[omk3]

Originally Posted by pdurrant

You are certainly thinking along the right lines, even if you do not have the answer yet. They do indeed get out on the 5th evening.

Spoiler:

Heh. I thought I had examined all possibilities, but I hadn't. A number being a possibility and it being a certainty changes everything. I had conveniently avoided 6 or 7 in my previous thinking, but these are the toughest numbers to see.

So let's say I see 6 towns.

I see 6.
I know he sees either 7 or 4.

If he saw 4 he would think I see either 9 or 6.

If I saw 9, I would think he would either see 1 or 4.

If he saw 1, he would think I saw either 9 or 12.

If I saw 12, we'd be out on the 1st day, with me stating there are 13 towns.

We are not out on the 1st day, so if he saw 1, he would know I saw 9, and we would be free on the 2nd day, with him stating there are 10 towns.

We are not free on the 2nd day, so he doesn't see 1. So he must be seeing 4. We would be out on the 3rd day, with me stating there are 13 towns.

We are not free on the 2nd day, so he knows I don't see 9. If he sees 4, he knows I must see 6, and we would be out on the 4th day, with him stating there are 10 towns.

We are not free on the 4th day, so he doesn't see 4. It follows that he sees 7, and we are out on the 5th day, with me stating there are 13 towns.