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?

The outermost layer of cubes is painted, hence, say: x^3-(x-1)^3=217.

3x^2-3x =216, x(x-1) =72 = 9*8, so x=9 is a solution.

The problem calls for 'the total number of unit cubes' so a unique solution is expected, which we have already.

Hence, there are 729 cubes in all.

No brain cells were harmed in deriving this solution.

Edited on July 1, 2016, 11:34 am

Posted by broll on 2016-07-01 11:22:50