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 solution to problem
Are you saying only 3-sides painted results in an odd number of cubelets with paint? Regardless of how many sides are painted, there are always possibilities of an odd number of cubelets getting paint, starting with a 1x1x1 cube.
If two adjacent faces of the large cube are painted, a 3x3x3 cube will have 15 smaller cubes with paint; a 9x9x9 would have 153; an 11x11x11, 231.
With four faces painted, leaving two adjacent faces unpainted, with a 3x3x3 cube, 23 small cubes get paint; with 7x7x7, 163 get painted and with 9x9x9, 281 get painted.
With five faces painted, a 3x3x3 cube will have 25 small cubes painted; 7x7x7, 193; 9x9x9, 337.
So odd numbers are possible with other than 3 sides of the large cube painted.
Posted by Charlie
on 2004-02-03 08:54:17