Magic constant
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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.
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
- <math>M_2(n) = \frac{n(n^2+1)}{2}<math>.
This formula is a consequence of the formula for the sum of the first n integers
- <math>1 + 2 + ... + n = \frac{n(n+1)}{2}<math>
applied to the case n=n2, yielding <math>\frac{n^2(n^2+1)}{2}<math>, 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):
The term magic constant is similarly applied to other "magic" figures such as magic stars and magic cubes.
External link
- 260 as magic constant for 8-queens problem and 8x8 magic square (http://www.muljadi.org/EightQueens.htm)