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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?

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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.

Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Cubes Within Cubes Revisited

Age 11 to 14
Challenge Level

This problem demonstrates how a formula can be derived from the context (rather than "pattern-spotting") in a variety of ways.
It also provides practice in the manipulation of algebraic symbols and generates generalities that can promote discussions on equivalent symbolic forms.

A teacher comments:
This was a good opportunity to practise simple and more difficult sequences and generation of formulae, ranging from linear to cubic. All pupils were able to access the problem up to (and beyond) their own "natural" ability (Year 9 set 2).