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.
The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes.
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[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
.
This formula is a consequence of the formula for the sum of the first n integers
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):
The numbers in any row, column, or diagonal of a normal magic square form a magic series.
[edit] Magic cubes
Similarly, if a magic cube consists of the numbers 1, 2, ..., n³, then it has magic constant (sequence A027441 in OEIS)
[edit] Magic stars
The magic constant of an n-pointed normal magic star is M = 4n + 2.
