Beyond that, the only advice I can give is to start in Truro and finish in Thurso. Unless, of course, you want to do a closed loop and finish at your start point, in which case it is irrelevant. But a closed loop will be significantly longer.
Is there a proof of that?
Not sure about "significantly", but you can easily show that for any closed loop visiting all towns, there are many shorter, non-closed paths through the same towns. (just take the closed loop through all towns and delete any link between two adjacent towns on the path:- voila - a shorter, non-closed path that still visits all the same towns).
Well of course what you say is correct; but I was making a feeble joke about "proofs"/theorems etc, and also making a pedantic point about that word "significantly".
(I assume we're just tossing around silly geometry chat at this stage, and not getting into a bitter forum point-scoring battle? Good, then I'll proceed ... )
Take this route through some fictional towns:
Clearly you could save distance by cutting out any one of the links ... but it would hardly be a "significant" saving. And of course our cycling salesman does need to get home, so a closed loop does have other advantages (including aesthetics?? )
Now, imagine Truro and Thurso are at the bottom/top of that map; clearly the closed loop will be far shorter than any route with those 2 fixed-end points.
[I'll try to find a more suitable graphic later ... ]
