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Problem of the Week 1090

A Painted Cube

Alice has a solid wooden cube which measures the same whole number of inches in length, width and height. Alice paints the entire surface of the cube black.

With slices parallel to the faces of the cube, Alice cuts the cube into one inch cubes. A certain number of the small cubes are painted black only on one side. A certain number of the small cubes are completely free of paint.

If these two numbers are the same, what was the size of the original cube?

Source: Konhauser's Kuestions (1963). The Macalester Problem of the Week was started in 1968 by the late Joe Konhauser, a longtime professor at the college. Prior to joining Macalester, Joe worked for the Ford Motor Company, where he contributed a weekly puzzle column called "Konhauser's Kuestions."

© Copyright 2008 Andrew Beveridge and Stan Wagon. Reproduced with permission.

14 February 2008