019: Magic Squares and Sequences

Proposed by Toby Walsh

An order $n$ magic square is a $n$ by $n$ matrix containing the numbers $1$ to $n^2$, with each row, column and main diagonal equal the same sum. As well as finding magic squares, we are interested in the number of a given size that exist. There are several interesting variations. For example, we may insist on certain values in certain squares (like in quasigroup completion) and ask if the magic square can be completed. In a heterosquare, each row, column and diagonal sums to a different value. In an anti-magic square, the row, column and diagonal sums form a sequence of consecutive integers.

A magic sequence of length $n$ is a sequence of integers $x_0 \ldots x_{n-1}$ between $0$ and $n-1$, such that for all $i$ in $0$ to $n-1$, the number $i$ occurs exactly $x_i$ times in the sequence. For instance, $6,2,1,0,0,0,1,0,0,0$ is a magic sequence since $0$ occurs $6$ times in it, $1$ occurs twice, etc.