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Putnam 2001 B1

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February 27, 2012
Putnamcollege contests

Problem Statement

Let nn be an even positive integer. Write the numbers 1,2,⋯ ,n21, 2, \cdots, n^2 in the squares of an n×nn \times n grid so that the kkth row, from left to right, is (k−1)n+1, (k−1)n+2, ⋯ , (k−1)n+n. (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.
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