You may also like

problem icon

All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

problem icon

Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

problem icon

Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

Christmas Boxes

Age 11 to 14 Challenge Level:

Teachers may want to provide students with squared paper for this problem.

One way of introducing this problem is to challenge students to first find all the different ways of arranging two squares (dominoes), three squares (triominoes), four squares (tetrominoes), five squares (pentominoes) and six squares (hexominoes) - all the squares must touch at least one other square along one of its edges (with the edges lining up exactly).

Number of Squares Number of Arrangements
2 1
3 2
4 5
5 12
6 35

This assumes that rotating or reflecting an arrangement does not produce a new arrangement.

Students could then be asked to look at the pentominoes to decide which will make cubes without lids.
They could then be asked to look at the hexominoes to decide which are the nets of cubes.

This problem could be followed up with Christmas Presents which asks students to consider cuboids.