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:
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)
(ii) Abe Gakuho's irregular squares:
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
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):
Then, let our interests be focused on the following octagonal pairs;
The pairs are exchanged to another, by holding the same sum;
0+1 ---> 1+0
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
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.
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