This is a tricky question. Partly because of the fact that the order in which people speak does make a difference.

Here's a solution.

• Player one: Mixed pair
• Player two: mixed pair
• Player three: double-ace

First of all note that player three will never be able to figure out that she has two aces (because a game where she has two eights is completely symmetrical; nothing can ever make her distinguish between these two options).

The game develops as follows.

Round 1:

• Player 1 says "I don't know" (because the only way she could have known is if there were two identical pairs)
• Player 2 says "I don't know" (roughly the same reason)
• Player 3 says "I don't know" (always)

At the end of round 1, everyone knows there must be at least one mixed pair. And in fact, it is common knowledge that there is at least one mixed pair. (if there is no mixed pair, someone figures out their cards in round 1)

Round 2:

• Player 1 says "I don't know".

• Now player 2 knows that player 1 sees at least one mixed pair! So... player 2 says "I know". Crucially (and this is where Jaap's answer breaks): Player 1 does not know why player 2 knows that she has a mixed pair. As far as Player 1 is concerned, this could be

  • because Player 2 is the only one with a mixed pair, and player 2 figured this out after the first round.
  • or because of the actual reason and player 2 only figured this out after the first answer in the second round

• Player 3 says "I don't know" (always)

Round 3:

I think that at this point, player 1 still does not have enough information to conclude her mixed pair. The reason is: if you play the first six steps of a game where player 1 has 88, player 2 has A8 and player 3 has AA, the result is exactly the same.

• So player 1 says "Don't know again"know" again
• So does player 3 (always)

etcetera.