The Chessboard From Greece!

For an even positive integer $n<10000$ , we write down all the numbers $1, 2, 3, \ \ldots\ , n^2$ on the squares of an $n\times n$ chessboard.

Let $S_1$ be the sum of all the numbers written on black squares and $S_2$ be the sum of all the numbers written on white squares.

How many possible values of $n$ are there such that it is possible to achieve

$\frac{S_1}{S_2}=\frac{39}{64}?$

Details and assumptions:

Each number appears exactly once on the board.

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This problem is from the Greece MO - 2014.

This problem is from the set "Olympiads and Contests Around the World -3". You can see the rest of the problems here .