[Sparrow]

Originally Posted by GeoffC

Spoiler:

What happens if both can see 1 ..... neither knows what the other sees.

Spoiler:

So neither gives an answer on the first day - but they both know they haven't been released or executed; therefore their partner didn't answer either.
The only way to answer correctly on the first day is if either saw 2 or 3 towns, in which case there are 3 towns - as no answer was given, they know neither of them saw 2 or 3 towns.
So they now know each saw either 0, or 1.
But both can't see 1, because they're told the answer isn't 2.
Hence 1 is the correct answer to give on day 2.