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Grogono Magic Squares Home Page

Same Author:
Napkin Folding
Animated Knots
Stereo Art

in Most


This Magic Square website shows how the abundance of Magic Squares can sometimes be reduced to a few underlying patterns. It is dedicated to my father E.B. Grogono (1909 - 1999) and much of it was created at his bedside during his final illness. My fondest memories of him center on his ability to transmit his love for, and fascination with, mathematics and science.


A Magic square is intriguing; its complexity challenges the mind. For order 4 and above the number of different magic squares is astonishing - and the number remains large even if we limit consideration to Regular Pan-Magic squares. This website presents techniques that reduce the huge numbers to a much smaller number of underlying patterns or Magic Carpets.

Make Your Own Pan-Magic Squares

Do it yourself! This allows you to Make your own magic square of any size up to 97x97.


If you want to check on the meaning of the terms used on this website, please review the Glossary

Regular Pan-Magic Squares.

The Main focus of this website is Regular Pan-Magic squares where even the broken diagonals add up to the Magic Sum, e.g., 60 in the square above (diagonal 13, 2, 16, 5, 24). Pan-magic squares have also been called Pan-Diagonal and Nasik. All these squares are based on regular underlying patterns.

Why start Magic Squares using zero?

A best answer is that it allows all the Component squares to start at zero. This is particularly useful when the Magic Carpet approach is used to analyze or construct a magic square, e.g., to construct an order four magic square, four magic Components would be required using: 8 & 0; 4 & 0; 2 & 0; and 1 & 0.

The Formula.

I am frequently asked to provide a Formula for Magic Squares. At the risk of spoiling some teacher's classwork assignments I have worked out a satisfactory answer and have devoted a page to this topic. (Thank you Danny Lawrence for making me do this and for sitting with me while I worked it out.) Two formulae are included, one for the prime-number orders, e.g., 5, 7, 11, 13, etc., and one for an order 4 square.

3x3 Square

How Many Squares?

My fascination with magic squares grew from experimental attempts to count the total number of possible squares when squares which, apparently different, were really identical when appropriately reflected or rotated. A separate page lists the Number of Pan-Magic Squares for the Prime Number Order squares.

Unique Identifiers.

The process of counting and comparing regular panmagic squares generated a need to identify squares to facilitate ranking and comparison. Out of this grew a scoring system to uniquely identify any order 4 or order 5 pan-magic square.

The method I developed assigns a Unique Identifier to each square and is applicable to regular panmagic squares of orders 4 and 5. It depends on summing defined cells which have been multiplied by successively higher powers of the square's order. Although the technique could be extended to larger squares, the length of the expression, and the resulting magnitude of the numbers, makes it too unwieldy.

By the Same Author

If you have found this website useful, you are invited to visit one of my other teaching sites.

Two of these other sites are mounted on my main website but all three are treated as an independent website:


Copyright © Mar 2010 Magic Squares
Mar 6, 2010