In a game of Texas Hold'em, all 5 community cards are dealt, and the three remaining players simultaneously say, "Well, there's only one hand that can beat me."

How can this situation arise? Assume that the players do not lie.

Here, "one hand" means a unique combination of 2 cards, out of the (52 choose 2) = 1326 possible ones.

For those unfamiliar with the basic rules of Texas Hold'em: each player has two face down cards, and there are five face up cards on the table. Each player makes the best possible 5-card poker hand using any of the 5 community cards and his 2 private cards.

(In reply to Another solution by tomarken)

OK, after some reflection I realized that the problem was asking for a solution where only one possible combination of two cards out of the deck would beat their hand, which would exclude my solution.

However, in reality this is a pointless difference.  In the example I gave in my previous post, you would not be able to predict the rank AND suit of the two cards that could beat you, but the suit is irrelevant because no flush is possible anyway.  Regardless of the suit the hands are exactly equivalent.

Setting reality aside for a moment, this is a good puzzle.  Although I agree with the previous comments, that the 2nd part of the posted solution is impossible.

Posted by tomarken on 2006-03-15 14:23:03