Math Forum - Geometry Project of the Month

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December 1999 - Painting a Cube

Back in October, for the Geometry Problem of the Week, we used a problem about painting a cube. You can read about it in our archives. The gist of the problem is this:
A bunch of unit cubes are put together to form a larger cube. Some of the faces of the larger cube are painted. Then the large cube is taken apart and 24 of the small cubes have no paint on them at all. How many small cubes were in the large cube and which of the large cube's faces were painted?
As you can read in the solutions, there is only one possible solution to this problem. Here's your task. Instead of 24 cubes, what is the smallest number of unpainted cubes for which there is more than one possible answer to the question (about how many small cubes were in the large cube and how many of the faces were painted)?

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