NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

GENERATION OF MAGIC SQUARES WITH SEVEN SQUARES (Part IG)

Picture of a square

Magic Squares of Seven Squares with Imaginary Numbers-Section I

Previously using non imaginary integers, the magic square below containing seven squares was discovered by Andrew Bremner and independently by Lee Sallows:

373228925652
3607214252232
20525272222121

This section deals with novel magic squares of squares containing at least one imaginary number having the form (ni)2, which when squared becomes the square of a negative number, -n2. It has been found that there exists an infinite number of tuples (ai,b,c) containing at least one imaginary square which can be used as the right diagonal in magic squares. Methods for construction of these tuples are listed in Part IA, Part IA2, Part IB, Part IC, Part IIC, Part IIIC, Part ID, Part IE and Part IF.

I have listed below three tables each of which contain one tuple (in color) which was used to generate the magic square of seven squares on the right. The first table G1 containing the line n = 1 was obtained from line 1 of Part IB, Table V (odd number 1) and is incremented below in columns 2, 3 and 4 of table G1 by 1, 2 and 3, respectively, with Δ and Sum defined as:

Δ = c2 − b2 = b2 − a2
Sum = Magic sum
Table G1 (Diagonal Tuples)
ni(a × n)i (b × n) (c × n) n
1i-i231
2i-2i462
3i-3i693
4i-4i8124
5i-5i10155
6i-6i12186
7i-7i14217
8i-8i16248
9i-9i18279
10i-10i203010
11i-11i223311
12i -12i 243612
Δ = 720, Sum=1728
412-1249362
191242312
(12i)2492 (23i)2
 
1681-12491296
191576961
-1442401-529

After doing an exhaustive search for each line of Table G1 using the equation:

c2 = a2 + 2b2

only line 12 and its multiples generated the square on the right.

The second table G2 contains the line n = 1 was obtained from line 2 of Part IB, Table V (odd number 1) and is incremented below in columns 2, 3 and 4 of table G2 by 7, 4 and 9, respectively:

Table G2 (Diagonal Tuples)
ni(a × n)i (b × n) (c × n) n
1i-7i491
2i-14i8182
3i-21i12273
4i-28i16364
5i-35i20455
6i-42i24546
7i-49i28637
8i-56i32728
9i-63i36819
10i-70i409010
11i-77i449911
12i -84i 4810812
Δ = 9360, Sum=6912
972(119i)21082
455948272
(84i)21372-4801
 
9409-1416111664
4559230449
-705618769-4801

Again after doing an exhaustive search for each line of Table G2 using the equation:

c2 = a2 + 2b2

only line 12 and its multiples generated the square on the right.

The third table G3 contains the line n = 1 was obtained from line 4 of Part IB, Table VI (odd number 2) and is incremented below in columns 2, 3 and 4 of table G3 by 7, 6 and 11, respectively:

Table G3 (Diagonal Tuples)
ni(a × n)i (b × n) (c × n) n
1i-7i6111
2i-14i12222
................
12i -84i 72 132 12
Δ = 12240, Sum=15552
1132(121i)21322
9839722232
(84i)225009 (49i)2
 
12769-1464117424
98395184529
-705625009-2401
Δ = 12240, Sum=15552
(121i)21132 1322
1932722-26881
(84i)2(49i)2 25009
 
-146411276917424
372495184-26881
-7056-240125009

This time line 12 and its multiples affords two magic squares having the same right diagonal but differing in two of the numbers in the center row.

Magic Squares with Imaginary Numbers-Section II

A second type of Magic squares with Imaginary numbers consists of the type (-ni, 0, n) whuch is constructed as follows:

Square P1
a2(di)2b2
(ci)202c2
(bi)2d2(ai)2
Square P2
a2-(a2+b2)b2
-(a2-b2)02a2-b2
-b2a2+b2-a2

whereby d2 = a2+b2 and c2 = a2-b2. In addition, both squares have Δ = b2 and Sum=0.

If we use for example n = 8, the following two magic squares containing ((8i)2, 0, 82) as the right diagonal can be generated. Furthermore, increasing n produces larger amounts of magic squares of seven squares as shown for the first four squares for n = 60. In addition, Squares D, E and F are divisible by 52, 42 and 152, respectively.

Square A(Δ = 64, Sum=0)
102-16482
(6i)20262
(8i)2164(10i)2
Square B(Δ = 64, Sum=0)
172-35382
(15i)202152
(8i)2353(17i)2
Square C(Δ = 3600)
612-7321602
(12i)202122
(60i)27321(61i)2
Square D(Δ = 3600)
652-7825602
(25i)202252
(60i)27825(65i)2
Square E(Δ = 3600)
682-8224602
(32i)202322
(60i)28224(68i)2
Square F(Δ = 3600)
752-9225602
(45i)202452
(60i)29225(75i)2

We can determine if it is possible to construct magic squares having more than seven squares of the type shown here by analyzing the equations in Square P2:

d2 = -(a2 + b2)(a)
d = √-(a2 + b2) = √(a2 + b2) i(b)
c2 = a2 - b2(c)
c = √(a2 - b2) (d)
For c2 and d2 to be both perfect squares c2 = -d2, i.e.,
(a2 - b2) = (a2 + b2)(e)

However, this is only possible when a = any number and b = 0, or when b = any number and a = 0 or when a = b = 0. Thus, we have either a square where all cells are 0 or one where either the right diagonal contains all zeros or the left diagonal contains all zeros and the rest, each row, column and the other diagonal consists of non zero integers as shown below in Squares G and H:

Square G(Δ=0,Sum=0)
a2-a20
-a20a2
0a2-a2
Square H(Δ=b2,Sum=0)
0-b2b2
b20-b2
-b2b20

Pythagorean theorem and Magic Squares of Seven Squares

Alternatively, we can use the Pythagorean theorem to generate the table below using Square P1 and 2 as template:

We can generate the following series of squares based on the Pythagorean theorem: a2 + b2 = d2 and force all the numbers in row 1 to be squares. Listed below are three examples. Example I uses the lowest set of squares 32 + 42 = 52, example II uses 62 + 82 = 102 and example III uses 92 + 122 = 152:

Examplecd
I(42 - 32) (32 + 42) i
II(82 - 62) (82 + 62) i
III(122 - 92) (92 + 122) i
Ex I, Δ = 9, Sum=0
42-5232
-707
-3252-42
Ex II, Δ = 36, Sum=0
82-10262
-28028
-62102-82
Ex III, Δ = 81, Sum=0
122-15292
-63063
-92152-122

Examining these squares, c2 is actually equal to 7n2 where n is a multiple of a square and, therefore, c = √7n. Thus, c can never be an integer and only seven squares are possible with this types of magic square.

The Square Variables for Each Cell of a 3x3 Square with a Negative a2

A 3x3 magic square of squares, having a -a2 in the left lower corner cell, is composed of nine cells having the structure as shown in Square I when four variables a2, b2, c2 and d2 are used to specify the square. Since the sum of the four corner cells equals the sum of the four outside central cells, e.g.,:

c2 + d2 + 2a2 + 4b2 = 2c2 + 2d2 + 4a2 + 4b2
2a2 = -(c2 + d2)
d2 = -c2 - 2a2 = -(c2 + 2a2)

then substituting c2 - 2a2 for d2 affords Square II in terms of only a2, b2 and c2:

Square I
c2 + a2 + b2d2a2 + 2b2
d2 + 2a2 + 2b2b2c2
-a2c2 + 2a2 + 2b2d2 + a2 + b2
Square II
c2 + a2 + b2-2a2 - c2a2 + 2b2
-c2 + 2b2b2c2
-a2c2 + 2a2 + 2b2 -a2 + b2 - c2

Furthermore Δ, the variable added to those cells having a b2, c2 and d2 as shown in the picture at the beginning of the page, is equal to a2 + b2 and is part of cells 1, 3, 4, 5, and 6 moving in a horizontal direction (→). Thus, all we need to know is a, b and c to generate a magic square of squares having an imaginary a in its major diagonal. In addition, the Magic Sum for each line or diagonal is 3b2. And finally we can conclude that we can generate magic squares having at least 7 squares, but finding one with 8 or 9 squares is a major effort.

This concludes Part IG.

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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com