## Hollow Cube

#### Problem

A cube measuring 3 3 3 is made up of 27 smaller cubes and has a 1 1 square hole pushed right through the centre of each face so that you can see straight through the cube from every side.

The number of small cubes remaining is 20.

If a 5 5 5 cube has 3 3 square holes pushed through the centre of each face, how many smaller cubes would remain?

#### Solution

A solid cube measuring 5 5 5 consists of 125 cubes.

A cube measuring 3 3 3 = 27 cubes is removed from the centre and six faces of 3 3 = 9 cubes are removed.

That is, 125 (27 + 6 9) = 125 (27 + 54) = 125 81 = 44 cubes remaining.

What about a 4 4 4 cube having 2 2 squares pushed through each face?

Can you generalise for any sized cube?

Problem ID: 27 (Dec 2000) Difficulty: 1 Star