NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES
GENERATION OF MAGIC SQUARES WITH SEVEN SQUARES (Part IG)
Magic Squares of Seven Squares with Imaginary NumbersSection I
Previously using non imaginary integers, the magic square below containing seven squares was discovered by Andrew Bremner and independently by Lee Sallows:
373^{2}  289^{2}  565^{2} 
360721  425^{2}  23^{2} 
205^{2}  527^{2}  222121 
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, n^{2}. 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 = c^{2} − b^{2} = b^{2} − a^{2}
Sum = Magic sum
Table G1 (Diagonal Tuples)
ni  (a × n)i 
(b × n)  (c × n) 
n 
1i  i  2  3  1 
2i  2i  4  6  2 
3i  3i  6  9  3 
4i  4i  8  12  4 
5i  5i  10  15  5 
6i  6i  12  18  6 
7i  7i  14  21  7 
8i  8i  16  24  8 
9i  9i  18  27  9 
10i  10i  20  30  10 
11i  11i  22  33  11 
12i
 12i 
24  36  12 

 
Δ = 720, Sum=1728
41^{2}  1249  36^{2} 
191  24^{2}  31^{2} 
(12i)^{2}  49^{2} 
(23i)^{2} 

≡ 
1681  1249  1296 
191  576  961 
144  2401  529 

After doing an exhaustive search for each line of Table G1 using the equation:
c^{2} = a^{2} + 2b^{2}
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  7i  4  9  1 
2i  14i  8  18  2 
3i  21i  12  27  3 
4i  28i  16  36  4 
5i  35i  20  45  5 
6i  42i  24  54  6 
7i  49i  28  63  7 
8i  56i  32  72  8 
9i  63i  36  81  9 
10i  70i  40  90  10 
11i  77i  44  99  11 
12i
 84i 
48  108  12 

 
Δ = 9360, Sum=6912
97^{2}  (119i)^{2}  108^{2} 
4559  48^{2}  7^{2} 
(84i)^{2}  137^{2}  4801 

≡ 
9409  14161  11664 
4559  2304  49 
7056  18769  4801 

Again after doing an exhaustive search for each line of Table G2 using the equation:
c^{2} = a^{2} + 2b^{2}
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  7i  6  11  1 
2i  14i  12  22  2 
....  ...  ...  ...  ... 
12i
 84i 
72  132  12 

  
Δ = 12240, Sum=15552
113^{2}  (121i)^{2}  132^{2} 
9839  72^{2}  23^{2} 
(84i)^{2}  25009 
(49i)^{2} 

≡ 
12769  14641  17424 
9839  5184  529 
7056  25009  2401 

Δ = 12240, Sum=15552
(121i)^{2}  113^{2} 
132^{2} 
193^{2}  72^{2}  26881 
(84i)^{2}  (49i)^{2} 
25009 

≡ 
14641  12769  17424 
37249  5184  26881 
7056  2401  25009 

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 NumbersSection II
A second type of Magic squares with Imaginary numbers consists of the type (ni, 0, n) whuch is constructed
as follows:
Square P_{1}
a^{2}  (di)^{2}  b^{2} 
(ci)^{2}  0^{2}  c^{2} 
(bi)^{2}  d^{2}  (ai)^{2} 

≡ 
Square P_{2}
a^{2}  (a^{2}+b^{2})  b^{2} 
(a^{2}b^{2})  0^{2}  a^{2}b^{2} 
b^{2}  a^{2}+b^{2}  a^{2} 

whereby d^{2} = a^{2}+b^{2} and c^{2} = a^{2}b^{2}. In addition, both squares have
Δ = b^{2} and Sum=0.
If we use for example n = 8, the following two magic squares containing
((8i)^{2}, 0, 8^{2}) 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 5^{2}, 4^{2} and 15^{2}, respectively.
Square A(Δ = 64, Sum=0)
10^{2}  164  8^{2} 
(6i)^{2}  0^{2}  6^{2} 
(8i)^{2}  164  (10i)^{2} 


Square B(Δ = 64, Sum=0)
17^{2}  353  8^{2} 
(15i)^{2}  0^{2}  15^{2} 
(8i)^{2}  353  (17i)^{2} 

Square C(Δ = 3600)
61^{2}  7321  60^{2} 
(12i)^{2}  0^{2}  12^{2} 
(60i)^{2}  7321  (61i)^{2} 


Square D(Δ = 3600)
65^{2}  7825  60^{2} 
(25i)^{2}  0^{2}  25^{2} 
(60i)^{2}  7825  (65i)^{2} 


Square E(Δ = 3600)
68^{2}  8224  60^{2} 
(32i)^{2}  0^{2}  32^{2} 
(60i)^{2}  8224  (68i)^{2} 


Square F(Δ = 3600)
75^{2}  9225  60^{2} 
(45i)^{2}  0^{2}  45^{2} 
(60i)^{2}  9225  (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 P_{2}:
d^{2} = (a^{2} + b^{2})  (a) 
d = √(a^{2} + b^{2}) =
√(a^{2} + b^{2}) i  (b) 
c^{2} = a^{2}  b^{2}  (c) 
c = √(a^{2}  b^{2})  (d) 
For c^{2} and d^{2} to be both perfect squares c^{2} = d^{2}, i.e.,  
(a^{2}  b^{2}) = (a^{2} + b^{2})  (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)
a^{2}  a^{2}  0 
a^{2}  0  a^{2} 
0  a^{2}  a^{2} 


Square H(Δ=b^{2},Sum=0)
0  b^{2}  b^{2} 
b^{2}  0  b^{2} 
b^{2}  b^{2}  0 

Pythagorean theorem and Magic Squares of Seven Squares
Alternatively, we can use the Pythagorean theorem to generate the table below using Square P_{1 and 2} as template:
We can generate the following series of squares based on the Pythagorean theorem: a^{2} + b^{2} = d^{2} and force all
the numbers in row 1 to be squares. Listed below are three examples. Example I uses the lowest set of squares
3^{2} + 4^{2} = 5^{2}, example II uses
6^{2} + 8^{2} = 10^{2} and example III uses 9^{2} + 12^{2} = 15^{2}:
Example  c  d 
I  √(4^{2}  3^{2}) 
√(3^{2} + 4^{2}) i 
II  √(8^{2}  6^{2}) 
√(8^{2} + 6^{2}) i 
III  √(12^{2}  9^{2}) 
√(9^{2} + 12^{2}) i 
Ex I, Δ = 9, Sum=0
4^{2}  5^{2}  3^{2} 
7  0  7 
3^{2}  5^{2}  4^{2} 


Ex II, Δ = 36, Sum=0
8^{2}  10^{2}  6^{2} 
28  0  28 
6^{2}  10^{2}  8^{2} 


Ex III, Δ = 81, Sum=0
12^{2}  15^{2}  9^{2} 
63  0  63 
9^{2}  15^{2}  12^{2} 

Examining these squares, c^{2} is actually equal to 7n^{2} 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 a^{2}
A 3x3 magic square of squares, having a a^{2} in the left lower corner cell, is composed of nine cells having the structure as shown in Square I
when four variables a^{2}, b^{2}, c^{2} and
d^{2} 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.,:
c^{2} + d^{2} + 2a^{2} + 4b^{2} = 2c^{2} + 2d^{2} + 4a^{2} + 4b^{2}
2a^{2} = (c^{2} + d^{2})
d^{2} = c^{2}  2a^{2} = (c^{2} + 2a^{2})
then substituting c^{2}  2a^{2} for d^{2} affords Square II in terms of only a^{2}, b^{2} and c^{2}:
Square I
c^{2} + a^{2} + b^{2}  d^{2}  a^{2} + 2b^{2} 
d^{2} + 2a^{2} + 2b^{2}  b^{2}  c^{2} 
a^{2}  c^{2} + 2a^{2} + 2b^{2}  d^{2} + a^{2} + b^{2} 

→ 
Square II
c^{2} + a^{2} + b^{2}  2a^{2}  c^{2}  a^{2} + 2b^{2} 
c^{2} + 2b^{2}  b^{2}  c^{2} 
a^{2}  c^{2} + 2a^{2} + 2b^{2}  a^{2} + b^{2}  c^{2} 

Furthermore Δ, the variable added to those cells having a b^{2}, c^{2} and d^{2} as shown in the picture
at the beginning of the page, is equal to a^{2} + b^{2} 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 3b^{2}. 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. EMail: enaguti1949@gmail.com