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Colored cube

Source: Mexican Mathematical Olympiad 2013 Problem 4

December 1, 2013
geometry3D geometryanalytic geometryprismcombinatorics proposedcombinatorics

Problem Statement

A n×n×nn \times n \times n cube is constructed using 1×1×11 \times 1 \times 1 cubes, some of them black and others white, such that in each n×1×1n \times 1 \times 1, 1×n×11 \times n \times 1, and 1×1×n1 \times 1 \times n subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each n×1×1n \times 1 \times 1, 1×n×11 \times n \times 1 and 1×1×n1 \times 1 \times n subprism contains exactly one black cube.
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