„Leaky choice” game - Lycée Żmichowska (Varsovie) Lycée Vicat (Souillac)

Lycée Żmichowska (Varsovie)
Two players alternately pull stones out of a hat which contains 7 stones (our players know that there are 7 of them). With each move, each player has only two possibilities: he draws either 1 or 3 stones (he cannot pull out 2 stones – that is why the choice is “leaky”). The one who pulls out the last stone – wins. Does any of these players have a winning strategy? If yes – please find it. Let’s generalize the problem – instead of 7 stones, let’s introduce n of them. Therefore we analyze an analogous game, but with n stones in the hat. Please find the solution for every n natural. If you manage to solve these problems, we may think about similar games for slightly different “leaky choices”.
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