Two men play a game of draw poker in the following curious manner. They spread a deck of 52 cards face up on the table so that they can see all the cards. The first player draws a hand by picking any five cards he chooses. The second player does the same.

The first player can now keep his original hand or draw up to five cards. His discards are put aside out of the game. The second player may now draw likewise. The person with the higher hand then wins.

Suits have equal value, so that two flushes tie unless one is made of higher cards.

After a while the players discover the first player can always win if he draws his first hand correctly. What hand must this be?

This is from : My Best Mathematical and Logic Puzzles (Math & Logic Puzzles)

## Thursday, December 02, 2010

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The obvious answer is a royal flush in any suit, but I have a feeling that's not the answer!

ReplyDeleteOkay I amend my answer....

ReplyDeleteYou're first so you pick four aces and a kicker. The next person will pick a straight flush up to the king. First person will discard three of the aces and pick ten, jack, queen, king to match the ace he kept. Second person can't win!

All the first person needs is to draw 4 Aces and a King. No royal flushes can be made without an ace. So therefore player 1 would have an Ace Four of a Kind - now the highest possible hand.

ReplyDeleteSterling, the second highest hand is a straight flush up to the King, so four aces is third highest.

ReplyDeletePlayer 1 needs to draw all four 10s with his first draw. The fifth card drawn is irrelevant.

ReplyDeleteWhatever player 2 draws thereafter is inconsequential; player 1 will always be able to discard three of the tens (and maybe the fifth card) such that he can form a straight flush that includes a 10. The best player 2 can do is a 9-high straight flush.

If player 2 draws four jacks as their first hand, player 1's straight flush will be 10-9-8-7-6. If player 2 draws four nines, player 1 can draw a royal flush. Player 2 can only block one suit with their fifth card, so player 1 will always have at least three choices as to which suit to take.

Addendum: Annie, the problem with your "four aces and a kicker" strategy is that it assumes player 2 immediately chooses a straight flush. A winning strategy for player 2 would be to pick the next highest four-of-a-kind. If player 1's kicker is a king, player 2 goes queens; otherwise, he goes kings.

ReplyDeletePlayer 1 can now only make a straight flush at the next highest card (J-high in case 1, Q-high in case 2). Say, in the king-kicker case when player 2 chooses four queens, player 1 goes J-10-9-8-7 of hearts. Player 2 will already have the queen in one of the other three suits, so a Q-J-10-9-8 straight flush can be built.

Obviously, player 1 can't adopt the next-lowest-four-of-a-kind strategy on his second draw to block player 2's attempt at a straight flush, because player 2 would already have a higher four-of-a-kind.