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This monograph was written by Dr. Joyce Bowen as a companion document to The Law of Digit Balance, in order to demonstrate the efficacy of the unimath technique, Increment Analysis, introduced by that book. This new approach to analyzing number sequences reveals hidden patterns and relationships between numbers in a spectacular fashion, and demonstrates a remarkable symmetry that emerges in all number sequences. Ben Franklin’s most magical, magic square, which is a 16 x 16 magic square, was analyzed using this technique and incredible symmetries are revealed which are far beyond the realms of chance.
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A UNIMATH ASSESSMENT
2
Copyright 2008 Joyce P. Bowen All Rights Reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying,
recording, or otherwise) without prior written permission from the author.
The author may be contacted at [email protected]
Cover Design: Yaoundé Olu
ISBN 0-9615454-4-5
Astropoint Research Associates Chicago, IL
3
TABLE OF CONTENTS Page Introduction 4 Ben Franklin’s Most Magical Magic Square 6 Rules of Analysis 7 Glossary of Terms 8 Increment Square Across 9 Increment Square Across Analysis 10 Increment Square Sum Modulo 9 Across 12 Increment Square Sum Modulo 9 Analysis 13 Increment Square Down 14 Increment Square Down Analysis 15 Increment Square Sum Modulo 9 Down 16 Increment Square Sum Modulo 9 Down Analysis 17 Composite Square – Across and Down 18 Composite Square – Across and Down Analysis 19 Composite Square – Sum Modulo 9 A - J 20 Composite Square – Sum Modulo 9 A - J Analysis 30 Composite Square Row 16 and Column P – Synthesis and Analysis 31 Conclusions and Implications 32
4
INTRODUCTION
The notion that numbers are things in themselves and have deeper import is not always
considered seriously by many people in contemporary society. Sometimes the question
even arises as to whether or not numbers were invented or discovered. Mathematician
Gregory J. Chaitin of the IBM Thomas J. Watson Research Center, author of The Limits of
Mathematics, suggests that the structure of arithmetic is random. He states, “Although almost
all numbers are random, there is no formal axiomatic system that will allow us to prove this fact.”
In this analysis of Ben Franklin’s “most magical, magic square,” I have examined the
numbers in that 16 x 16 square, and have uncovered deeper relationships and an
evidence of an underlying order in arithmetic not immediately evident. Increment
Analysis, assessing number relationships between adjacent numbers in numerical
sequences, is a technique that I introduced in my book, The Law of Digit Balance,
wherein it was found that all numerical sequences exhibit incredible symmetries.
Increment Analysis is used to identify and synthesize number relationships. It differs
from the calculus of sequence differences in that the result of analysis is a charged
number line. Increment Analysis, a Unimath technique used in Uniphysics, the Science
of Synthesis, reveals previously unobserved patterns in ALL number sequences that
were heretofore hidden, and where the increments of all whole numbers, or fragments of
irrational or transcendental numbers, sum to zero while exhibiting beautiful symmetries.
There is a logic and order between adjacent digits in number sequences wherein they increase
positively or negatively to each other on face value, even though each digit differs by a
magnitude of 10, 100, 1,000 units, etc… This seems to be counterintuitive, even though it has
been demonstrated to be true in all cases.
In this monograph, the Increment Analysis approach has revealed new symmetries in
Ben Franklin’s 16 x 16 magic square. Furthermore, the derivation of subsequent squares
from the original squares which still hold these symmetries, whether assessing across
5
columns or down rows, lends credence to the efficacy of the Increment Analysis
approach as outlined in The Law of Digit Balance.1
Benjamin Franklin (1706 – 1790) was a famous eighteenth - century scientist, inventor,
statesman, printer, philosopher, musician, and economist. 2 Regarding Franklin’s
remarkable 16 x 16 square, the sum of the numbers in any 4 x 4 subsquare is 2056. The
sum of the numbers in any 2 x 2 subsquare is 514. The various sums can be formed by
simple expressions of powers of 2:2056 = 2 11 + 23 and 514 = 29 + 21. The square also
has many bent row properties.3
According to Dr. Clifford A. Pickover, author of The Zen of Magic Squares, Circles, and
Stars, Franklin’s square does not totally satisfy the main diagonal sums, which means
they cannot be strictly considered magic squares. However, it is found that one diagonal
is 2056 – 128 and the other is 2056 + 128. This is still a beautiful symmetry and is in
keeping with my findings wherein all number patterns in numerical sequences and magic
squares exhibit positive and negative balanced symmetries. Moreover, Dr. Pickover has
pointed out a number of other symmetries, and predicted that more would be discovered.
Certainly, Increment Analysis has uncovered some of them.
Finally, the squares within squares in this monograph maintain beautiful symmetries and
intra or extra-dimensional properties. Certainly, this should challenge the notion of
randomness in arithmetic.
1 Bowen, Joyce P. The Law of Digit Balance. Astropoint. Chicago, IL. 2003.
2 Pickover, Clifford A. The Zen of Magic Squares, Circles, and Stars. Princeton University Press. Princeton, NJ. 2002.
3 Ibid.
6
BEN FRANKLIN’S MOST MAGICAL MAGIC 4SQUARE
4 Ibid.
7
CONJECTURE
RULES OF INCREMENT ANALYSIS 1. Select a number sequence and evaluate the increments between
adjacent numbers so that they become positive or negative in relation to each other. Ignore decimal points and treat the number as a continuous sequence. For example, the reciprocal of seven is a series of repeating numbers 0.01428571…:
1 4 2 8 5 7 1…: The increment between 1 and 4 is +3 “ “ “ 4 and 2 is –2 “ “ “ 2 and 8 is +6 “ “ “ 8 and 5 is –3 “ “ “ 5 and 7 is +2 “ “ “ 7 and 1 is -6 The resulting number string, therefore, is
+3 -2 +6 -3 +2 -6… 2. Isolate the resulting binary patterns and cancel them to zero,
leaving one or more remainders. In the above case, +3 -2 and –3 +2 cancel each other, and the +6 and –6 also cancel each other, resulting in 0.
3. To see the underlying patterns of any sequence or sequence
segment that is not a circular number, you add zeros before and after the sequence and proceed with the methodology outlined in number 1 above. The digits will always cancel to zero. (The same is true for any sequence that begins and ends with the same number).
4. Though it is important to keep in mind that each digit is an
increment that is ten times more precise than the last in the decimal expansion of numbers, the numbers are taken at face value.
5. For further details on analysis and extracting 2nd, 3rd, or N order
sequences from non-circular number segments without adding zeros, see the chapter on Pi Patterns in The Law of Digit Balance.
8
GLOSSARY OF TERMS
Digit Balance Formula - Σ ∆n = 0 – The Sum (Σ) of the increment changes (∆) between numbers (n) = 0 Increment Analysis – Assessing the increments between adjacent numbers and creating a positive or negative number depending upon whether the increment increases or decreases. Increment Square Across – Assessing the increments between adjacent numbers in Ben Franklin’s most magical, magic square from columns A through P Increment Square Sum Modulo 9 Across – Reducing the resultant numbers to a single digit between one and nine by adding them together. Increment Square Down - Assessing the increments between adjacent numbers in the magic square from rows 1 and 16. Increment Square Sum Modulo 9 Down – Reducing the resultant numbers to a single digit between one and nine by adding them together. Composite Square Across and Down – This is a square that is a synthesis of the across and down numbers. Each cell is a result of adding together the numbers in the corresponding cross and down cells and then assessing the resultant increments. Composite Square – Sum Modulo 9 – Reducing the resultant numbers in the composite square to a single digit between one and nine by adding them together. Mirrors – Rows or Columns that exhibit the same numbers but the reverse sign, summing to zero. Pseudo-Pairs – Numbers that are seemingly separated by two, but are of alternating signs, such as +2 -4 +6 -8, etc. Unimath – The mathematical tool utilized in Uniphysics, the Science of Synthesis. Increment Analysis is one element of Unimath.
9
INCREMENT SQUARE - ACROSS A B C D E F G H I J K L M N O P
Σ ∆n = 0 Reverse Twin Patterns
with anomalies at Column D
Total = 100%
1
+17
+15
+17
-241
+17
+15
+17
+15
+17
+15
+17
+15
+17
+15
+17
-15
2
-19
-13
-19
+243
-19
-13
-19
-13
-19
-13
-19
-13
-19
-13
-19 +13
3
+21
+11
+21
-245 +21
+11
+21
+11
+21
+11
+21
+11
+21
+11
+21 -11
4
-23
-9
-23
+247
-23
-9
-23
-9
-23
-9
-23
-9
-23
-9
-23 +9
5
+15
+17
+15
-239
+15
+17
+15
+17
+15
+17
+15
+17
+15
+17
+15 -17
6
-13
-19
-13
+237
-13
-19
-13
-19
-13
-19
-13
-19
-13
-19
-13 +19
7
+11
+21
+11
-235
+11
+21
+11
+21
+11
+21
+11
+21
+11
+21
+11 -21
8
-9
-23
-9
+233
-9
-23
-9
-23
-9
-23
-9
-23
-9
-23
-9 +23
9
+7
+25
+7
-231
+7
+25
+7
+25
+7
+25
+7
+25
+7
+25
+7 -25
10
-5
-27
-5
+229
-5
-27
-5
-27
-5
-27
-5
-27
-5
-27
-5 +27
11
+3
+29
+3
-227
+3
+29
+3
+29
+3
+29
+3
+29
+3
+29
+3 -29
12
-1
-31
-1
+225
-1
-31
-1
-31
-1
-31
-1
-31
-1
-31
-1 +31
13
+25
+7
+25
-249
+25
+7
+25
+7
+25
+7
+25
+7
+25
+7
+25 -7
14
-27
-5
-27
+251
-27
-5
-27
-5
-27
-5
-27
-5
-27
-5
-27 +5
15
+29
+3
+29
-253
+29
+3
+29
+3
+29
+3
+29
+3
+29
+3
+29 -3
16
-31
-1
-31
+255
-31
-1
-31
-1
-31
-1
-31
-1
-31
-1
-31 +1
S
u
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
INCREMENT SQUARE – ACROSS** ANALYSIS
**Please note that in this, and all subsequent analyses, color coding in the squares is such that like colors represent related patterns. The exception would be in white squares, unless otherwise noted.
� The sum of all columns = 0
� The sum of all rows A through D, 1 through 16 = alternating +192 or -192
� There are no even increments in the square
� The sum of all rows 1 - 16, columns F through P, exhibits alternating patterns as follows: -15 +13, -11 +9, -17 +19, -21 +23, -25 +27, -29 +31, -7 +5, -3 +1. These are alternating polarity pseudo-pairs.
� In all cases, the sum of each row is the mirror of one of the numbers in the
repeating number pairs. For example, in row one, the sum, -15, is the mirror of the +15; in row two, the sum +13 is the mirror of the -13; in row 3 the sum -11 is the mirror of the +11, etc…
� All rows exhibit an alternating pattern based on two numbers, with the exception of column D, as follows:
Row Repeating Pattern
1. +17, +15… 2. -19, -13… 3. +21, +11… 4. -23, -9… 5. +15, +17… 6. -13, -19… 7. +11, +21… 8. -9, -23… 9. +7, +25… 10. -5, -27… 11. +3, +29… 12. -1, -31… 13. +25, +7… 14. -27, -5… 15. +29, +3… 16. -31, -1…
11
� Regarding the aforementioned, note that Rows 1 and 5 are reverses of each other as are rows 2 and 6, rows 3 and 7, rows 4 and 8, rows; rows 9 and 13; rows 10 and 14, rows 11 and 15, and rows 12 and 16.
� ALL number pairs above add to (alternating) +32 or -32. � This square is a ZERO SUM INCREMENT SQUARE, with all columns
and rows summing to Zero:
The sum of all columns is zero (see page 9), and the sum of all rows is Zero (see table below)
ROW SUM 1 -30
2 +26
3 -22
4 +18
5 -34
6 +38
7 -42
8 +46
9 -50
10 +54
11 -58
12 +62
13 -14
14 +10
15 -6
16 +2
TOTAL 0
12
INCREMENT SQUARE Σ MOD 9 - ACROSS
A B C D E F G H I J K L M N O SUM Mod9
1
+8
+6
+8
-7
+8
+6
+8
+6
+8
+6
+8
+6
+8
+6
+8
+93
+3
2
-1
-4
-1
+9
-1
-4
-1
-4
-1
-4
-1
-4
-1
-4
-1
-23
-5
3
+3
+2
+3
-2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+34
+7
4
-5
-9
-5
+4
-5
-9
-5
-9
-5
-9
-5
-9
-5
-9
-5
-90
-9
5
+6
+8
+6
-5
+6
+8
+6
+8
+6
+8
+6
+8
+6
+8
+6
+91
+1
6
-4
-1
-4
+3
-4
-1
-4
-1
-4
-1
-4
-1
-4
-1
-4
-34
-7
7
+2
+3
+2
-1
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+33
+6
8
-9
-5
-9
+8
-9
-5
-9
-5
-9
-5
-9
-5
-9
-5
-9
-94
-4
9
+7
+7
+7
-6
+7
+7
+7
+7
+7
+7
+7
+7
+7
+7
+7
+92
+2
10
-5
-9
-5
+4
-5
-9
-5
-9
-5
-9
-5
-9
-5
-9
-5
-90
-9
11
+3
+2
+3
-2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+34
+7
12
-1
-4
-1
+9
-1
-4
-1
-4
-1
-4
-1
-4
-1
-4
-1
-23
-5
13
+7
+7
+7
-6
+7
+7
+7
+7
+7
+7
+7
+7
+7
+7
+7
+92
+2
14
-9
-5
-9
+8
-9
-5
-9
-5
-9
-5
-9
-5
-9
-5
-9
-94
-4
15
+2
+3
+2
-1
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+33
+6
16
-4
-1
-4
+3
-4
-1
-4
-1
-4
-1
-4
-1
-4
-1
-4
-34
-7
Reverse Patterns and Twins Total = 100%
13
INCREMENT SQUARE Σ MOD 9 - ACROSS ANALYSIS
� All columns A – O sum to zero with the exception of D, which sums to +18.
� TWIN ROWS ARE 2 and 12, 3 and 11, 4 and 10, 6 and 16, 7 and 15, 8 and 14, 9 and 13. The patterns of the sums of these rows are: -23 -23; +34 +34; +33 +33; -94 -94; +92 +92, with two anomalies: +93 and +91 (rows one and five), which are reverse pseudo twins with variations at column “D” (-7 and -5 respectively).
� The alternating +, - pattern continues throughout without exceptions. � The sum of all rows totals +20, but when considered mod 9 becomes -16. The difference between +20 and -16 is 4, a number that is of great underlying import throughout the square.
14
INCREMENT SQUARE – DOWN A B C D E F G H I J K L M N O P
1
-142
-178
-206
-242
+242
+206
+178
+142
+114
+78
+50
+14
-14
-50
-78
-114
= 0
2
+140
+180
+204
+244
-244
-204
-180
-140
-116
-76
-52
-12
+12
+52
+76
+116
= 0
3
-138
-182
-202
-246
+246
+202
+182
+138
+118
+74
+54
+10
-10
-54
-74
-118
= 0
4
+141
+179
+205
+243
-243
-205
-179
-141
-115
-77
-51
-13
+13
+51
+77
+115
= 0
5
-146
-174
-210
-238
+238
+210
+174
+146
+110
+82
+46
+18
-18
-46
-82
-110
= 0
6
+148
+172
+212
+236
-236
-212
-172
-148
-108
-84
-44
-20
+20
+44
+84
+108
= 0
7
-150
-170
-214
-234
+234
+214
+170
+150
+106
+86
+42
+22
-22
-42
-86
-106
= 0
8
+152
+168
+216
+232
-232
-216
-168
-152
-104
-88
-40
-24
+24
+40
+88
+104
= 0
9
-154
-166
-218
-230
+230
+218
+166
+154
+102
+90
+38
+26
-26
-38
-90
-102
= 0
10
+156
+164
+220
+228
-228
-220
-164
-156
-100
-92
-36
-28
+28
+36
+92
+100
= 0
11
-158
-162
-222
-226
+226
+222
+162
+158
+98
+94
+34
+30
-30
-34
-94
-98
= 0
12
+147
+173
+211
+237
-237
-211
-173
-147
-109
-83
-45
-19
+19
+45
+83
+109
= 0
13
-134
-186
-198
-250
+250
+198
+186
+134
+122
+70
+58
+6
-6
-58
-70
-122
= 0
14
+132
+188
+196
+252
-252
-196
-188
-132
-124
-68
-60
-4
+4
+60
+68
+124
= 0
15
-130
-190
-194
-254
+254
+194
+190
+130
+126
+66
+62
+2
-2
-62
-66
-126
= 0
Mirrors Each column is a mirror of the column with the same color.
Total = 100%
15
INCREMENT SQUARE – DOWN ANALYSIS
� All of the increments are even with the exception of rows 4 and 12, where all of
the numbers 4A – P and 12A -P are odd.
� The Sum of All rows is 0.
� The Sum of All columns is 0.
� Essentially, regarding the columns, this square behaves as two zero sum squares with the following patterns:
*Columns A & H = -136 +136 B & G = -184 +184 C & F = -200 + 200 D & E = -248 +248 I & P = +120 - 120 J & O = +72 - 72 K & N = + 56 - 56 L & M = + 8 - 8
*Note that in the first set of 8 columns the negative number leads, and in the second set of 8 columns the positive number leads.
16
INCREMENT SQUARE Σ MOD 9 - DOWN A B C D E F G H I J K L M N O P
1
-7
-7
-8
-8
+8
+8
+7
+7
+6
+6
+5
+5
-5
-5
-6
-6
2
+5
+9
+6
+1
-1
-6
-9
-5
-8
-4
-7
-3
+3
+7
+4
+8
3
-3
-2
-4
-3
+3
+4
+2
+3
+1
+2
+9
+1
-1
-9
-2
-1
4
+6
+8
+7
+9
-9
-7
-8
-6
-7
-5
-6
-4
+4
+6
+5
+7
5
-2
-3
-3
-4
+4
+3
+3
+2
+2
+1
+1
+9
-9
-1
-1
-2
6
+4
+1
+5
+2
-2
-5
-1
-4
-9
-3
-8
-2
+2
+8
+3
+9
7
-6
-8
-7
-9
+9
+7
+8
+6
+7
+5
+6
+4
-4
-6
-5
-7
8
+8
+6
+9
+7
-7
-9
-6
-8
-5
-7
-4
-6
+6
+4
+7
+5
9
-1
-4
-2
-5
+5
+2
+4
+1
+3
+9
+2
+8
-8
-2
-9
-3
10
+3
+2
+4
+3
-3
-4
-2
-3
-1
-2
-9
-1
+1
+9
+2
+1
11
-5
-9
-6
-1
+1
+6
+9
+5
+8
+4
+7
+3
-3
-7
-4
-8
12
+3
+2
+4
+3
-3
-4
-2
-3
-1
-2
-9
-1
+1
+9
+2
+1
13
-8
-6
-9
-7
+7
+9
+6
+8
+5
+7
+4
+6
-6
-4
-7
-5
14
+6
+8
+7
+9
-9
-7
-8
-6
-7
-5
-6
-4
+4
+6
+5
+7
15
-4
-1
-5
-2
+2
+5
+1
+4
+9
+3
+8
+2
-2
-8
-3
-9
Mirrors, Reverse Mirrors, Homogenous Pairs and Mixed-Sign Pairs Total = 99.1%
(If we consider the -2 in 5A as coupled with the -2 in 5P, the incidence is 100%)
17
INCREMENT SQUARE Σ MOD 9 - DOWN ANALYSIS
� Sum of all columns is +3, with anomalies occurring at columns D and E (-2,+5), whose sum is +3.
� First row, 1A – P: all couplets sum to Zero. � Sum of Rows 1 – 15, A – P, sum to Zero.
All rows have mirrors, summing to zero; all columns have mirrors.
18
COMPOSITE SQUARE – Σ ACROSS AND DOWN
A B C D E F G H I J K L M N O P
1
-125
-163
-189
+1
+259
+221
+195
+157
+131
+93
+67
+29
+3
-35
-61
-114
2
+121
+167
+185
+487
-263
-217
-199
-153
-135
-89
-71
-25
-7
+39
+57
+116
3
-117
-171
-181
-491
+267
+213
+203
+149
+139
+85
+75
+21
+11
-43
-53
-118
4
+118
+170
+182
+490
-266
-214
-202
-150
-138
-86
-74
-22
-10
+42
+54
+115
5
-131
-157
-195
-477
+253
+227
+189
+163
+125
+99
+61
+35
-3
-29
-67
-110
6
+135
+153
+199
+473
-249
-231
-185
-167
-121
-103
-57
-39
+7
+25
+71
+108
7
-139
-149
-203
-469
+245
+235
+181
+171
+117
+107
+53
+43
-11
-21
-75
-106
8
+143
+145
+207
+465
-241
-239
-177
-175
-113
-111
-49
-47
+15
+17
+79
+104
9
-147
-141
-211
-461
+237
+243
+173
+179
+109
+115
+45
+51
-19
-13
-83
-102
10
+151
+137
+215
+457
-233
-247
-169
-183
-105
-119
-41
-55
+23
+9
+87
+100
11
-155
-133
-219
-453
+229
+251
+165
+187
+101
+123
+37
+59
-27
-5
-91
-98
12
+146
+142
+210
+462
-238
-242
-174
-178
-110
-114
-46
-50
+18
+14
+82
+109
13
-109
-179
-173
-499
+275
+205
+211
+141
+147
+77
+83
+13
+19
-51
-45
-122
14
+105
+183
+169
+503
-279
-201
-215
-137
-151
-73
-87
-9
-23
+55
+41
+124
15
-101
-187
-165
-507
+283
+197
+219
+133
+155
+69
+91
+5
+27
-59
-37
-126
16
-31
-1
-31
+255
-31
-1
-31
-1
-31
-1
-31
-1
-31
-1
-31
Column M Reverse Sign Triplets, Column P Pseudo-Pairs, Row 16 alternating -31 and -1.
19
COMPOSITE SQUARE – Σ ACROSS AND DOWN ANALYSIS
� Sum of rows, with the exception of row 1, and rows 12 & 13, are alternating
positive and negative pseudo pair increments of 2 as follows:
Row 1 = +469 2 = + 13 3 = -11 4 = + 9
5 = - 17 6 = +19 7 = -21 8 = +23 9 = -25 10 = +27 11 = -29 12 = +31 13 = -7 14 = +5 15 = -3 16 = +1
� The sum of columns generate mirrors as follows
Column A and H = -136 + 136 B and G = -184 + 184 C and F = -200 +200 -520 +520 I and P = +120 -120
J and O = +72 - 72 K and N = +56 – 56 L and M = + 8 -8 +256 -256
� Furthermore, the sum of D and E = +484 and the sum of all columns = +484
o Column P consists of Pseudo-Pairs, with the exceptions of Rows 4 and 12, (115
and 109).
20
COMPOSITE SQUARE Σ MOD 9A A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
Mirrors and Twins… with an anomaly in column M at rows 2 and 7 and rows 3 and 6. In both cases the anomalies
sum to +9 and -9
Total = 58.2%
21
COMPOSITE SQUARE Σ MOD 9B A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
6 – 1 Couplings
Total = 18.2%
22
COMPOSITE SQUARE Σ MOD 9C A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
Homogenous Pairs
Total = 22.2%
23
COMPOSITE SQUARE Σ MOD 9D
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
Position and/or Sign
Triplets and
Rows 11 and 12 Mirrors (Red Squares are Shared Numbers)
Total = 96%
24
COMPOSITE SQUARE Σ MOD 9E
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
9 – 4 Couplings Total =15%
25
COMPOSITE SQUARE Σ MOD 9F
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
8 – 3 couplings
Total = 18%
26
COMPOSITE SQUARE Σ MOD 9G
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
8 – 4 Couplings
Total = 14.2%
27
COMPOSITE SQUARE Σ MOD 9H
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
7-3 Couplings
Total = 20%
28
COMPOSITE SQUARE Σ MOD 9i
A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
7 – 2 Couplings
Total = 20%
29
COMPOSITE SQUARE Σ MOD 9J A B C D E F G H I J K L M N O
1
-7
-1
-9
+1
+7
+5
+6
+4
+5
+3
+4
+2
+3
-8
-7
2
+4
+5
+5
+1
-2
-1
-1
-9
-9
-8
-8
-7
-7
+3
+3
3
-9
-9
-1
-5
+6
+6
+5
+5
+4
+4
+3
+3
+2
-7
-8
4
+1
+8
+2
+4
-5
-7
-4
-6
-3
-5
-2
-4
-1
+6
+9
5
-4
-4
-6
-9
+1
+2
+9
+1
+8
+9
+7
+8
-3
-2
-4
6
+9
+9
+1
+5
-6
-6
-5
-5
-4
-4
-3
-3
+7
+7
+8
7
-4
-5
-5
-1
+2
+1
+1
+9
+9
+8
+8
+7
-2
-3
-3
8 +8
+1
+9
+6
-7
-5
-6
-4
-5
-3
-4
-2
+6
+8
+7
9
-3
-6
-4
-2
+3
+9
+2
+8
+1
+7
+9
+6
-1
-4
-2
10
+7
+2
+8
+7
-8
-4
-7
-3
-6
-2
-5
-1
+5
+9
+6
11
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
-9
-5
-1
12
+2
+7
+3
+3
-4
-8
-3
-7
-2
-6
-1
-5
+9
+5
+1
13
-1
-8
-2
-4
+5
+7
+4
+6
+3
+5
+2
+4
+1
-6
-9
14
+6
+3
+7
+8
-9
-3
-8
-2
-7
-1
-6
-9
-5
+1
+5
15
-2
-7
-3
-3
+4
+8
+3
+7
+2
+6
+1
+5
+9
-5
-1
Special Case Triplets (Colored Squares Only)
Total = 17.3%
30
COMPOSITE SQUARE Σ MOD 9A - J ANALYSIS
� The sum of columns A – O = +65 � The sum of rows 1 – 15 = +65
� 96% of the cells are Position and/or Sign Triplets with Row 11 and 12 Mirrors � 58.1% of the cells consists of Mirrors and Twins � 22% are Homogenous Pairs � 20% are 7 – 3 Couplings � 20% are 7 – 2 Couplings � 18% are 8 – 3 Couplings � 18.2% are 6 – 1 Couplings � 17.3% are Special Case Triplets � 15% are 9 – 4 Couplings � 14.2% are 8 – 4 Couplings
31
COMPOSITE SQUARE ROW 16 AND COLUMN P SYNTHESIS AND ANALYSIS
P
Synthesis Sum Mod 9
Row 16 + Column P = N
-145 (-31 -114 = -145) -1
+115 (-1 + 116 = +115) +7
-149 (-31 -118 = -149, etc.) -5
+370 +1
-141 -6
+107 +8
-137 -2
+103 +4
-133 -7
+99 +9
-129 -3
+108 +9
-153 -9
+123 +6
-157 -4
• All Synthesized Numbers Are Alternating polarities, i.e., - + - + - + - , etc…
• The Numbers in Column P are alternating Pseudo Pairs based on
differences of two, with the exception of +115 and +109.
• In Row 16, the alternating -31 and -1’s sum to -254; when added to column D,
+255, the resultant is +1.
• The sum of Mod 9 resultants is +7
16
-31
-1
-31
+255
-31
-1
-31
-1
-31
-1
-31
-1
-31
-1
-
31
-114
+116
-118
+115
-110
+108
-106
+104
-102
+100
-98
+109
-122
+124
-126
32
CONCLUSIONS AND IMPLICATIONS
This analysis of Ben Franklin’s Most Magically Magic Square was originally conceived as a
companion document to The Law of Digit Balance to demonstrate the efficacy of Increment
Analysis. The results show that Increment Analysis does pinpoint a new way of analyzing
number relationships on a multi-dimensional basis.
It has been found that the numbers in Ben Franklin’s are arranged in a Grand Chessboard of
alternating polarities, twins and mirrors when the increments are analyzed. 100% of the square
exhibits these characteristics, along with interesting triplet pairs (+7 -2 + 2; +2 -7+7) and
other fascinating patterns. Of particular interest is the Composite Square, which, counter
intuitively, exhibits these same characteristics with an ordering of patterns even after several
permutations. Because of this, it can be clearly seen that the positive and negative increments
between numbers bear some import in number relationships and, when these meanings are
eventually determined, may lead to broad, new vistas in number theory and consequent
applications.
Σ ∆n = 0
33
34
MATHEMATICS/NUMBER THEORY $10.00
INCREMENT ANALYSIS IS A REVOLUTIONARY UNIMATH TECHNIQUE
USED FOR UNCOVERING HIDDEN PATTERNS IN NUMBER SEQUENCES. IT
REVEALS NEW SYMMETRIES IN BEN FRANKLIN’S MOST MAGICAL, MAGIC
SQUARE AND HAS IMPLICATIONS FOR A DEEPER UNDERSTANDING OF
NUMBER DIMENSIONALITY.
AASSTTRROOPPOOIINNTT
IISSBBNN:: 00--99661155445544--44--55
