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A game of 2014 cards

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There is a table with a row of 2014 cards.  Each card has a red side and a blue side.  We’ll say that a card is red if the color on its visible face is red, and analogously for blue.  Two players take turns to do the following move:  select any 50 consecutive cards where the left-most card is red, then flip each of those 50 cards (thus, for those 50 cards, turning red cards into blue cards and blue cards into red cards).  Both players look at the cards from the same side of the table, so “left-most” means the same to both of the players (that is, you can think of one of the ends of the table as being designated as the left end).  When it is a player’s turn, if that player cannot make a move (that is, if there is no way to select 50 consecutive cards the left-most one of which is red), then that player loses and the other player wins.  If you are one of the players and all cards are initially red, can you be sure to win, and if so, do you want to be the player who goes first or second?

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ronret45 Expert Asked on 1st August 2015 in Microsoft Interview Puzzles.
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    • 1 Answer(s)

    If all the cards are initially red, the first player can always win the game.

    To see why, note that the game can be divided into rounds, where each round consists of both players making a move. The first player will always start each round, and the second player will always finish each round.

    In the first round, the first player can select the left-most 50 cards and flip them, which will leave the left-most 50 cards blue and the remaining 1964 cards red. In subsequent rounds, the first player can always choose the left-most 50 red cards, because there will always be at least 50 red cards to the left of any blue card. By doing this, the first player will always ensure that there are at least 50 consecutive red cards at the left end of the table, which means that the game can continue.

    Eventually, the first player will flip the left-most 50 red cards and win the game, because there will be no more red cards left and the second player will not be able to make a move.

    Therefore, the first player can always win the game if all the cards are initially red, and it is preferable to be the first player to make the first move.

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    Moshe Expert Answered on 23rd March 2023.
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