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Is it possible to have an $n \times n$ magic square with a another magic square of $\frac n 4 \times \frac n 4$ magic square inside it? If so provide an example, if not prove if impossible.

Rules:

  1. $n$ cannot be 4

Clarifications:

  1. A magic square is an $n \times n$ grid filled with numbers 1, ..., $n \times n$ (this goes for the inside square too)
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    $\begingroup$ I'm assuming $n$ cannot be $4$? (Otherwise, this would be very easy ;) ) $\endgroup$
    – Doorknob
    Dec 16, 2014 at 23:45
  • $\begingroup$ @Doorknob Oh yeah, forgot to add that in lol! $\endgroup$
    – warspyking
    Dec 17, 2014 at 0:01

1 Answer 1

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It's possible for any $n$ divisible by 4, other than $n=8$.

First, find a magic square of size $n/4$. (This is impossible when $n=8$.) Call it $\mathcal{M}$. Now consider the following 4 by 4 magic square $\mathcal{S}$, using the numbers 0 through 15:

$\left( \begin{array}{ccc} 8 & 5 & 2 & 15 \\ 3 & 14 & 9 & 4 \\ 13 & 0 & 7 & 10 \\ 6 & 11 & 12 & 1 \end{array} \right)$

For each number $k$ in this square, replace it by a copy of $\mathcal{M}$ with every number increased by $kn^2/16$. One can easily check that the resulting square must be magic, because both of the starting squares are.

If you've heard of a tensor product or Kronecker product, then this should feel like the product of two magic squares.

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  • $\begingroup$ Normally a magic square is filled with 1, ..., n (so if a magic square is inside it, it would contain 1, ..., n/4) $\endgroup$
    – warspyking
    Dec 17, 2014 at 0:36
  • $\begingroup$ If $\mathcal{M}$ contains the numbers 1 through $n^2/16$, then the procedure in this answer should produce a new square containing the numbers 1 through $n^2$. Unless I screwed up. $\endgroup$
    – Lopsy
    Dec 17, 2014 at 0:39
  • $\begingroup$ Could you fix this then? Also look at the clarification I added in the question. $\endgroup$
    – warspyking
    Dec 17, 2014 at 0:42
  • $\begingroup$ How do you fill an n by n magic square with the numbers 1 through n? Do you use each number n times? I've never seen a magic square where you're allowed to repeat numbers. $\endgroup$
    – Lopsy
    Dec 17, 2014 at 1:06
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    $\begingroup$ I'll suggest several improvements: 1) you might want to replace "tensor product" with "Kronecker product" since a tensor product is more general, 2) your example $\mathcal{M}$ should have all elements incremented by 1 to meet the spec, 3) for the programmers in the crowd, the MATLAB code snippet big_M = kron( M - 1, ones( n/4, n/4 )*n ) + kron( ones( n/4, n/4 ), S ), where $\mathsf{M} = \mathcal{M}$ and $\mathsf{S} = \mathcal{S}$, might help to clarify the concept. +1 for your answer, though. $\endgroup$
    – COTO
    Dec 17, 2014 at 1:21

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