### Delia's Routes

A little mouse called Delia lives in a hole in the bottom of a tree.....How many days will it be before Delia has to take the same route again?

### 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?

### 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.

# Diagonal Trace

### Why do this problem?

This challenge is accessible to all pupils - everyone can have a go - but explaining why it is impossible with certain shapes is more difficult. It is a good context in which to encourage pupils to make conjectures and to verify them.

### Key questions

Do you have a system fortrying to trace over the diagonals?
What do the shapes that work have in common?
Which other shapes do you think it will work for?

### Possible extension

The problem Networks and Nodes would be a good follow-up challenge to this one.

### Possible support

This sheet , with six of each shape and their diagonals drawn, will be useful to print off for some pupils.