Some unit cubes are assembled to form a larger cube. Some of the faces of the larger cube are then painted. The cube is taken apart and it is found that 217 of the unit cubes have paint on them. What is the total number of unit cubes?

If one face were painted then the number would be a perfect square, but 217 is not a perfect square. If two opposite faces were painted, the number would be twice a perfect square, which it isn't either.

So, depending on n, the edge length of the larger cube, the number of unit cubes with paint could be:
If two adjacent faces:
2 * n * n - n

If 3 faces meeting at a corner:
3 * n * n - 3 * n + 1

If 3 faces not meeting at a corner:
3 * n * n - 2 * n

If 4 faces, leaving two opposite faces unpainted:
4 * n * n - 4 * n

If 4 faces, leaving two adjacent faces unpainted:
4 * n * n - 5 * n + 2

If 5 faces:
5 * n * n - 8 * n + 4

The only value of n for which any of these results in 217 is 9, for which the formula for three adjacent sides being painted produces 217. And 9^3 = 729.

Posted by Charlie on 2004-02-02 14:45:36