> So what knowledge have you actually learned after the first round?

That your friend sees 2 or more, and your friend knows you see 18 or less. By itself, this is trivially deducable; but each day walks the limits closer to revela a fact that isn't trivial. You start the puzzle knowning that your friend can see either 6 or 8 statues, and each day you add a new fact about how many statues she can't see.

I'm more concerned that the problem as stated can be solved in 2 days: you say 18 on day one, then 20 on day two…

Judging from: > [...] and so on, until you get it right or wrong. The caretaker cackles, “If you need me, I’ll be out preparing your graves.”

My impression is that once you have the wrong answer, you're not going to have another day.

Therefore you say I see more than (y - 2) statues while he says morethan (x + 2) until you both repeat the same value.

You know you cannot lie so a simple iteration on inequalities is enough to find the right value.

This relies on your friend figuring out the linear addition scheme.

Oh thanks, I missed this crucial caveat.

So, you don't know how many statues your friend can see.

On day 1, imagine you go second. If your friend can see 19 or 20 statues, they know there is 20. Therefore, if you go second on the first day, you know your friend couldn't see 19 or 20 statues.

But you already know your friend sees 6 or 8 based on what you can see, and that you know that between you, you see 20. Both of you knew already, that this was not a possibility. I think I'm missing something from the setup.

You know that your friend sees 6 or 8 and you know that if your friend sees 6 then they know you see 12 or 14 and (you know that they know that) if you see 14 then you know that they see 4 or 6 and (you know that they know that you know that) if they see 4 then they know that you see 14 or 16 and ((you know that they know that)^2) if you see 16 then you know that they see 2 or 4 and ((you know that they know that)^2 you know that) if they see 2 then they know that you see 16 or 18 and ((you know that they know that)^3) if you see 18 then you know that they see 0 or 2 and ((you know that they know that)^3 you know that) if they see 0 they know you see 18 or 20 ((you know that they know that)^4) if you see 20 then you know the correct answer and it's over

So when it's not over on the first day, that updates your knowledge about each other's knowledge (about each other's knowledge ...) by eliminating that particular possibility. That goes on until only one possibility remains and you know the answer.

Ah, nice. This is the answer I was looking for.

This bugged me too, the process works but they’re throwing away a lot of information at the start, which should mean it can be done faster.

I made another answer that works the same way but doesn’t throw that away, and it seems it’s possible to finish on Day 2.

Good question. I suspect the answer is something like "that they know that you know that they know that you know that they see at least one statue".