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?
(In reply to Extension (A deeper question)
I'm not sure I understand what your final question in the last paragraph is.
At first I thought you were asking if it was possible to replace the number 217 in the exact same problem statement, but come up with more than one possibility of cube size and painting pattern. But the way you mentioned that scenario in your 5th statement, it seemed like you were explaining that's not what you meant.
Are you asking if there is a way to do your 91 example, but with more than two possible solutions? If so, I haven't found one yet. I checked all the ways to paint a cube, up to a 500x500x500 cube, and I found forty-two numbers that would give more than one possible solution. 1 and 8 gave more than two solutions, but the other forty all only gave two (and for just those 40 I checked that a cube bigger than 500x500x500 wouldn't have a solution either).
For those that are curious, the ones I found were 1, 8, 23, 91, 98, 120, 169, 218, 225, 288, 728, 946, 1156, 1225, 1352, 1657, 1680, 2465, 3176, 8911, 8978, 9800, 11781, 18818, 24571, 29800, 32761, 34133, 37153, 43681, 54289, 63656, 64898, 71288, 78408, 139128, 262088, 298936, 326040, 332928, 765118, 837224.
Edited on February 4, 2004, 6:05 pm
Posted by nikki
on 2004-02-04 18:03:26