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What kind of sports and games use mathematics beyond simple arithmetic? How is math applied to build strategies for these games?

Sailing could use mathematics in terms of astronavigation, tying knots, and force vectors. Billards also uses geometry and basic physics to line up shots and control the cue. Chess seems it would be related to Math but I'm not too sure how. Any form of gambling will take into account a betting strategy based on probability. I heard bridge is a game popular among math enthusiasts although I don't know how to play it.

Are there any more you can think of?

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    $\begingroup$ I cannot help mentioning use of categories in boxing. $\endgroup$ – Start wearing purple May 4 '13 at 20:28
  • $\begingroup$ I've never seen billiards players whip out a calculator, so they must be doing all the math in their heads. $\endgroup$ – Kaz May 4 '13 at 20:44
  • $\begingroup$ Tic tac toe, connect four, and checkers have been completely broken. $\endgroup$ – Fixed Point May 4 '13 at 21:10
  • $\begingroup$ Do war games count? Militaries extensively use math to make decisions involving logistics, strategies, tactics, and of course my absolute favorite cryptography for communications. Coaches (those with high stakes sports/games such as soccer where it can cost 75 million euros just to transfer a player) use headgear for communication with some sophisticated encryption. $\endgroup$ – Fixed Point May 4 '13 at 21:18
  • $\begingroup$ @Kaz, I have read a book on billiards that illustrates the physics in billiards and explains how players can control their shots. Most professionals probably know the technique but are unaware of where they came from. $\endgroup$ – cowchee May 4 '13 at 23:52
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In the game of darts, a player wins (checks out) by reducing their score to zero. This requires a knowledge of the integer partitions of the current score, restricted by the scores achievable by single darts, the fact that up to 3 darts are available, and that the final dart must land on a double or bull.

For example, if your score is 132, you could checkout with treble 20, treble 16 and double 12; 25 (outer bull), treble 19 and bull; or bull, bull and double 16.

According to Project Euler, there are 42336 distinct ways of checking out. I suspect in practice most pro darts players memorize the commonest checkout combinations.

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There's an extensive literature on the statistical analysis of baseball strategy.

One technique is to apply multivariate regression to determine the number of runs a team expects to score as a linear function of their singles, doubles, walks, and so forth. One can then use this to assign a run value to each individual player based on the individual accumulations of singles, doubles, walks, and so forth. This achieves a fairer and more reliable estimation of player value than the traditional measures of individual hitting value, such as runs scored or batted in.

Other methods allow one to discover that some baseball parks strongly favor the pitcher, while others strongly favor the batter, and to adjust the estimates of player skill to compensate for this. For example, Greg Maddux pitched half his 1993 games in Fulton County Stadium, a notorious hitter's park. By comparing his home pitching with his road pitching, and by comparing his pitching in Fulton County with other pitchers', we can get some understanding of how well he would have pitched in a more neutral park.

Similarly, one can use statistical methods to compare players of different eras who may never have played at the same time.

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Have you heard of the game Go? There is a great deal of mathematical and probabilistic analysis of strategy in Go, especially in combinatorial game theory. Go, chess, and checkers are all zero-sum, perfect-information, partisan, deterministic strategy games. Go is often considered one of the most complicated games in the world due to the vastness of its move set and variance between individual games. Also, the concepts of infinity and repeated loops come up in the basic rules, see The Ko Rule.

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In football (American), arithmetic could be used to calculate or determine whether a 2-point conversion is attempted or not. The choice would depend on how the future point accumulations on both teams "aligned". Also, regarding the choice of attempting 4-th down conversion, on-side kick, etc.. it depends on the odds or the probability of the action succeeding or failing relative to the gain or loss from the current time to the end of the game (so the time of the action is taken into account).

In tennis, under a certain model, there's a way to analytically calculate the probability of a player winning a match, if the probability of a player winning a single point is known.

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