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.

I think Kardo's and Ajosin's solutions are both valid, but I don't think they are topologically different.  This problem looks like enough fun for me to try looking for that other solution.

First, there are a few limitations on what could have happened.  The three players do not necessarily tie, as both Ajosin's and Kardo's solutions show.  However, it is not possible for one of the hands to be beaten by both of the others, for there is only one hand that can beat it.  It seems to me unlikely that all three would tie, unless they all used straights, also very unlikely.

With that in mind, if there is one player beating the other two, he must think there is one hand that is better yet.  The other two players must each have one of the cards of this better hand in order to eliminate the possibility in their minds.

I wanted to have a solution following these thoughts, but my time is cut short.

Posted by Tristan on 2005-04-13 00:04:20