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Only One Hand (Posted on 2005-04-11) Difficulty: 2 of 5
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.

See The Solution Submitted by David Shin    
Rating: 4.2857 (7 votes)

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Solution Another solution | Comment 29 of 31 |

I know this is an old problem, but I had to comment.  I'm an avid poker player and there is a much simpler solution that everyone missed. 

In all the comments I read, everyone was focused on really great hands like Aces full and Royal Flushes, but that is not necessary.  For example (this is just one of many like it):

Community Cards - Ace hearts, 4 spades, 8 diamonds, 9 hearts, 10 spades.

All three players hold 7, J (suit is irrelevant).  They have each made a straight (7 8 9 10 J), and can only be beaten by someone who holds J, Q, giving them the nut straight (8 9 10 J Q).

When you play hold'em, there is what is known as the "nuts", a slang term meaning the best possible hand, given what is available on the board.  One of the first things you learn when you start playing hold'em is to recognize the nuts immediately - for example, if there are two cards of the same rank in the community cards, it is possible for someone to have made 4 of a kind, or a full house; if there are three or more cards of the same suit on the board, it is possible for someone to have made a flush, etc. 

Once you recognize the "nuts", you can also see what I call the "2nd nuts", i.e., the second-best possible hand.  In the example I gave above (or anything like it with three cards to a straight and no flush or better possibilities available) holding J, Q would give you the nuts; holding 7, J would give you the 2nd nuts.  You can carry this on all the way down, too, which is essential when calculating the strength of your own hand (3rd nuts is 6, 7; 4th nuts is A, A; 5th nuts is 10, 10, etc...)

Edited on April 24, 2006, 2:59 pm
  Posted by tomarken on 2006-03-15 14:14:50

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