There are 2 players, who we will call A and B. A stack of cards is given initially. It may contain an arbitrary number of cards. A player whose turn comes may take between 1 and 5 cards; how many they take is their choice. Player A begins. The last player who can make a move has won the game. How should the players behave? Is there a strategy by which one of the players can force a win, no matter what the opponent does? How does it depend on the initial numbers of cards? For given numbers of cards, how many rounds must at least be played before the game ends?


If $A$ takes $k$ cards and $B$ replies by taking $6-k$ cards, then $B$ will win whenever there are a multiple of $6$ cards at the start.

Suppose the starting number is not a multiple of $6$. What should $A$ do?

  • $\begingroup$ I can't get my head around how to work it out? $\endgroup$ – a. macklin Apr 26 '17 at 17:41
  • $\begingroup$ @a.macklin Let's play: start with 15 cards, I'll let you go first. $\endgroup$ – Lord Shark the Unknown Apr 26 '17 at 17:43
  • $\begingroup$ I'll take 3 cards then $\endgroup$ – a. macklin Apr 26 '17 at 17:46
  • $\begingroup$ @a.macklin Leaving me 12. Hmm. I'll take 2, leaving you 10. $\endgroup$ – Lord Shark the Unknown Apr 26 '17 at 17:48
  • $\begingroup$ I'll go for 5 then $\endgroup$ – a. macklin Apr 26 '17 at 17:49

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