As 2021 comes to an end, I thought I’d use this month’s blog post to share one of the cooler bits of math I’ve learned this year. It’s a paradox that comes from combinatorial game theory which, for those not familiar with the field, Wolfram Mathworld describes as “the theory of two player games of perfect knowledge such as go, chess, or checkers”. Before I can get into the paradox, I need to start with a definition:<\/p>\n
Definition: <\/strong>A game G is somewhat finite <\/em>if it satisfies the following four conditions:<\/p>\n To make the above definition more concrete, here are a few examples of somewhat finite games<\/em>:<\/p>\n So despite the silly examples, somewhat finite <\/em>games seem fairly well-behaved. However, consider a game with the following rules:<\/p>\n The game described above is called the hypergame<\/em>. You can check that the hypergame is somewhat finite<\/em>, which means that, during play of the hypergame, it is a valid move for Player 1 to choose G = hypergame. This means that, on Player 2’s next move, they must behave as the first player of the hypergame, and it’s again a valid move for Player 2 to choose G = hypergame. Then Player 1 is once again the first player of the hypergame, and can choose G = hypergame, and so on ad infinitum<\/em>. But this implies that the hypergame is not <\/em>somewhat finite, since we can force a game that goes on forever. Thus we have a paradox.<\/p>\n Paradoxes like the above are one of the things that first fascinated me when I got into math, and I hope you found this interesting, too. See you all in 2022!<\/p>\n (If my explanation of the paradox is unclear or you’d like to learn more, I’d suggest reading this paper<\/a> by William S. Zwicker, who was the first to discover the paradox.)<\/p>\n <\/p>\n\n
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