### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Christmas Boxes

##### Stage: 3 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.

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.