Friday, March 30, 2018

game - Face Up Poker with Alice and Bob


Alice and Bob are going to have a poker night. They have invented a variant where there are no secrets, and no randomness, called Face-Up Poker. Here's how it works.


A standard 52 card deck is spread face up on a table.



  1. Alice picks up any 5 cards from the table.

  2. Bob does the same.


  3. Alice discards any number of cards, throwing them on the floor. She then picks up the same number of cards from the table.

  4. Bob does the same (Bob can't pick up cards that Alice discarded).


They then compare hands, and the best hand wins, with Bob winning ties.



Which player can force a win? What is their winning strategy?



This puzzle assumes familiarity with the rankings of poker hands, but no other knowledge of the rules of poker. For a refresher these rankings, see this helpful page (the hands are listed worst to best from top to bottom).



Answer



There is a winning strategy for




Alice. For her starting hand, she will select all four 10's, along with another lower card (let's say the 2 of clubs for concreteness). This will make it impossible for Bob to end up with a royal flush. Bob needs to prevent Alice from making a royal flush when she discards, so he must select a starting hand containing a card of rank higher than 10 in each of the four suits. This means that for at least three of the four suits, Bob's starting hand cannot contain a card of rank lower than 10 in that suit.

When it is Alice's turn to discard, she chooses one of the suits in which Bob did not take a card of rank lower than 10. She keeps the 10 of that suit, discards her other four cards, and takes the 6, 7, 8, and 9 of the chosen suit (Bob did not select a card of rank lower than 10 in the chosen suit). Alice's final hand will be a straight flush to the 10. Since all of the other 10's are now on the floor, Bob cannot possibly make a straight flush whose high card is 10 or above, so Bob's hand cannot possibly tie or beat Alice's.



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