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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 Solution | Comment 4 of 31 |

This situation arises with the following card distribution,

Comunity; A,10,J of spades, together with two other unimportant cards that do not contain another A, 10, or J, nor the Q, K,9, 8, 7 of spades (to avoid other straight flush posibilities other than royal flush), and that do not form a pair (to avoid the posibility of poker by other players).

The other players have two spades each, but none of them has the Q or K of spades.

In this situation, they all have flushes in spades with A as the high card. Since there are no poker or full house posibilities, the only way anyone can beat their hand is if they hold the Q and K of spades, forming a royal flush. 

PS; This can be extented to 4 players; 3 Community spades + 8 spades in players hands + Q and K of spades in the deck = 13 spades.

  Posted by ajosin on 2005-04-11 15:38:41
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