Combinatorics Poker Puzzle

Today we have a special puzzle for you that will test your intuition as a poker player, and challenge your understanding of combinatorics.

Six players go all-in preflop with a range of (QQ+, AK).

Questions:

Can QQ win the pot? If so, how?
Can KK win the pot? If so, how?
Would you rather hold QQ, KK, or AKo in this scenario?
A spectator places a side bet. They win the side bet if any A/K/Q lands on the board postflop. What are the chances that they win?

Please note that you do not need an equity calculator or any special software to answer these questions.

Answers:

Correct answer

The trick to answering these questions is to ask yourself how it is possible that 6 different players ended up with (QQ+, AK). When you consider card removal, the possibilities become quite limited.

There are only 12 cards in the deck that contain a Queen, King or an Ace.
6 players with two cards each will use up every king, queen, and ace in the deck.

Now let’s ask how many ways can 6 players actually have these ranges. There are only 3 possibilities:

Can QQ win the pot? If so, how?

Yes, but only if it makes a straight flush! Two players are guaranteed to have QQ in all scenarios. All the aces and kings are in play, so winning with a regular flush is impossible. All queens are in play, making it impossible to outdraw better pairs.

Can KK win the pot? If so, how?

No, KK cannot win the pot. KK is always trying to outdraw AA. To do this, it must make a straight, a flush, or use a 3rd king. All the kings are in play, so it can’t use a 3rd king. All the queens are in play, so a straight is impossible. All the Aces are in play, so it can’t win with a flush. Therefore, KK cannot win the pot.

Would you rather hold QQ, KK, or AKo in this scenario?

QQ has substantially more equity than the other hands. Winning isn’t the only way to win the pot, you can also chop. ⅓ of the time QQ ends up in scenario 1, where you’re up against another QQ and 4 players with AK. This is very profitable for QQ since the AK block each other’s outs.

The only way KK can chop is if all 6 players chop via a straight on board.

AK is always up against 1 or 3 other AK’s. It can win with an ace-flush, and it can chop with a wheel straight. However, these events are far less likely to happen compared to QQ’s 1 in 3 chance of running into scenario 1.

If we run a simple equity calculation, we see that QQ has substantially more equity than KK or AKo. However, this comes from the fact that QQ chops with the other QQ over 20% of the time.