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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

Age 7 to 11
Challenge Level

Why do this problem?

This problem has a story line that will encourage learners to become involved in Delia's nocturnal adventures. To answer the question How many days before Delia has to take the same route again? requires the children to find all of the possibilities. Given the number of pathways, this problem could be ongoing asking the children if they could find another three or four routes each day over a couple of weeks. This sheet gives 6 copies of Delia's garden plan.

Key questions

How many tiles along and how many tiles up must Delia go on each journey?
Can you think of a way to record all the different pathways Delia could take?
How many ways can you go when you get to this junction?
Which tiles can Delia not run along because they go into the pond?

Possible extension

Learners could repeat the problem with a differently-sized pond and/or garden.

Possible support

Suggest using different colours to find different routes on this sheet.