Cut cube into small cubes
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?
When you cut the 4-inch cube into the 1-inch cube, there will be a 4X4 grid on all sides.
So there will be 8 cubes who will be having 3 sides with paint on them.
24 cubes with two painted side
24 cubes with one painted side
8 cubes with no painted side.
Where n > 1, if you cut a painted n-inch cube into 1-inch cubes, you will have:
n^0: 8 * n^0 cubes with 3 sides painted
n^1: 12 * (n – 2)^1 cubes with 2 sides painted
n^2: 6 * (n – 2)^2 cubes with 1 side painted
n^3: (n – 2)^3 cubes with 0 sides painted
So plugging in 4 for n gives:
8 * 4^0 = 8 * 1 = 8 cubes with 3 sides painted
12 * (4-2)^1 = 12 * 2^1 = 12 * 2 = 24 cubes with two sides painted
6 * (4-2)^2 = 6 * 2^2 = 6 * 4 = 24 cubes with one side painted
(4-2)^3 = 2^3 = 8 cubes with no side painted
The cubes with 3 side paint will be 8 – 1 on every corner with 8 corners in a cube.
The cubes with 2 side painted will be 24. – 2 on each edge with 12 edges in a cube.
The cubes with 1 side painted will be 24- 4 on each face with 6 faces in a cube.
With no paint- 8 cubes- 64 minus (8+ 24+ 24)
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