5
12
1
N
14
5
5
-
-
-
3
-
5
-
-
-
-
O
=
6
11
1
O
15
6
6
-
-
-
3
-
-
6
-
-
-
G
=
7
13
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
G
=
7
3
1
G
7
7
7
-
-
-
3
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
-
-
-
-
8
-
R
=
9
1
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
I
=
9
2
1
I
9
9
9
-
-
-
3
-
-
-
-
-
9
R
=
9
10
1
R
18
9
9
-
-
-
3
-
-
-
-
-
9
-
-
-
-
13
RIGHT AND WRONG
158
77
77
-
1
2
3
4
15
6
14
8
27
-
-
-
-
1+3
-
1+5+8
7+7
7+7
-
-
-
-
-
1+5
-
1+4
-
2+7
-
-
-
-
4
RIGHT AND WRONG
14
14
14
-
1
2
3
4
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
RIGHT AND WRONG
5
5
5
-
1
2
3
4
6
6
5
8
9

 

 

L
=
2
-
5
LIGHT
56
29
2
A
=
6
-
3
AND
19
10
1
L
=
6
-
4
LIFE
32
23
5
A
=
7
-
3
AND
19
10
1
L
=
2
-
4
LOVE
54
18
9
S
-
23
4
19
First Total
180
90
18
-
-
2+3
-
1+9
Add to Reduce
1+8+0
9+0
1+8
-
-
9
-
10
Second Total
9
9
9
-
-
-
-
1+0
Reduce to Deduce
-
-
-
-
-
9
-
1
Essence of Number
9
9
9

 

 

L
=
3
-
5
LIGHT
56
29
2
I
=
9
-
2
IS
19
10
1
L
=
3
-
4
LIFE
32
23
5
I
=
9
-
2
IS
19
10
1
L
=
3
-
4
LOVE
54
18
9
S
-
27
4
17
First Total
198
90
18
-
-
2+7
-
1+7
Add to Reduce
1+9+8
9+0
1+8
-
-
9
-
8
Second Total
18
9
9
-
-
-
-
1+0
Reduce to Deduce
1+8
-
-
-
-
9
-
8
Essence of Number
9
9
9

 

 

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

 

 

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

 

 

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

 

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
PHOENIX
-
-
-
-
-
-
-
-
-
-
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
X
=
6
7
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
-
-
46
-
7
PHOENIX
91
46
46
-
1
2
3
4
10
12
7
8
9
-
-
4+6
-
-
-
9+0
4+6
4+6
-
-
-
-
-
1+0
1+2
-
-
-
-
-
10
-
7
PHOENIX
10
10
10
-
1
2
3
4
1
3
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
9
-
7
PHOENIX
1
1
1
-
1
2
3
4
1
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
PHOENIX
-
-
-
-
-
-
-
-
-
-
-
-
-
P
=
7
1
1
P
16
7
7
-
1
2
3
4
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
1
2
3
4
-
-
-
8
-
O
=
6
3
1
O
15
6
6
-
1
2
3
4
-
6
-
-
-
E
=
5
4
1
E
5
5
5
-
1
2
3
4
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
1
2
3
4
5
-
-
-
-
I
=
9
6
1
I
9
9
9
-
1
2
3
4
-
-
-
-
9
X
=
6
7
1
X
24
6
6
-
1
2
3
4
-
6
-
-
-
-
-
46
-
7
PHOENIX
91
46
46
-
1
2
3
4
10
12
7
8
9
-
-
4+6
-
-
-
9+0
4+6
4+6
-
-
-
-
-
1+0
1+2
-
-
-
-
-
10
-
7
PHOENIX
10
10
10
-
1
2
3
4
1
3
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
9
-
7
PHOENIX
1
1
1
-
1
2
3
4
1
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
PHOENIX
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
4
1
E
5
5
5
-
1
2
3
4
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
1
2
3
4
5
-
-
-
-
O
=
6
3
1
O
15
6
6
-
1
2
3
4
-
6
-
-
-
X
=
6
7
1
X
24
6
6
-
1
2
3
4
-
6
-
-
-
P
=
7
1
1
P
16
7
7
-
1
2
3
4
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
1
2
3
4
-
-
-
8
-
I
=
9
6
1
I
9
9
9
-
1
2
3
4
-
-
-
-
9
-
-
46
-
7
PHOENIX
91
46
46
-
1
2
3
4
10
12
7
8
9
-
-
4+6
-
-
-
9+0
4+6
4+6
-
-
-
-
-
1+0
1+2
-
-
-
-
-
10
-
7
PHOENIX
10
10
10
-
1
2
3
4
1
3
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
9
-
7
PHOENIX
1
1
1
-
1
2
3
4
1
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
5
6
7
8
9
-
-
-
-
-
PHOENIX
-
-
-
-
-
-
-
-
-
E
=
5
4
1
E
5
5
5
-
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
5
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
6
-
-
-
X
=
6
7
1
X
24
6
6
-
-
6
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
8
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
9
-
-
46
-
7
PHOENIX
91
46
46
-
10
12
7
8
9
-
-
4+6
-
-
-
9+0
4+6
4+6
-
1+0
1+2
-
-
-
-
-
10
-
7
PHOENIX
10
10
10
-
1
3
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
9
-
7
PHOENIX
1
1
1
-
1
3
7
8
9

 

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
SPHINX
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
4
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
P
=
7
5
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
3
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
1
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
X
=
6
2
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
-
-
36
-
6
SPHINX
90
45
36
-
1
2
3
4
5
6
7
8
9
-
-
3+6
-
-
-
9+0
3+6
3+6
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
SPHINX
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

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