Magic constant

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The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15.

Image:MagicSquare-ExplicitSums.png

The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes.

[edit] Normal magic squares

If a magic square of order n is normal (i.e., it contains the numbers 1 to n2), then the magic constant depends only on n; its value is

M_2(n) = \frac{n(n^2+1)}{2}.

This formula is a consequence of the formula for the sum of the first n integers

1 + 2 + ... + k = \frac{k(k+1)}{2}

applied to the case k = n2, yielding n2(n2+1)/2, which is then divided by n because there are n rows, all of which sum to the same value.

The magic constants of normal magic squares of order n = 3, 4, 5, … are (sequence A006003 in OEIS):

15, 34, 65, 111, 175, 260, 369, 505, 671, 870

The numbers in any row, column, or diagonal of a normal magic square form a magic series.

[edit] External link

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