Is there a quick method/ heuristic to solve for the length of the optimal route for a simple travelling salesman problem?

Question

E.g take this problem:

You would have about 1 minute and 40 seconds to solve this in an exam. I tried to solve this by listing the permutations of Arford, Yewton and Teechester in between going from and back to Essover.

E.g.

ATY

AYT

YAT

Is this the fastest method? Also, how can you tell what the number of routes you can take will be?

Answer 1

This is the traveling salesman problem, not the Chinese postman problem. For $n$ cities, there are $(n-1)!/2$ possible tours, so that is only three tours here. Just try all three and take the smallest.