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?
Subtract the outside squares, which will all have some paint on them
8 more left
8 more right
4 more front
4 more back
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
No blue side = 8
1 side = 24
2 sides = 24
3 sides = 8
Total = 64 cubes