My mathematically talented father introduced me to Magic Squares very early in my life. Through his enthusiasm, the patterns underlying the prime number squares, and the 4x4 square, stayed with me and provided me with material to idle away many lectures, and nearly all meetings, experimenting with magic squares. I realized that there were elegant patterns underlying the simple "Knight's move" method. As I explored these patterns I formulated the idea of underlying patterns as the basis for many magic squares.
When the Internet started growing I quickly found the accumulated list of sites provided by Dr. Suzuki . I was struck by the lengthy lists of distinct order 4 and order 5 squares. What I needed was a way of identifying each square to eliminate duplicates - particularly the many rotations, reflections, and translations of what, essentially, was the same square. Unique identifiers would make it possible to ennumerate squares conveniently.
While doing this I found that very simple patterns were the foundation of all order 4 and 5 pan magic squares. The example to the right is the pattern for the 4x4 pan-magic squares. These patterns seemed to invite me to repeat them in both axes, like a patterned carpet - hence the name magic carpets. These magic carpets provide us with an excellent tool to understand and make pan-magic squares of any size.
The sequence of squares below shows how this magic carpet can be used to create four little carpets. By selecting four pieces of the original carpet, with two of the pieces rotated, four distinct patterns can be found and multiplied by different powers of two ( 8, 4, 2, 1). These four numbers could have been used in any order. Here, they are used in descending order and the resulting squares are added together to make the final pan-magic square.
Initially, I had a happy, naive dream that all pan-magic squares would have a Magic Carpet structure; that every pan-magic square could be broken down into a series of simple carpets, each easily understood and recognized. Initial experiments appeared to confirm this idea. Pan-magic squares of order 7, 8, 9, and 11 were constructed using the carpet approach. They were easy to understand. The structures and the processes had a pleasing logic and symmetry, and each component carpet was pan-magic.
And, then in October of 1996, Mutsumi Suzuki reported on his visit to Akita University where he borrowed a Japanese book: "Researches In Magic Squares", by A. Hirayama and G. Abe, (1983). On page 278 he found a description of some 7x7 pan-magic squares which were called "irregular complete squares" which could not be decomposed into two orthogonal Latin squares. This news effectively destroyed my dream of a universal magic carpet construction for all pan-magic squares. The structure of these squares is described in more detail on the page about 7 x 7 magic Squares.
The existence of irregular squares does, of course, define the regular ones. They are the Pan-Magic Squares which can be constructed from a series of component Magic Carpets, each one of which has Pan Magic Properties. The associated pages describe the construction and properties of individual pan-magic squares in more detail:
|Copyright © Mar 2010
Mar 6, 2010