# magic constant

Given a magic square, magic cube, etc., the sum of any row, column or diagonal is called the magic constant of that magic square, cube, etc.

In the case of a standard $n\times n$ magic square that uses the integers from 1 to $n^{2}$, the magic constant is

 $\frac{1}{n}\sum_{i=1}^{n^{2}}i,$

while that for a magic cube is

 $\frac{1}{n^{2}}\sum_{i=1}^{n^{3}}i.$

We can then generalize to higher dimensions $d$ thus:

 $\frac{1}{n^{d-1}}\sum_{i=1}^{n^{d}}i.$

So, for dimension $d$ the magic constant is $\frac{n(n^{d}+1)}{2}$. For instance, a Franklin magic square ($n=8,d=2$) has magic constant $\frac{8(8^{2}+1)}{2}=260$.

In a trivial sense, an $n\times n$ sudoku puzzle has a magic constant of $n^{2}$.

Title magic constant MagicConstant 2013-03-22 16:24:57 2013-03-22 16:24:57 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 05B15