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?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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.
If the circle now rolls around an equilateral triangle, can you describe the locus of the centre of the circle and its length?
Can you generalise your findings?