  ## Introduction

Magic squares are arrangements of numbers into squares, cubes, or hyper-cubes where the sum of each row and each column is identical.  Additionally, the sum along the "principal diagonals" is also equal to the same number.  The magic square below was known in antiquity in China, Greece, and Egypt.

 2 9 4 7 5 3 6 1 8

It is interesting to note that subtracting a constant from each cell in a magic square or hyper-cube does not change its essential property: namely, that the sum of any row, column, or principal diagonal is identical.  For example, the following magic square is obtained by subtracting 5 from each cell in the magic square above:

 -3 4 -1 2 0 -2 1 -4 3

An interesting property about this square is that all rows, columns, and principal diagonals sum to 0.  Also, it is easy to visually recognize symmetry in the square.

This web site contains: Examples Instructions Applications Theory Algorithm Symmetry - examples using "tuples" Generator - program for generating magic squares and Hyper Cubes Other Web Sites for Magic Squares and Cubes

Please see the Instructions Page before viewing the Magic Squares and Hyper Cubes Generator.

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