Poker Riddle

Yesterday, a guy named Jesse, his friend (can't remember his name), and the actor Michael Muhney were in the 500 NL game discussing probabilities and IQs and other nerdy things. I learned that Michael is in MENSA (actually I knew this from his IMDb page), Jesse has a 168 IQ, and his friend went to a school that required an IQ of at least 150. They had a discussion of how many times you'd need to double up to get $1 million if you started with $3. Jesse concluded that the probability was 1-(1/2)^18. He was emphatic on this point for a while even after it was suggested to him that the real answer was more likely (1/2)^18.

Anyway, Jesse had a riddle that he was sure nobody could figure out: you are heads-up in a holdem game, and you're ahead preflop, on the flop, and on the turn, but you have no chance to win on the river (only fold or chop). Jesse offered Michael 3-1 odds on a bet that he couldn't figure it out in 15 minutes, but Michael refused because, as Jesse put it, he is a "life nit," meaning that he is unwilling to gamble on things outside of poker. (A "nit" in poker is someone who is unwilling to gamble much, mostly only playing the nuts.) Michael confirmed that yes, he is a "life nit," if, by that, Jesse meant "a responsible person with two kids who doesn't want to have to explain to his wife in 15 years that he can't pay for their kids to go to college because he gambled all his money away to some guy at the casino with a 168 IQ who can't even do simple algebra."

I asked Jesse's friend, who had heard the riddle and was sitting next to me, how they define "being ahead on the turn." My definition would be that "being ahead" means you have the best chance to win the pot. Clearly, they had a different definition, since the problem is set up such that the hand that is "ahead" supposedly has no chance to win the pot. Jesse's friend said that a hand was defined as being "ahead" if it would win without any more cards coming. Jesse offered me the deal as well ($25 to $75), but, being a life nit myself, I turned him down.

I thought about the problem for a minute but did not figure it out. I eliminated the possibility that the answer involved flushes, and decided that it probably involved low cards. With low cards, it's much easier for kickers to get counterfeited on the river. Anyway, after playing for another half hour, I asked Jesse's friend what the answer was.

"You have to figure it out for yourself, man. Jesse! He wants us to just tell him the answer!"
"Okay, let me think about it," I said.

I thought about 32 against 42, but it didn't quite work. I thought about 43 against 42, with a flop of QQ4. Then a 2 on the turn. So far, so good. 43 is ahead preflop, on the flop, and on the turn, but I can't think of any river card that would result in a win for 43. If the river is a 2, then 42 wins with a full house. Any X higher than 2, and both players have QQ44 with a X kicker. Any Q or 4 and both players have the same full house. After about 1 minute, I told Jesse's friend I thought I figured it out.

"Did you think through all the possibilities? Just think about it." Jesse told me he still wanted to bet me 3-1, and he would give me twelve minutes and let me discuss it with anyone in the casino except his friend, who knew the answer. I hesitated, partly due to his confidence (although he had been similarly confident about the 1-(1/2)^18 formulation), but mostly just because I do not like making proposition bets, especially at poker tables. After a minute, I took the bet anyway. I asked Tony, another prop at the 500NL table, to confirm my answer. We discussed it for about five minutes, and he thought it looked good. Meanwhile, Jesse, having heard my discussion with Tony, proclaimed that I was "a million miles off" and wanted to double the stakes and give me a hint and let me call someone on the phone. I didn't want to escalate the situation any further, so I declined. I gave him my answer. After studying it incredulously for ten or fifteen minutes (and briefly trying to argue that 43 was not "ahead" on the turn), he conceded that it looked right and gave me my $75. I gave $15 to Tony for helping. Jesse said he had seen this question in a magazine and thought there was only one answer, which he told me. He said he had been asking that question at poker tables for five years and nobody had figured it out. I suspect he left something important out of the question, but I can't think what it would be. Anyway, Tony now thinks I'm a genius.

My answer can actually be generalized quite a bit. Instead of 43 and 42, I think any X3 and X2 will work, except X=2 or 3. Similarly, in addition to QQ on the flop, any YYX flop will work, unless it gives X3 a backdoor straight draw. So, X=Ace and Y=4 does not work, but X=7 and Y=4 does. The key is that a 2 has to come on the turn in all cases.

Jesse's answer does not fall into this category. Can you think of it? I'll give clues in the comments if anyone asks. I haven't thought of any solutions besides Jesse's answer and mine.