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  • the game's structure,
  • optimal strategies,
  • practical strategies,
  • analysis of the game results/performance.

Which popular games are particularly mathematical by this definition?


Other examples:

  • go
  • bridge
  • Set.






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53 Answers 53

Diplomacy was pretty much invented to illustrate points in cooperative game theory. At least, that's how Martin Shubik told me the story. :)

Actually, I think he invented a card game called "So long, Sucker", and claimed Diplomacy was based on that.

That said, the core points are psychological -- if somebody knows he is losing but has the power to choose the winner, on what basis does he choose?

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I would be tempted to include Blackjack in the list of mathematical games. It's supposedly a game of chance, but gambling establishments routinely forbid card counting, because a mathematical approach to the game gives the player too big of an advantage. This rule is typically enforced through a mathematical analysis of the player's bets.

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Well, most card games have mathematical implications, of course.



(added later)


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There is a popular game in current cellphones called Pixelated (in BlackBerry) or Flood-It (in iPhone) that has a very interesting analysis (its generalization is equivalent to the Shortest superstring problem):

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There's also Nine Men's Morris, which is a very ancient game. My understanding is that it has been effectively solved in recent years with the help of computer analysis.

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Battle Line (originally published under the name Schotten-Totten) is a simple 2-player strategy card game designed by noted boardgame designer and math PhD Reiner Knizia. The goal is to try to capture a majority of the flags, which are contested regions where each player assigns troops to try and dominate. It has a number of mathematical elements, the most interesting of which, to me, is that during the course of the game you are allowed to capture a flag at any time that you can prove that your troops will be the strongest possible based on public information (the cards that have been played already).

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Peg solitaire which I played on chess board in childhood. Let me explain this solitary game; consider the following table consisted of 33 holes.

There is a peg in all except one of the holes.(Assume that one is in the center). You can move a peg horizontally and vertically. The permissible move consists of jumping one peg over another into an empty hole and simultaneously removing the peg has been jumped over. One can continue this process to reach a situation than cannot have an acceptable move. And the player is winner if the last situation has only one peg.

Thus we can ask some question:

1) Does this play has a winning strategy?

2) What are the possible last situations?

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The interesting point is the analysis of this game relates to the finite field with 4 elements, GF(4). The answer to question 1 is yes. For question 2 please consult the papaer "A solitaire game and its relation to a finite field" by N. G. de Bruijin.

It seems the result of the paper can be extended. I think it is useful in a course in algebra, I exposed the paper when I was a TA in algebra 2.

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There's also a discussion of this game in the beginning of Winning Ways, Volume 4, by Berlekamp, Conway, and Guy. – Todd Trimble Nov 23 '15 at 6:00

Khet is a great new game awarded by Mensa. There is even a master thesis dedicated to it:

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Rummikub ? It encourages some logical thought and analysis. It seems to have at least one mathematical paper on it

and it's popular and fun.

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The page claims that the popular trading card game Magic: The Gathering is Turing complete. Some mathematician who knows the rules should recheck the proof.

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I’d like to emphasize that typical sports games (eg. played at Olympic games, NFL, MLB, NBA, NHL) often fit at least three criteria mentioned in the question. Before me, Barry Cipra mentioned baseball. These games are very useful to ‘popularize mathematics’, more precisely applied mathematics (operations research). Being a mathematician can help you play the game better (sometimes as a coach because we think of optimal strategies). And ‘mathematics is involved in a surprising way’ as people (students) generally don’t expect mathematics can help in such situations. The techniques used involve eg. dynamic programming, probability trees, game theory and Monte Carlo methods. As examples, I could give the following papers: a) Beaudoin, D., & Swartz, T. B. (2010). Strategies for pulling the goalie in hockey. The American Statistician, 64(3), 197-204; b) Clarke, S. R., & Norman, J. M. (2012). Optimal challenges in tennis. Journal of the Operational Research Society, 63(12), 1765-1772 c) Annis, D. H. (2006). Optimal end-game strategy in basketball. J Quant Anal Sports, 2(2). d) Kostuk, K. J., & Willoughby, K. A. (2006). Curling's paradox. Computers & operations research, 33(7), 2023-2031. e) Chiappori, P. A., Levitt, S., & Groseclose, T. (2002). Testing mixed-strategy equilibria when players are heterogeneous: the case of penalty kicks in soccer. American Economic Review, 1138-1151. f) Tibshirani, R. J., Price, A., & Taylor, J. (2011). A statistician plays darts. Journal of the Royal Statistical Society: Series A (Statistics in Society), 174(1), 213-226.

For more on the topic, I recommend (Wright MB. “50 years of OR in Sport”. Journal of the Operational Research Society (2009) 60:S161-S168) or the short intro to this paper to be found here: Moreover, Journal of Quantitative Analysis in Sports can also be of interest.

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Heads or Tails - Is it popular game? There is a lot of mathematics related with this game. For example, it's non-transitive variant - Penney's game.

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Untangle (available in Debian based distros and easy online HTML5 version

From the documentation:

You are given a number of points, some of which have lines drawn between them. You can move the points about arbitrarily; your aim is to position the points so that no line crosses another.

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There is the amazing set of one-player games presented by S. Tatham (especially loopy and towers):

You should try it !


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The two-player single suit whist has been analyzed completely in this paper by Johan Wastlund. This was mentionned by Alison Miller in her answer to my MO question Bridge game with only one suit: strategy.

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There are various map/graph colouring games which are more subtle than determining the chromatic number. For example take a planar graph or map (particular or restricted to some category) - how many colours can P1 force P2 to use: or who wins if the first person forced to use a fifth colour loses?

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Just a two cents worth here. :) Chess itself might perhaps not be too mathematical, but the chess evaluation functions of any chess-playing computer program seems like a mathematical object. After all, these are maps from the set of chess positions to $\mathbb{R}$ and they are bound to satisfy various properties. Given any two chess programs that are both strong and might be expected to be decent (in terms of current technology) approximations to objective truth, one might probably expect them to be "close" in some meaningful way that one could perhaps attempt to define.

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Blood Bowl! All about managing probability.

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I am a bit surprised that Dominion has not been mentioned yet. I am referring less to the gameplay itself rather than to the analyzes that people do in order to assess the "intrinsic worth" or "situational worth" (my terms) of a card or a strategy, using a rather complicated simulator. I perceive it as a kind of Monter Carlo analysis.

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Othello, since the number of the opponent's pieces you can flip highly depends on where you put yours.

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Being a mathematician allows you to play the game of chess better.

Mathematicians perform calculations quickly and accurately. Mathematicians are creative. Mathematicians are good at identifying the important component to a problem. Mathematicians are accustomed to encountering new problems and challenges. Mathematicians are good at comparing alternatives.

Now consider the game of chess. At any level 90%+ of chess games end as a result of an oversight in calculation. This often happens because one misses a surprising or unconventional move that requires creativity. In chess there are many strategic components, pawn structure, piece activity, material imbalance, king safety etc. but often one of these takes center stage. Almost always in a game of chess there will be a situation you have never encountered before. 90+% of grandmaster chess moves would be listed in the top 5 choices of any tournament player but consistently play the third or fourth choice move is a recipe for disaster even in scholastic chess.

I can not think of a profession better suited to playing chess than that of mathematician!

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The best mathematician chess players in history are: Emmanuel Lasker John Nunn (who appears to have published in "Topology" as J. D. M. Nunn). There are some people who were strong masters once upon a time (Sarnak,Formanek) I don't agree with Doug's claim that time pressure is the culprit (many good mathematicians were good at mathematics contests), but the real point (imho) is that the amount of self-discipline and effort needed to become a strong tournament chess player is much greater than the intrinsic interest of the game (if you want to work that hard, why not work on mathematics? – Igor Rivin Jan 16 '11 at 21:58
Well, in my college chess teams, most of the chess players were studying mathematics or computer science; but it was very noticeable that the best mathematicians were only average chess players, and vice versa, with few exceptions. (And, several of the top mathematicians didn't play at all). Former World Chess Champion Karpov gave up mathematics at university because mathematics and chess were "incompatible". These are not statistically significant samples, of course! – Zen Harper Jun 8 '11 at 1:24
Former world champion Dr. Botvinnik was also an electrical engineer. – Zen Harper Jun 8 '11 at 1:28
Former world champion Dr. Euwe was a mathematician and a computer scientist. – Keep these mind Jul 8 '13 at 13:51

Has no one mentioned Finchley Central and the other games discussed in Volume 3 (1969) of Manifold? Available here, starting on page 31.

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I'm afraid I don't understand the payoffs of that game. Is/was it popular? – Douglas Zare Apr 29 '15 at 13:48
Variants of Finchley Central have been popular, e.g., Mornington Crescent, "Finchley Central and Mornington Crescent became popular in the United Kingdom as a play-by-mail pastime, and in the 1980s were played by post in a number of play-by-mail magazines. One format involved a series of elimination rounds, with everyone except the winner of the current round going forward onto the next. Mornington Crescent is now played widely online, in the spirit of the radio series. Games are played by fans on Usenet, in diverse web forums, and ... – Gerry Myerson Apr 30 '15 at 1:49

Games are mathematical. That's not something you can avoid. If there are rules for moves and goal states, you've already entered mathematics through logic and proof theory. And even for the simplest games, there is a rich mathematical theory studying it's structure and analysing results.

Mathematicians aren't just interested in finding optimal strategy and assigning a solution to a game. They also look to generalise and abstract theories of the gameplay and find ways to evaluate statements in the ontology of the game. Even simple games have rich logical calculi, and there are questions of rule independence and game extensions that can always be asked.

Dynamic Epistemic Logic is one of the general settings in which more advanced theories of game play have been formulated. Whenever two or more people are involved in gameplay, there are important distinctions between belief, knowledge, shared knowledge, and strategy that must be made to fully understand the dynamics. DEL (and a number of related settings) have been used to evaluate many common games, from simple pebble and card games to complicated board games and beyond, as a means to understand the evolution of belief systems and evaluate strategy during the play.

The Dynamic Epistemic Logic of Games

Epistemic Logic and the Foundations of Decision and Game Theory

Selection Monads and the Relation Between Game Theory and Proof Theory

I think the real mathematical answer to your question, then, is every popular game. There's no "most mathematical". Proof theory, models, modal logics, agent calculi and bisimulation lie at the heart of all games.

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How would you start an analysis of the party game Charades? Gerhard "Or Worse Yet, Of Pictionary" Paseman, 2013.01.30 – Gerhard Paseman Jan 31 '13 at 1:57

protected by quid Apr 29 '15 at 10:49


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