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I was so convinced that the "correct" answers to this excellent question were wrong that I took out a deck of cards to prove it. Then I realised that I was changing one of the rules. So here's the alternate version with Bob's rules:

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. Today, however, the table is full of work and so they're playing on the floor Here's how it works.

A standard 52 card deck is spread face up on the floor.

  • Alice picks up any 5 cards from the floor. Bob does the same.
  • Alice discards any number of cards, throwing them on the floor. She then picks up the same number of cards from the floor.
  • Bob does the same. (He can pick up cards Alice discarded. This is the difference from the other question.)
  • 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).

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  • $\begingroup$ Clarification needed: The cards Alice throws on the floor, are they counted as "face-up" being available to Bob or are they "lost" to the game? $\endgroup$ – BmyGuest Jun 5 '15 at 10:58
  • $\begingroup$ To b eclear, the only difference with the other game is that Bob can take the cards that Alice discarded, correct? $\endgroup$ – oerkelens Jun 5 '15 at 12:34
  • $\begingroup$ @oerkelens - that's right the only difference is that Bob can take the cards that Alice discarded. This however changes the entire game. $\endgroup$ – Akshay Rangnekar Jun 5 '15 at 14:19
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I don't think Alice can win this version.

All 52 cards are available to Bob, apart from the 5 in Alice's hand.

If she starts with a royal flush, so too will Bob, making it a tie, meaning Bob wins

If she acts in any way to block a royal flush, then it doesn't matter what Bob takes. Since she discards first, Bob always has the option to equal or better her hand.

She either has 4 of a kind and Bob has a straight flush, or they both have a royal flush. Either way, Bob wins

To summarise enormously:

There are is no five card poker hand you can remove from a deck of 52 cards that can't at least be equalled by another hand chosen from the remainder

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    $\begingroup$ Right. Actually, with the lack of discarded cards, it doesn't matter what either of them choose in the first round. In the second round, the highest unique (non-replicable) hand that Alice can choose is 4 Aces, which Bob can beat with a straight flush. $\endgroup$ – Akshay Rangnekar Jun 5 '15 at 14:21
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Bob, by selecting a royal flush, and no discarding of cards? He has the highest combination possible, and he wins when Alice has the same combination.

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  • $\begingroup$ I probably miss something essential, but I'm not sure what. $\endgroup$ – Mathias711 Jun 5 '15 at 9:25
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    $\begingroup$ You are only missing what happens if Alice blocks Bob's royal flush. $\endgroup$ – LeppyR64 Jun 5 '15 at 10:02

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