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There are 7 coins in a heap. Two players A and B play, take turns. At each step, a player can put a coin from the heap and put it on the table on a head or tail, or turn over a previous coin on the table.

The winner is a player who first collect 4 heads or tails.

Question. Which player has a winning strategy?

Attepmt. Here is cycle, and nobody can not win.

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  • $\begingroup$ Players may not draw all coins. The winner is a player who first time collect 4 heads or tails. $\endgroup$
    – Nick
    Commented May 6, 2020 at 12:32

2 Answers 2

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Nobody wins. If a player would take the third coin from the heap he would lose. That move would either make it 3:0 and the other player can win immediately or the move makes it 2:1. Then the other player could take another coin to make it 2:2 and force that three of a kind are opened up in the following round.

Therefore, no player wants to take the third coin from the heap and nobody wins. The players just keep turning around the first two coins.

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  • $\begingroup$ Thanks for the answer. How can you reformulate the rules of the game so that a player has a winning strategy? The number of coins cannot be changed $\endgroup$
    – Nick
    Commented May 7, 2020 at 3:12
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    $\begingroup$ There would be a winning strategy for one player if it was not allowed to turn around an existing coin instead of taking a new one or if it was only allowed to turn around existing coins a limited number of times. You can find out which player wins using the argumentation from the answer. $\endgroup$
    – DFL
    Commented May 7, 2020 at 10:53
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It is, of course possible and actually strictly necessary that by the end of the game there will be 4 heads or 4 tails face up, this could happen as early as the fourth or as late as the seventh turn.

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  • $\begingroup$ Players can also turn around an existing coin instead of taking one from the heap. $\endgroup$
    – DFL
    Commented May 6, 2020 at 12:44
  • $\begingroup$ I apologise for missing this out. As long as there is no restriction on turning the same coin over or the number of consecutive coins you can turn over, there is not necessarily a winner, however a winner is of course still possible as your question asks. $\endgroup$
    – mathemagic
    Commented May 6, 2020 at 12:49

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