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.