Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
You can trace over all of the diagonals of a pentagon without
lifting your pencil and without going over any more than once. Can
the same thing be done with a hexagon or with a heptagon?
This drawing shows the train track joining the Train Yard to all
the stations labelled from A to S. Find a way for a train to call
at all the stations and return to the Train Yard.
Think about the number of choices there are at each point. Or
simply count how many there are.