An Order-5 Pandiagonal, Associative, Compact, and Self-similar Magic Square

Introduction

The example on this page is used to show some of the number patterns that sum to the magic constant.
Also some definitions to help you understand my explanation of this square.

The Order-5 Magic Square

See how five numbers can combine in 1128 (?) ways to sum to 65.

Some Definitions

Definitions

Pandiagonal

Associative

A magic square where all pairs of cells diametrically equidistant from the center of the square equal the sum of the first and last terms of the series, or m² + 1. Also called Symmetrical or Regular. The center cell of odd order associated magic squares is always equal to the middle number of the series. Therefore the sum of each pair is equal to 2 times the center cell and the sum of any 2 symmetrical pairs plus the center cell is equal to the constant. This permits a great many combinations (the order 5 square above has 58 of this type).
In an even order magic square, the sum of any 2 symmetrical pairs will equal the constant.
There are NO singly-even Associated magic squares.
The one order-3 magic square is associative.

Compact means corner and center cells of all small squares (including rhombics) sum to the constant. This is an extension of the formal definition of Compact as written for double even order pandiagonal magic squares. See Glossary. Self-similar means that when each number is changed to its complement (i.e. subtracted from m+1), an identical magic square is formed but rotated 180 degrees. This term was coined by Mr. Mutsumi Suzuki who discovered six such squares. See his excellent site at http://www.pse.che.tohoku.ac.jp/~msuzuki/  (This is now available at http://www.archive.org/  (the Wayback Machine) Wrap-around means when you go off of one edge, continue (wrap-around) to the corresponding cell on the opposite edge. The easiest way to accomplish this is to lay out the magic square in a repeating pattern on the plane as shown to the right. Then any 5 x 5 array is an equivalent pan-diagonal magic square. So, to complete a line, you can just keep going in the same direction. The red cell values  illustrate wrap-around when applied to a broken diagonal pair. Wrap-around works only with pan-diagonal magic squares.

Five Symmetrical cell pairs

A Symmetrical cell pair is defined as 2 cells that are symmetrical around the center cell i.e. 1 & 25, 8 & 18, 4 & 22, 7 & 19 etc. Wrap-around doesn't work with these. The 3 squares illustrate the 5 basic  pairs. Each of these may be rotated and/or reflected and used in any unique combination of 2 pairs plus the center cell.

Note that some of these combinations are duplicates of the basic patterns counted at the start. Example; if the 2 patterns in the center diagram are lined up, this combination constitutes a main diagonal which has already been counted.

Eight Non-symmetrical patterns

The seven  patterns on the left (below) each appear 25 times because of wrap around and 4 times due to rotation ( i.e. 7 x 25 x 4 = 700).
They are symmetrical about a horizontal, vertical or diagonal axis and so cannot be reflected for unique solutions.
That pattern 1 will always be magic in an order-5 pandiagonal magic square, if it is rotated so the 3 adjacent cells are in any of the 4 corners was proven by Rosser and Walker in 1939. [1]

The pattern on the right is not symmetrical around a vertical, horizontal, or diagonal axis, so appears 25 times because of wrap around, 4 times due to rotation and 2 times because of reflection (i.e. 25 x 4 x 2 = 200).
Starting a given pattern on a different cell, in combination with a rotation or a reflection, may result in the same 5 cells being used. Example, start the first pattern with 9, 23, 1, 15, 17. Now rotate the pattern 90 degrees clockwise and start with 15,17, 1, 23, 9. Both combinations use the same 5 cells, but in a different arrangement. I consider them different combinations.
Of course, we could say the same about the use of wrap-around when computing the number of combinations for rows, columns and the diagonals. However, by convention, these extra combinations are not counted.

[1] R. Rosser and R.J Walker, The Algebraic Theory of Diabolic Magic Squares, Duke Mathematical Journal, Vol. 5, No, 4, Dec. 1939 pp 705-728 (p.717)

Addendum

On June 28/99, I received an e-mail from Aale de Winkel advising me that he thought  non-symmetric pattern 1, above, was actually equivalent to the wrap-around 2 by 2 rhombic, and patterns 2 and 8 were equivalent to the wrap-around 2 by 2 square.

Subsequent investigation on my part confirmed these equivalents, and also that four of  the other five patterns are also equivalent to symmetric patterns already counted.
1. is equivalent to wrap-around 2 by 2 rhombic ( now called plusmagic --- see Quadrant Magic Squares)
2. is equivalent to wrap-around 2 by 2 square ( now called crosmagic)
3. is equivalent to wrap-around 3 by 3 rhombic corners.
4. is equivalent to wrap-around 5 by 5 square corners.
5. is just a reflected 4. (another goof!).
6. is unique, so add 100 combinations to the total.
7. is equivalent to wrap-around 5 by 5 square corners.
8. is equivalent to wrap-around 2 by 2 square

The three diagrams below demonstrate the equivalence of  the 2 x 2 rhombic (plusmagic) and non-symmetric pattern 1.
Diagram A shows pattern 1 within the magic square and the plusmagic with two cells outside of the square. diagrams B and C show the patterns shifted down and to the right. In each of these cases, the plusmagic pattern is within the square and two cells of the pattern 1 outside.

Yet to be determined. Should the rotated non-symmetric patterns be counted and included in the total?
For now, let us say that there are 328 different combinations!
By the way. There are 1394 sets of 5 numbers, from the first 25 consecutive numbers, that sum to 65!

	This page was originally posted June 1998 It was last updated February 24, 2011 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz