ON SQUARES OF COMPLEX SQUARES - AN OFF THE WALL APPROACH

Picture of a square

A Discussion of the Method

Magic squares whose row, column and diagonals sums are equal has been of academic interest and to date squares having all these attributes have only achieved only partial success. Although a magic square containing 7 square entries Andrew Bremner-On Squares of Squares has been accomplished, attempts to increase that number to 8 or 9 is still some way off. An alternative approach is to employ as a square a complex number and its conjugate. It is known that multiplying a complex number and its conjugate always produces a real number. For example (3 + 4i) (3 - 4i) produces the real number 25.

The construction of these magic squares utilizes the approach used in The wheel method:Variant 1 with a slight variation.

  1. Normally the left diagonal is filled with the group of numbers ½(n2 - 1) to ½(n2 + 3) in consecutive order (top left corner to the right lower corner) from the numbers listed in the 3x3 complementary table. For a 3x3 square the numbers in the left diagonal correspond to 4 → 5 → 6.
  2. Add the right diagonal in reverse order from bottom left corner to the right upper corner.
  3. This is followed by the central column in regular order.
  4. Then by the central row in reverse order.

This produces the well known 3x3 square:

438
951
27 6

where the difference between successive numbers is one, i.e, Δ = 1. By successive is meant using the figure at the top of the page and moving along each line to the appropriate color ball (blue to blue, orange to orange and magenta to magenta. The distance from one colored ball to the same colored ball is Δ. In order to measure Δ the square must be in the right configuration, i.e., the right diagonal has the numbers ½(n2 - 1) to ½(n2 + 3).

In order to generate the requisite square or squares a different approach is taken. 3x3 squares are to be generated from complementary tables where n = 17 and n = 25 where the total numbers in each table are 289 and 625, respectively. See modified wheel and Loubère method for a similar approach. The reason for choosing these large squares is that the numbers ranging from ½ (n2 - 1) to ½(n2 + 3) have a Δ = 1, i.e,

(144, 145, 146) for an n = 17
and
(312, 313, 314) for an n = 25

and a second very important property which will be mentioned shortly.
Because Δ = 1 many 3x3 squares can be generated from one complementary table as long as the conditions listed above are adhered to.

Conditions for Building a Square of Complex Numbers

If we treat the complementary table as a table of complex numbers (second important property) conforming to the following rules:

Rule 1: an imaginary part that is not equal to 0
or
Rule 2: an imaginary part equal to 0 and a real part which when multiplied by its complex conjugate produces a real square

then the following complex numbers along with their complex conjugates are produced. Numbers that do not conform to this rule (although equal to their complex conjugates) are not used in building a 3x3 square. These complex numbers, as mentioned previously, when multiplied with their conjugates are real numbers equivalent to the real numbers above.

{(11 + 3i)(11 - 3i), (12 + i)(12 - i), (17 + 4i)(17 - 4i)} for n = 17
and
{(17 + 3i)(17 - 3i), (12 + 13i)(12 - 13i), (18 + 2i)(18 - 2i)} for n = 25

However, when an attempt is made to produce squares using these three set of complex numbers and a Δ = 1 the most complex squares that can be generated are 8. It is only in both cases when a Δ = 15 that a square of 9 complex numbers obeying the above 2 rules is possible.

Setting up the Complementary Tables and Magic Squares

17x17 Complementary tables

The complementary tables for the typical 17x17 square is shown below along with the complementary table in complex numbers.

1 . . . 16 17 18 . . . 34 . . . 49 . . . 64 . . . . . 130 . . . 144
145
289 . . . 274 273 272 . . . 256 . . . 241 . . . 226 . . . . . 160 . . . 146
1+0i . . . 4+0i 4+i 3+3i . . . 5+i . . . 49+0i . . . 64+0i . . . . . 11+3i . . . 12+0i .
12+i
17+0i . . . 15+7i 273 16+4i . . . 16+0i . . . 15+4i . . . 15+i . . . . . 12+4i . . . 11+5i

Multiplication of the complex numbers in the latter table produce with their complex conjugates generates the square of complex numbers I using the green entries above where the Δ is 15. Inspection of the three adjacent cells 15+7i, 273 and 16+4i shows that the middle entry 273 does not conform to Rule 1 or Rule 2. In fact the six adjacent entries along with the entries adjacent to 145 (and 145) produce a square of eight complex numbers that conform to Rules 1 and 2 and one that does not. The following two magic squares are produced when the green entries are entered according to the list of rules at the beginning of the page. Magic square 1 is identical to magic square 2. The latter magic square is the traditional way the square is drawn and allows one to calculate Δ as shown in the figure at the top of the page.

Complex No.
(11 + 3i)(11-3i)(8 + 0i)(8 - 0i)(15 + 4i)(15 - 4i)
(4 + 0i)(4 - 0i)(12 + i)(12 - i)(5 + 3i)(5 - 3i)
(7 + 0i)(7 - 0i)(15 + i)(15 - i) (12 + 4i)(12 - 4i)
Real No.
13064241
25614534
49226160
Real No.
49256130
22614564
16034241

25x25 Complementary Tables

The complementary table for the typical 25x25 square is shown below along with the complementary table in complex numbers. In this case one of he cells adjacent to the center of the table, viz, ½(n2 - 1) does not conform to either Rule 1 or 2 and, therefore, Δ = 1 cannot be constructed. However, two sets of magic squares are possible both having Δ of 15 using the same values for the left diagonal. A third square composed entirely of odd numbers is also included at the end with a Δ of 20.

1 . . . . . 34 . . . 49 . . . 64 . . . . . 106 . . . 121 . . . 136 . . . . . 298 . . 312
313
625 . . . . . 592 . . . 577 . . . 562 . . . . . 520 . . . 505 . . . 490 . . . . . 328 . . 314
1+0i . . . . . 5+3i . . . 11+0i . . . 10+6i . . . . . 9+5i . . . 11+0i . . . 8+0i . . . . . . 17+3i . . 312
12+13i
25+0i . . . . . 24+4i . . . 21+4i . . . 21+11i . . . . . 16+0i . . . 15+4i . . . 15+i . . . . . . 18+2i . . 17+5i

Below are the two sets of magic squares generated from the complementary tables:

Complex No.
(17 + 3i)(17-3i)(8 + 0i)(8 - 0i)(21 + i)(21 - i)
(24 + 4i)(24 - 4i)(12 +13i)(12 - 13i)(5 + 3i)(5 - 3i)
(7 + 0i)(7 - 0i)(21 + 11i)(21 - 11i) (18 + 2i)(18 - 2i)
Real No.
29864577
59231334
49562328
Complex No.
(17 + 3i)(17-3i)(10 + 6i)(8 - 6i)(21 + 8i)(21 - 8i)
(22 + 6i)(22 - 6i)(12 +13i)(12 - 13i)(9 + 5i)(9 - 5i)
(11 + 0i)(11 - 0i)(21 + 7i)(21 - 7i) (18 + 2i)(18 - 2i)
Real No.
298136505
520313106
121490328

Below is the square having a Δ of 20 and composed entirely of odd numbers:

1 . . . . . . . 41 . . . 61 . . . 81 . . . . . 293 . . 312
313
625 . . . . . . . 585 . . . 565 . . . 545 . . . . . 333 . . 314
1+0i . . . . . . . 15+4i . . . 5+6i . . . 8+0i . . . . . 17+i . . 312
12+13i
25+0i . . . . . . . 24+3i . . . 23+6i . . . 23+4i . . . . . 18+3i . . 17+5i
Complex No.
(17 + 4i)(17-4i)(9 + 0i)(9 - 0i)(23 + 6i)(23 - 6i)
(24 + 3i)(24 - 3i)(12 +13i)(12 - 13i)(15 + 4i)(15 - 4i)
(5 + 6i)(5 - 6i)(23 + 4i)(23 - 4i) (18 + 3i)(18 - 3i)
Real No.
29381565
58531341
61545333

This concludes the complex square method.
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Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com