[omk3]

Originally Posted by GeoffC

Spoiler:

I'm going to have to walk on by on this one, there is a level of logic (or even guess work) going on that makes it difficult to understand how each wise one knows what the other is thinking, and why it takes 5 days before one of them gives an answer to the gaoler ..... and upon what that answer is based.....

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...