# Solution May 19, 2008

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### Problem

Consider a completely filled Sudoku, written as a 9x9 matrix. Show that the determinant of this matrix is divisible by 405.

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### Solution

This is true for any 9x9 latin square. We can add the second, third, ..., ninth column of the matrix to the first column without changing its determinant. The resulting matrix has all entries of the first column equal to 45. We can then divide the first column by 9, which divides the determinant by 9 as well.

Now we do the same operation on the rows. This time, the first row will end up with all entries equal to 45, without changing the determinant. So the determinant is still divisible by 45, and the original determinant was divisible by 405 as claimed.