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Let a_i, i=1,...,8, be the number of checkers in the i-th column of the checkerboard. If there are no checkers in the i-th column, set a_i=1 (so that for such i, a_i-1 contributes nothing to the sum below). There are a_i-1 different distances between the highest checker in the i-th column and the other checkers in that column. Hence there are
a_1-1+a_2-1+...+a_8-1=16-8=8
such distances, not necessarily all distinct. However, those distances must be in the range 1 square up to 7 squares. Hence the top checker, A, in one column must be the same distance from another checker, B, in that column as the top checker, C, in some other column is from another checker, D, in that column. But then the checkers A, B, C, and D form the vertices of a parallelogram with vertical sides AB and CD and parallel, but not vertical, sides AC and BD. This parallelogram has area, in squares, the distance between A and B times the distance between the two columns containing A and C.
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