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.


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.
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