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?

I like this problem a lot, although I've seen variations before.

The most interesting variations (In my opinion) have a missing number that gives multiple possibilities.

217 has only a single possibility for cube size and method of side painting.

If you are told that 8 sides have paint on them, there are 5 possible ways of painting a 2x2x2 cube, so this isn't very interesting.

There are plenty of examples with two possible size & pattern cominations. For example: 91

My question are there any numbers that could yield a lot of solutions? There are 8 ways of painting up to 5 sides of a cube, but some are mutually exclusive, so there can't be 8 solutions. How many can there be?

-Jer

Posted by Jer on 2004-02-04 11:26:12