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Grogono Magic Squares Home Page

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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 Pan-Magic squares. This website reflects my own fascination with these large numbers and presents techniques aimed at explaining and reducing the huge numbers by showing how this abundance can be reduced to a small number of underlying patterns or Magic Carpets.

Make Your Own Pan-Magic Squares

Do it yourself! This is now working again. Changes in the language had stopped it working for a while. Make your own magic square of any size up to 97x97.


The development of this website was associated with several intriguing discoveries. Please look at the pages for the Order 4, Order 5, Order 6 magic squares.


This Magic Square website is dedicated to my father E.B. Grogono (1909 - 1999) and was originally created at his bedside during his last illness. My fondest memories of him, from my earliest childhood to the final days of his life, center on his ability to transmit his love for, and fascination with, mathematics and science.


This revision uses up to date technology to make the website easier to manage and the material has been re-arranged to make it more accessible. A glossary has been added and the index system has been revised.

5x5 Square

Now, Belatedly, Welcome!

Visit, play, learn about Magic Squares and Magic Carpets, make your Own Magic squares, and explore the techniques devised to understand pan-magic squares.


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

Pan-Magic Squares.

The Main focus of this website is 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.

Why start Magic Squares using zero?

A glib answer might be because I like to and this is my site. A mathematical answer is that analysis (and construction) of magic squares is more logical, and the results easier to analyze, when the smallest number is 0. This is particularly true 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 carpets would be required using: 8 & 0; 4 & 0; 2 & 0; and 1 & 0.

The Formula.

I am frequently asked to provide the 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