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?
How many different ways can I lay 10 paving slabs, each 2 foot by 1
foot, to make a path 2 foot wide and 10 foot long from my back door
into my garden, without cutting any of the paving slabs?
This solution came from Helen of Madras
College who knew what to do with the goats:
The solution would be to get one piece of rope 15 metres long,
loop it through the rings and attach the goats to either end of the
rope. The goats could reach every corner of the field without
difficulty. The worst that could happen would be that their horns
are just touching. The rope slides through the rings at the top of
the posts so at some point one goat may have only 1 metre of space
whilst the other may have 14 metres. They will never be close
enough to fight though. (Helen means
circles of diameters 1 metre and 14 metres)
With this arrangement the goats will be pulling on each other
which is enough to make even the gentlest old goat irritable! To
check that the goats really can reach the corners of the field,
using Pythagoras theorem you find the exact distance from the
centre of the post to the corner is 5$\sqrt 2$ m which is between
7m and 7.1m. Using a single rope 14.6m long which is threaded
through both the rings on the tops of the posts and tied to the
collars on the goats, each goat can reach the corner of the field
at his end but only when the other goat is right up against the
other post. The goats can graze overlapping circles of radius
7.1m.The nearest they can get to each other is along the centre
line when they are always at least 0.4 metres apart.
For the goats to be able to meet the rope would have to be at
least 15m long so if the rope is between 14.6 and 15 metres the
problem is solved.