The 7x7 Irregular Pan-Magic Squares

Notes from Prof. Suzuki:

While I was preparing the first draft for this web site and writing the material about regular 7x7 pan-magic squares, I received the following e-mail from Prof. Mutsumi Suzuki (Sun, Oct 27, 1996) about irregular squares:

To all;

I visited Akita University last week to give a lecture. At the library of the Univ I borrowed a Japanese book on the magic squares: "Researches In Magic Squares", by A. Hirayama and G. Abe, (1983). This book was very exciting. One of the authers was a professor of my own University, the other was not a mathematician but an owner of a company of Japanese Lacqer ware.

I found strange squares in the book (p.278), which they called "irregular complete squares". These pan-magic squares are not decomposed into two orthogonal Latin squares!

(i) This "semi-irregular" case was discovered by A.L.Candy (1940)

1  8 19 25 35 39 48    0 1 2 3 4 5 6    0 0 4 3 6 3 5
31 41 44  2 12 17 28    4 5 6 0 1 2 3    2 5 1 1 4 2 6
11 21 24 33 37 43  6    1 2 3 4 5 6 0    3 6 2 4 1 0 5
36 47  4 14 18 26 30    5 6 0 1 2 3 4    0 4 3 6 3 4 1
20 23 29 40 46  7 10    2 3 4 5 6 0 1    5 1 0 4 3 6 2
49  3 13 16 22 34 38    6 0 1 2 3 4 5    6 2 5 1 0 5 2
27 32 42 45  5  9 15    3 4 5 6 0 1 2    5 3 6 2 4 1 0

      Square-A             Square-B         Square-C

                A = B x 7  + C  + 1

The matrix-B has simple (Grog's spread-sheet) pattern (Latin square) but C does not.

(ii) Abe Gakuho's irregular squares:

19  7  5 41 29 43 31   2 0 0 5 4 6 4   4 6 4 5 0 0 2
46 40 22  1  4 34 28   6 5 3 0 0 4 3   3 4 0 0 3 5 6
37 16 45 33 21 10 13   5 2 6 4 2 1 1   1 1 2 4 6 2 5
3  6 42 25 48 36 15   0 0 5 3 6 5 2   2 5 6 3 5 0 0
35 24 12  9 30 18 47   4 3 1 1 4 2 6   6 2 4 1 1 3 4
27 44 23 17 11 14 39   3 6 3 2 1 1 5   5 1 1 2 3 6 3
8 38 26 49 32 20  2   1 5 3 6 4 2 0   0 2 4 6 3 5 1

      Square-A           Square-B         Square-C

A is pan-magic but both B and C are not Latin squares! According to Abe's study, there are 64 classes 127073856 squares of such irregular squares! He also confirmed 3 kind 480090240 semi-irregular type squres. He wrote that his calculation was not completed, so there must be more squares.

- Mutsumi Suzuki, Sun, Oct 27, 1996

Pairwise Exchange.

His message destroyed my happy, simple thought that pan-magic squares are always based on magic carpets. He had not, however, provided any details of how such irregular squares might be constructed. This was provided in his follow-up message ( Mutsumi Suzuki, Tue, Oct 29, 1997) which gave some details about Abe Gakuho's "Pairwise exchange"

Followings are a summary of the note on the irregular pan-magic squares written in the book: "Researches In Magic Squares", by A. Hirayama and G. Abe, (1983)

Let us consider the following two Latin squares (Combined square Ax7+B+I is pan-magic):

5 1 0 2 4 6 3    5 2 3 0 6 1 4
2 4 6 3 5 1 0    1 4 5 2 3 0 6
3 5 1 0 2 4 6    0 6 1 4 5 2 3
0 2 4 6 3 5 1    2 3 0 6 1 4 5
6 3 5 1 0 2 4    4 5 2 3 0 6 1
1 0 2 4 6 3 5    6 1 4 5 2 3 0
4 6 3 5 1 0 2    3 0 6 1 4 5 2

   Square A        Square B

Then, let our interests be focused on the following octagonal pairs;

* 1 0 * * * *    * 2 3 * * * *
2 * * 3 * * *    1 * * 2 * * *
3 * * 0 * * *    0 * * 4 * * *
* 2 4 * 3 5 *    * 3 0 * 1 4 *
* * * 1 * * 4    * * * 3 * * 1
* * * 4 * * 5    * * * 5 * * 0
* * * * 1 0 *    * * * * 4 5 *

   Square A        Square B

The pairs are exchanged to another, by holding the same sum;

0+1 ---> 1+0
2+3 ---> 3+2
3+0 ---> 2+1
2+4 ---> 3+3
3+5 ---> 4+4
1+4 ---> 0+5
4+5 ---> 5+4
1+0 ---> 0+1

Then we obtain;

* 0 1 * * * *   * 3 2 * * * *
3 * * 2 * * *   0 * * 3 * * *
2 * * 1 * * *   1 * * 3 * * *
* 3 3 * 4 4 *   * 2 1 * 0 5 *
* * * 0 * * 5   * * * 4 * * 0
* * * 5 * * 4   * * * 4 * * 1
* * * * 0 1 *   * * * * 5 4 *

   Square A'       Square B'

The resulting squares A' and B' are not Latin squares, but the pan-magic property is unchanged. Thus the new square A'x 7 + B' + 1 is pan-magic.

Mr. Abe constructed various pattern of pairs: 6-pairs, 7-pairs, 8-pairs, 9-pairs, 10-pairs and 12-pairs. Combination of such exchange pairs yields various irregular squares.

- Mutsumi Suzuki, Tue, Oct 29, 1997

Comment

The occurence of irregular pan-magic squares can hardly be a surprise: the larger the square the more likely it must be that pan-magic squares could be made by various methods. It was a little surprising to find examples in squares as small as order 7 and it was disappointing because their existence. and nature, makes it impossible to employ simple arithmetic to calculate the total number of such squares.

Nevertheless, I am grateful to Prof. Suzuki for his providing this interesting information and it is a pleasure to thank him.

The 7x7 Magic Square Index

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Updated
Mar 6, 2010