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Consider the following game. The rules are as follows: There is an unlimited supply of identical coins, each with radius s > 0. Each player takes turns to place coins inside a shaded region. The coins cannot overlap, and they must lie entirely inside the shaded region. The first player who cannot put down a coin will lose the game. Consider the shaded region B, which is an annulus bounded by two circles with radius r(> s) and R(> 3r) respectively. Is it true that one of two players has a winning strategy ?

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I know that there is not a winning strategy, but how to prove it?

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    $\begingroup$ You know that "there is not a winning strategy"? What do you mean by that? Do you mean that neither player has a winning strategy? How do you know that? It seems obvious to me that one player or the other must have a winning strategy. $\endgroup$ – bof Dec 17 '16 at 2:21
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    $\begingroup$ By the way, did you delete your first draft of this question and post it as a new question? Why did you do that instead of editing your question? $\endgroup$ – bof Dec 17 '16 at 2:22
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    $\begingroup$ By the way, I think the position of the small circle might also influence the strategy. $\endgroup$ – Jason Dec 17 '16 at 2:29
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    $\begingroup$ If it's a standard annulus, with the circles of radius $r$ and $R$ sharing a common center, then the second player has a winning strategy, simply by "mirroring" whatever the first player does. For certain nonstandard annuli, the first player has a winning strategy. $\endgroup$ – Barry Cipra Dec 17 '16 at 2:31
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    $\begingroup$ I know that by the Zermelo's Theorem, once the radius r and R is fixed, there must be some winning strategy for someone. Good, but you should have put that in your question. You should tell us what you know about the problem, so we know we don't have to explain Zermelo's theorem to you. $\endgroup$ – bof Dec 17 '16 at 2:31

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