[LazyScot]

Originally Posted by GeoffC

You know, I'm still not convinced ....

Spoiler:

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.


Each night provides the same scenario .....
How does that give them the answer of 10 or 13 - it still leaves them each night not knowing ..... there is no new information between night one and any of the succeeding nights .... !?

A different version of the explanation pdurrant gave above...

Spoiler:

The new information on the 2nd night is that they are still imprisoned. This sounds like it isn't information, but it is, just encoded in a strange way (double dutch, perhaps....). The reason that it is information is that it tells you that the other wise man couldn't work out whether the answer 10 or 13.

This means you, on the second night, now know that the other wise man cannot see certain numbers of towns. This is the information that is interchanged on the second night.