Consider any completed Sudoku puzzle as a 9x9 matrix, and
take its determinant. Show that the determinant is
divisible by 405.
If you're not familiar with Sudoku puzzles, you can just
consider any 9x9 matrix with the following properties:
We'll use the following two properties of determinants, both of
which you can find in any book on linear algebra:
- The value of the determinant is not affected if you add the entries
in one row to another row, or if you add the entries in one column to
another column
- If all of the values in one row (or one column) have a common
factor k, then the determinant is equal to k times the determinant
of the matrix in which all of those values are divided by k.
So, consider a 9x9 Sudoku matrix, and let D be its determinant:
| D = |
a11 |
a12 |
... |
a19 |
| a21 |
a22 |
... |
a29 |
| : |
: |
|
: |
| a91 |
a92 |
... |
a99 |
By the first property, we can add one row to another without changing the
value of the determinant, so we add Row 2 to Row 1, then add Row 3 to Row 1,
and so on, ending up by adding Row 9 to Row 1.
Since each column of a Sudoku matrix contains the numbers from 1 to 9,
the result will be 45 in each entry of Row 1:
| D = |
45 |
45 |
... |
45 |
| a21 |
a22 |
... |
a29 |
| : |
: |
|
: |
| a91 |
a92 |
... |
a99 |
By the second property, we take the 45 out of each entry in Row 1:
| D = 45 |
1 |
1 |
... |
1 |
| a21 |
a22 |
... |
a29 |
| : |
: |
|
: |
| a91 |
a92 |
... |
a99 |
Now we add Column 2, Column 3, ..., Column 9 to Column 1. In the first
row, the result is 9, and we get 45 for the other rows, since each row
of a Sudoku matrix contains the numbers from 1 to 9:
| D = 45 |
9 |
1 |
... |
1 |
| 45 |
a22 |
... |
a29 |
| : |
: |
|
: |
| 45 |
a92 |
... |
a99 |
Now we can take a common factor of 9 out of the first column:
| D = 405 |
1 |
1 |
... |
1 |
| 5 |
a22 |
... |
a29 |
| : |
: |
|
: |
| 5 |
a92 |
... |
a99 |
Since the remaining determinant is clearly an integer, we see that D is
divisible by 405.