Magic constant

The magic constant 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.

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MagicSquare-ExplicitSums.png
Image:MagicSquare-ExplicitSums.png

If a magic square of order n is normal (i.e. it contains the numbers 1 to n²), then any such sum is known as a magic series, and its magic constant depends only on n having the value

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

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

[itex]1 + 2 + ... + n = \frac{n(n+1)}{2}[itex]

applied to the case n=n2, yielding [itex]\frac{n^2(n^2+1)}{2}[itex], which is then divided by n because there are n rows, each 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 term magic constant is similarly applied to other "magic" figures such as magic stars and magic cubes.

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