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Is There a Theorem?

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Christmas Boxes

Stage: 3 Challenge Level: Challenge Level:1

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.

These Maths Films may help.

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.