[GeoffC]

Originally Posted by omk3

Spoiler:

Let's take the simplest case:
One of them sees 11,12 or 13 towns. He is sure the towns are 13, and asks to be freed.

Now the simplest-but-one case:
One of them can see 0,1 or 2 towns. He knows that if the other one could see more than 10 (simplest case above) they would be out on the 1st day. Since they are still in on the 2nd day, he knows the other one sees 10 towns or less.

You know the number of towns you see, and you know the 2 possible numbers the other one can see. If you can discard one of them, you can be sure of the other.
So if you see 2 towns for example, the other one sees either 11 or 8.
But if you haven't been freed on the 1st day, you know he can't see 11. So he must see 8.
And so it goes on.

I hope this helps...

Spoiler:

Why ?

He sees 8 - he doesn't know whether it is 10 or 13
He sees 11, therefore it must be 13. They are both out.

However on the next day, he still only sees 8 - and still he doesn't know whether it is 10 or 13.

and the same for each of the remaining days until old age creeps in or one of them takes the plunge and guesses 50:50 at being right.....