### Redblue

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?

### Diagonal Trace

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?

### Rail Network

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.

# Delia's Routes

##### Stage: 2 Challenge Level:

A little grey mouse called Delia lives in a hole in the bottom of a tree in a small square garden (A).

The garden is paved with 6 large square paving stones in each direction and has a circular pond right in the middle that has a diameter of 3 of the paving stones. Delia's tree is at the left hand corner at the bottom of the garden. At the top right hand corner of the paved area there is a bird table (B).

Each night Delia runs along the edges of the paving stones, avoiding the pond, to collect up the bits that have been dropped by the birds. She likes to go a different way each night but she always takes as short a route as she can, towards the top of the garden and to the right.

How many days will it be before Delia has to take the same route again?