# Cut cube into small cubes

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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?

DrAPSingh Scholar Asked on 4th November 2015 in

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.

ketul Expert Answered on 5th November 2015.

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

dougbell Genius Answered on 9th November 2015.

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)

Ayushi Curious Answered on 15th November 2015.

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