# Determinant of a standard magic square

• What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order $$n$$?
• Are there singular standard-magic-square-matrices of any order greater than $$3$$?

First of all, the determinant of a standard-magic-square-matrix must be a multiple of $$\frac{n^2(n^2+1)}{2}$$ for odd $$n$$. This follows easily by the following process:

• Add all the columns to the last column. Then every entry in the last column is $$\frac{n(n^2+1)}{2}$$,the constant of the standard-magic-square.

• Now extract this constant and add all the rows to the last one. Then every entry in the last row is again the constant, beside the last entry, which is $$n$$.

• Since $$n$$ is for odd n a divisor of the constant, it can be extracted as well. For even $$n$$, only $$\frac{n}{2}$$ can be extracted, so the determinant is only a multiple of $$\frac{n^2(n^2+1)}{4}$$. This gives lower bounds for the absolute value of the determinant of regular magic-square-matrices.

• For size $$3$$, the only possible determinant (ignoring the sign) is $$360$$.

• For size $$4$$, my personal minimum for the absolute non-zero determinant is $$2176$$ and my maximum is $$17408$$.

• For size $$5$$, my best results are $$325$$ and $$6\ 547\ 775$$.

• For sizes $$4$$ and $$5$$, I also found matrices with determinant $$0$$, but for $$n = 6$$ I found none. OEIS claims that the magic square of order $$6$$ produced by Matlab has determinant $$0$$ (By the way, the sequence seems to contain a typo because in the list $$-360$$ appears for $$n=2$$ instead of $$n=3$$).

My pascal program generating random magic squares did not find a magic square with order $$6$$ and determinant $$0$$. Since I do not have Matlab, I cannot verify the magic square produced by it.

• $\pmatrix{ 13 & 11 & 6 & 4 \\ 12 & 2 & 15 & 5 \\ 1 & 7 & 10 & 16 \\ 8 & 14 & 3 & 9}$ has determinant $0$ – Peter Aug 15 '14 at 13:10
• $\pmatrix { 11 & 10 & 8 & 5 \\ 6 & 3 & 13 & 12 \\ 15 & 14 & 4 & 1 \\ 2 & 7 & 9 & 16}$ has determinant $2176$ – Peter Aug 15 '14 at 13:14
• $\pmatrix { 7 & 22 & 1 & 10 & 25 \\ 16 & 18 & 24 & 5 & 2\\ 9 & 13 & 6 & 17 & 20 \\ 12 & 8 & 23 & 19 & 3\\ 21 & 4 & 11 & 14 & 15}$ has determinant $0$ – Peter Aug 15 '14 at 13:17
• $\pmatrix{ 15 & 1 & 8 & 10\\ 2 & 12 & 13 & 7\\ 11 & 5 & 4 & 14\\ 6 & 16 & 9 & 3}$ has determinant $17408$ – Peter Aug 15 '14 at 13:24
• I found a $6x6$-magic square in wikipedia, constructed with the strachey-method, which has determinant $0$ – Peter Aug 15 '14 at 16:37