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Sum in black and white squares

Source: Greek Mathematical Olympiad 2014

March 14, 2014
Eulermodular arithmeticcombinatorial geometrycombinatorics proposedcombinatoricsDiscrete intermediate value theorem

Problem Statement

For even positive integer nn we put all numbers 1,2,...,n21,2,...,n^2 into the squares of an n×nn\times n chessboard (each number appears once and only once). Let S1S_1 be the sum of the numbers put in the black squares and S2S_2 be the sum of the numbers put in the white squares. Find all nn such that we can achieve S1S2=3964.\frac{S_1}{S_2}=\frac{39}{64}.
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