[omk3]

Originally Posted by GeoffC

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.....

Spoiler:

Okay, I started from the simplest case, but let's take this one:

A sees 8. So A knows that B sees either 2 or 5.
A knows that if B sees 2, he will know that A sees either 8 or 11.

Working on the assumption that B sees 2:
A knows that B will wait one day to see if A sees in fact 11, because if he does, they will be free on day 1.
As they are not, A knows that B now knows that A does not see 11.
Therefore, B, if he sees 2, will know on day 2 that A must see 8 instead of 11, so he will tell the guard the towns are 10 and they will be freed.
If day 2 passes without event, A knows that B does not see 2 towns after all.
Assumption disproved.

As B could only see either 2 or 5 towns, and 2 is out, he must see 5. So A now asks to be freed, and tells the guard the towns are 13.

No guess work, just a waiting game.