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Today I came across an interesting question

Consider a completely filled Sudoku, written as a $9 \times 9$ matrix. Show that the determinant of this matrix is divisible by $405$.

The solution proves this in a neat way, but I think it leaves open another question: is $405$ the biggest number with this property?

One could prove the answer "Yes" by taking the intersection of the prime factorization of determinants until it becomes just $\{3,3,3,5\}$. Any better approach?

William
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  • I doubt there is a better appoach - after all you must make use of the fact that it is a sudoku, not just a laitin square. If you try an artificially symmetric sudoku, you are likely to end up with $\det=0$, so I suggest you pick a handful of them from your localnewspaper and compute the gcd. If after three sudokus you are still $>405$, we might have to contemplyte if there's not a reason behind it. :) – Hagen von Eitzen Feb 11 '14 at 16:49

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