TABLE OF RIGHT DIAGONALS GENERAL METHOD

GENERATION OF RIGHT DIAGONALS FOR MAGIC SQUARE OF SQUARES (Part IIA)

Picture of a square

Square of Squares Tables

Andrew Bremner's article on squares of squares included the 3x3 square:

Bremmer's square
5824621272
942113222
972822742

The numbers in the right diagonal as the tuple (972,1132,1272) appears to have come out of the blue. But I will show that this sequence is part of a larger set of tuples having the same property, i.e. the first number in the tuple when added to a difference (Δ) gives the second square in the tuple and when this same (Δ) is added to the second square produces a third square. All these tuple sequences can be used as entries into the right diagonal of a magic square.

We will show a general method for generating the squares for a right diagonal of a magic square. Beginning with the the tuple (1,b1, c1) we can generate the tuple (a, b, c) which when squared gives the diagonal numbers. Initially either b1 or c1 will be equal to ± k where k is any natural number 1,2,3,4.... Again the end result is that a12 + b12 + c123b120 = S is converted to a2 + b2 + c23b2 = 0 which is a necessary condition for the square to be magic.

To summarize the tuples of Table II below will be used as entries into a right diagonal of a magic square. Knowing the difference (b2a2) or (c2b2) will give us a value Δ which can be used to produce other entries into the magic square. To date only one magic square containing 7 entries has been found. Most other squares will contain 6 entries.

As to the reason for the picture of a square, the entries to the square occur as three tuples,viz, (a,b,c), (l,m,n) and (x,y,z) showing their connectivity. In addition, six or more of these entries are present as their squares.

Generation of Tables where c1 = 3

  1. The object of this exercise is to generate a Table I with a set of tuples that obey the rule: a12 + b12 + c123b120 and convert these tuples into a second set of tuples (Table II) that obey the rule: a2 + b2 + c23b2 = 0.
  2. In addition, we need to know two numbers e and g where g = 2e which when added to the b1 and c1 numbers of Table I, produce the next line of numbers (n + 1) in the next row of Table I. The number a1 will always be 1.
  3. Two other numbers f and d are calculated using the equation
    f = [2e2n2 + (4c1 − 4b1) en +(1 − 2b12 + c12)] / {2(2b1 − c1 − 1)}
    where n is the line number of the tables. f can also be generated directly from Table II from S/d. However, the value of d is equal to the denominator of the general equation above.
  4. Finally Δs are calculated by taking the difference in Table II between (b2a2) or (c2b2), and the results placed under the Δ column. Both differences must be the same.
  5. As an example we begin with the tuple (1,1,3), where a1 = 1, b1 = 1 and c1 = 3 and use the equation to generate f.
    f = [2e2n2 + (12 −4)en + 8] / 2× (−2) = [2e2n2 + 8en + 8] / (−4)
    Setting e = 2 and g = 4 affords f = −(2n2 + 4n + 2)
    Substituting for f in (b) gives
    a = (−2n2 −4n − 1 )
    b = (−2n2 −2n − 1)
    c = (−2n2 + 1)
  6. Substituting the appropriate n into the equations for a, b, and c produces Table II below. Using a computer program and the requisite calculations produced the tables below. As can be seen taking the value of f from the middle table and adding to a1, b1, c1, produced a, b, c, respectively of Table II.
  
n
0
1
2
3
4
5
6
7
8
9
10
11
12
Table I
a1 b1 c1
113
137
1511
1715
1919
11123
11327
11531
11735
11939
12143
12347
12551
  
f = S/d
-2
-8
-18
-32
-50
-72
-98
-128
-162
-200
-242
-288
-338
Table II
a b c
-1-11
-7-5-1
-17-13-7
-31-25-17
-49-41-31
-71-61-49
-97-85-71
-127-113-97
-161-145-127
-199-181-161
-241-221-199
-287-265-241
-337-313-287
  
Δ
0
-24
-120
-336
-720
-1320
-2184
-3360
-4896
-9240
-600
-12144
-15600


  1. The magic square A was found by Bremner and has 7 square terms with the magic sum (Sm) 541875 and n = 11. Two other examples are B and C having the right diagonal tuple (97, 113, 127) and (337, 365, 391)as their squares. The magic sum, Sm, for these cases are 38307 and 399675, respectively and the n's are 7 and 13, respectively.
Magic square A
373228925652
3607214252232
20525272222121
  
Magic square B
742167021272
234221132462
97294220062
  
Magic square C
25521817693912
29108136522132
33722912201425

This concludes Part IIA. To go back to Part IE. To continue to Part IIB which treats tuples of the type (1,1,11).
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Copyright © 2012 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com