Same Author:
|
Revision:
|
A Magic square is intriguing; its complexity challenges the mind, certainly more than games like online bingo. 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.
Do it yourself! 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.
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
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.
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.
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.
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.
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.
People try to use counting techniques for practical reasons as well. In card games, for instance, people go looking for an online casino where they can try to make money using various counting techniques, partly because counting itself is actually an exciting and challenging practice.
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:
This website is aimed at physicians, physiologists, medical students, nurses, and other health care professionals. The Tutorial includes interactive diagrams and equations to make the material more interesting and more readily understood.
This website is aimed at yachtsmen, scouts, climbers, fishermen and anyone else who needs to know how to tie Practical Safe Knots. Each animated knot "ties itself" automatically and can also be "tied" and "untied" slowly to reveal its structure.
This website demonstrates how a vivid three-dimensional stereo image is created from a repeated stereo image pair. A collection of Stereo Art Images illustrates the technique.
| Copyright © Dec 2009 | Magic Squares Website |
Updated December 11th 2009 |