EMT 725

Philippa M. Rhodes

A 2 x 2 x 2 cube is made up of 8 1 x 1 x 1 blocks. If the cube is painted, then each of the 8 blocks has 3 of its faces painted. If a 3x3x3 cube is painted, a given block may have 0, 1, 2, or 3 of its faces painted. Count how many of each. Generalize to N x N x N.

3 x 3 x 3

There are 27 1x1x1 blocks.

The 8 corners are painted on 3 sides.

There are 12 with 2 sides painted. ( 4 on the top, 4 in the middle, and 4 on the bottom).

There are 6 with 1 side painted. (1 block in the center of each of the 6 sides.)

There is 1 with 0 sides painted. (Subtract the previous total from 27 or visualize the 1 block in the middle that can not be seen.)

4 x 4 x 4

There are 64 1x1x1 blocks.

The 8 corners are painted on 3 sides.

There are 24 with 2 sides painted. ( 8 on the top, 8 in the middle, and 8 on the bottom).

There are 24 with 1 side painted. (4 blocks in the center of each of the 6 sides.)

There are 8 with 0 sides painted. (Subtract the previous total from 64 or visualize the 2x2x2 block in the middle that can not be seen.)

For a 5 x 5 x 5, there are 125 1x1x1 blocks.

The 8 corners are painted on 3 sides.

There are 36 with 2 sides painted. (12 on the top, 12 in the middle, and 12 on the bottom).

There are 54 with 1 side painted. (9 blocks in the center of each of the 6 sides.)

There 27 with 0 sides painted. (Subtract the previous total from 125 or visualize the 3x3x3 block in the middle that can not be seen.)

For n x n x n, there are 8 corners with 3 sides painted.

There are (n - 2) * 4 (1 for each side - not top and bottom) * 3 (for each level: top, middle, and bottom) with 2 sides painted.

There are * 6 with one side.

There are (n - 2) x (n - 2) x (n - 2) with 0 sides painted.

So, the total is: