Saturday, July 5, 2008

Puzzle: Deck of 52 cards, two players, who will win?

Puzzle: The rules of a card game (having 52 cards) played between two peole are:- they will turn over two cards at a time. If both the cards are Black they will go the first player's pile and if both are Red they they will go to the second player's pile. If one is Red and one Black then both the cards will simply be discarded.

The process of turning over two cards at a time is repeated till all the 52 cards are exhausted. Whoever is having more number of cards in their pile wins the game. In case of tie the first player is declared winner. What's the chance of second player winning the game?

Solution: Zero ... yeah you read it right. Second player will never win the game. We can have only the following possibilities:-

  • If, All the pairs are perfectly matched - 13 Black pairs and 13 Red pairs and hence a tie and therefore First Player wins the game.
  • Else If, 12 pairs are matched - If we get 12 matched Black pairs then we'll have 12 matched Red pairs as well as if the rest two pairs are also matched then it'll be same as the above case. Okay... so in this scenario both the mixed pairs will be discarded and it'll be tie again and hence First Player wins in this case as well.
  • Else If, 11 pairs are matched - we'll have 4 mixed and hence discarded pairs and 11 matched pairs each for both Red and Black... again a tie and First Player wins the game.
  • Else If, 10 pairs are matched... tie and First player wins.
  • Else If, 9 pairs are matched... tie and First Player wins.
  • ... and so on.
Thus we that irrespective of how mixed the deck of cards is... it'll always be a tie and hence the First Player will always emerge as the winner.


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