Question

Would any one know how to solve this arithmetic series on the number of magic squares created per order. n=order, A=integer, D=difference between terms. The solution is n=1,2,... are 1, 0, 1, 880, 275305224, ...

Answer 1

Explanation:

Answer 2

1.7761 * 10^19 (5*5 magic square estimation)

Explanation:

Excluding rotations and reflections, there is exactly one 3×3 magic square, exactly 880 4×4 magic squares, and exactly 275,305,224 5×5 magic squares. 778-783 gives the 880 4 X 4 squares.

9

down vote

Given an n×nn×n magic square, write MnMn for its magic constant. There are nn rows, each of which has sum MnMn, so the sum of all the entries in the square is n⋅Mnn⋅Mn. Each whole number between 11 and n2n2 appears once, so

nMn=1+2+…+n2=n2(n2+1)2,Mn=n(n2+1)2.

nMn=1+2+…+n2=n2(n2+1)2,Mn=n(n2+1)2.

In particular, every n×nn×n magic square has the same constant.