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 $n^2$.