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?
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?
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't fight
each other but can reach every corner of the field?
A farmer has a flat field and two sons who will each inherit
half of the field. The farmer wishes to build a stone wall to
divide the field in two so each son inherits the same area. Stone
walls are expensive to build, so naturally the farmer wishes to
build the shortest wall he can.
Can you prove that the shortest wall is always straight,
whatever the shape of the original field? Or perhaps you can find a
shape where the shortest wall isn't straight!