[pdurrant]

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.....

I think the thing you've missed is that

Spoiler:

(given the first chap can see 2 towns)

if the second chap sees eight, and so can't get them both freed on the first night, this gives information to the first chap, who on the second day knows that the second chap doesn't see 11. It's the first chap who now knows that there are 10 towns in total and gets them free.

The information passed from prisoner to prisoner each night is "I don't know enough to say how many towns there are", passedin the form of the prisoners still being prisoners the next day.

And each day, the information that this conveys is different, reducing the number of possibilities for the number of towns that the other person can see and can think that the other prisoner can see.

HTH.