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We had a well presented solution from Sarah. She showed it as follows:

Two routes to Cheston:
Appleby - Zogborough - Barkton - Xanly - Cheston
Appleby - Yanbridge - Barkton - Worsby - Cheston

Two routes to Dorby:
Appleby - Zogborough - Barkton - Xanly - Cheston - Vilham - Dorby
Appleby - Yanbridge - Barkton - Worsby - Cheston - Underton - Dorby

This next solution came from Year 5 Maths Group in Parklands Primary School. They wrote:

We worked out the first answer as being 4 routes to Cheston by following the tracks which means that there is one crossover at Barkton.
The routes are
AZBXC,
AYBWC,
AYBXC
and AZBWC.

The different routes to Dorby: We already knew that there were 4 routes to Cheston and because there are two single tracks VD and UD we added these routes to the 4 existing routes to Cheston, making 8.

THEREFORE EVERY 2 SINGLE LINE STATIONS DOUBLES THE AMOUNT OF EXISTING ROUTES. EVERY CROSSOVER STATION MULTIPLIES THE AMOUNT OF ROUTES BY 4.
Andrew, Jason, James, Ibrahim, Verity and Rianna.

What fantastic reasoning! Thank you.

There were also solutions from Thomas, Leahye and Rowan, and a team solution sent in by pupils from Patcham Junior School. Well done to you all.