Cut Painted Cube Puzzle

Puzzle:

A solid, four-inch cube of wood is coated with blue paint on all six sides. Then the cube is cut into smaller one-inch cubes. These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?

Puzzle Solution:

Subtract the outside squares, which will all have some paint on them

16 top
16 bottom
8 more left
8 more right
4 more front
4 more back

= 56

So, only 8 will have no blue side.

The only cubes that will have blue on only one side will be the four center squares of each side of the 64-cube — so, 4 x 6 sides = 24 cubes

The only cubes that will have blue on three sides are the corner pieces — there are 8 corners, so 8 cubes.

The cubes with two sides blue are the edge cubes between the corners — two on each side of the top and bottom, so 2 x 4 sides x 2 (top and bottom) = 16, + the side edge/non-corner pieces, which will be another 2 x 4 = 8

So

No blue side = 8
1 side = 24
2 sides = 24
3 sides = 8

Total = 64 cubes

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