# 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)

### Your Answer

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