# Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in

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

Which popular games are particularly mathematical by this definition?

Motivation: I got into backgammon a bit over 10 years ago after overhearing Rob Kirby say to another mathematician at MSRI that he thought backgammon was a game worth studying. Since then, I have written over 100 articles on the mathematics of backgammon as a columnist for a backgammon magazine. My target audience is backgammon players, not mathematicians, so much of the material I cover is not mathematically interesting to a mathematician. However, I have been able to include topics such as martingale decomposition, deconvolution, divergent series, first passage times, stable distributions, stochastic differential equations, the reflection principle in combinatorics, asymptotic behavior of recurrences, $\chi^2$ statistical analysis, variance reduction in Monte Carlo simulations, etc. I have also made a few videos for a poker instruction site, and I am collaborating on a book on practical applications of mathematics to poker aimed at poker players. I would like to know which other games can be used similarly as a way to popularize mathematics, and which games I am likely to appreciate more as a mathematician than the general population will.

Other examples:

• go
• bridge
• Set.

Non-example: I do not believe chess is mathematical, despite the popular conception that chess and mathematics are related. Game theory says almost nothing about chess. The rules seem mathematically arbitrary. Most of the analysis in chess is mathematically meaningless, since positions are won, drawn, or tied (some minor complications can occur with the 50 move rule), and yet chess players distinguish strong moves from even stronger moves, and usually can't determine the true value of a position.

To me, the most mathematical aspect of chess is that the linear evaluation of piece strength is highly correlated which side can win in the end game. Second, there is a logarithmic rating system in which all chess players say they are underrated by 150 points. (Not all games have good rating systems.) However, these are not enough for me to consider chess to be mathematical. I can't imagine writing many columns on the mathematics of chess aimed at chess players.

Non-example: I would exclude Nim. Nim has a highly mathematical structure and optimal strategy, but I do not consider it a popular game since I don't know people who actually play Nim for fun.

To clarify, I want the games as played to be mathematical. It does not count if there are mathematical puzzles you can describe on the same board. Does being a mathematician help you to learn the game faster, to play the game better, or to analyze the game more accurately? (As opposed to a smart philosopher or engineer...) If mathematics helps significantly in a game people actually play, particularly if interesting mathematics is involved in a surprising way, then it qualifies to be in this collection.

If my criteria seem horribly arbitrary as some have commented, so be it, but this seems in line with questions like Real world applications of math, by arxive subject area? or Cocktail party math. I'm asking for applications of real mathematics to games people actually play. If someone is unconvinced that mathematics says anything they care about, and you find out he plays go, then you can describe something he might appreciate if you understand combinatorial game theory and how this arises naturally in go endgames.

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Yeah, well the rules of mathematics are chessly arbitrary. – Harry Gindi Feb 1 '10 at 11:19
I think it's a mischaracterization to say chess is nonmathematical; it's just that chess, like so many things one encounters in the real world, is neither elegant nor simple from the point of view of mathematics. That game theory can't tell us much about chess tells us more about the limitations of game theory than about the mathematical nature of chess. That said, your suggested examples are definitely better. – Mark Meckes Feb 1 '10 at 14:29
This is very far from 'give a list of all games.' One hope is to find other popular games whose play involves mathematics. Another is to learn more real mathematics about games I already know. A third idea is to see what resonates with other mathematicians. I'm sorry if you don't find these interesting, or if you find my criteria arbitrary--I don't see a huge difference between this and questions like, "What are neat applications of mathematics/this field?" – Douglas Zare Feb 1 '10 at 17:56
You disagree with my statement that Nim is not actually played for fun? I can show you go clubs, bridge clubs, backgammon clubs, even a "world championship of rock-paper-scissors," etc. I've never seen a Nim club or heard someone describe himself or herself as a Nim player. There are many theoretical games people don't actually play, and I don't think it's arbitrary to exclude those. I'll clarify my reasons for excluding chess later. – Douglas Zare Feb 1 '10 at 23:17
In "l'année dernière à Marienbad" ("last year in Marienbad"), a movie by Alain Resnais, you can see people playing Nim for fun. Now this just moves the problem, because I don't know anyone watching that movie for fun. (I just mean that peculiar movie. Resnais made a lot of very good movies. But that one is a serious contender for the prize of the most boring movie ever). – Joël Oct 7 '11 at 19:31

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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If you can choose the winner (other than yourself), you are called the kingmaker. That's a common phenomenon in games with $3$ or more sides. en.wikipedia.org/wiki/Kingmaker – Douglas Zare Jan 15 '13 at 20:12

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.

I'm disappointed at your considering chess non-mathematical. Wonder what Noam Elkies would think. :-)

When I was a teenager I would play a lot a board game named here "Risiko!" (I believe that the English name is "Risk"). My impression then was that there were some mathematical aspects that could be considered while planning a strategy.

Also Hex should be added to the list of mathematically interesting games.

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We don't play risk anymore at my house because it always ends in either a verbal argument or a physical confrontation. =\. – Harry Gindi Feb 1 '10 at 11:37
You mention card games. In Cryptonomicon book by Neal Stephenson ( great one!) there is a solitaire card game which gives You a cipher called SOLITARE strong enough to be "industry standard" whilst still may be performed without computer (only with a card deck only!). It was analysed by Bruce Schneier and this analysis is part of the book! en.wikipedia.org/wiki/Solitaire_%28cipher%29 – kakaz Mar 2 '10 at 14:24

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): http://arxiv.org/abs/1001.4420 http://www.cs.bris.ac.uk/Research/Algorithms/BAD10/Slides/Jalsenius.pdf

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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?

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
@ToddTrimble Thank you. I see it, that's interesting. – Rahman. M Nov 23 '15 at 8:15

Khet is a great new game awarded by Mensa. There is even a master thesis dedicated to it: http://www.personeel.unimaas.nl/Uiterwijk/Theses/MSc/Nijssen_thesis.pdf

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

http://comjnl.oxfordjournals.org/content/49/6/665.abstract

and it's popular and fun.

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The page http://www.toothycat.net/~hologram/Turing/index.html 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: http://ifors.org/web/ifors-september-2009-newsletter/. 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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Obverse or reverse! – Harry Gindi Feb 1 '10 at 11:36

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):

http://www.chiark.greenend.org.uk/~sgtatham/puzzles/

You should try it !

Damien.

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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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In some games, computer programs evaluate positions, and choose to play the position with the highest evaluation, perhaps among a restricted set of candidates. Backgammon is an example. However, many competitive chess programs don't fit this description. Instead of evaluating positions they often directly choose the move to make, based on partial searches of the game tree that are affected by the amount of time left. A computer might try to spend ten seconds, quitting early if it finds a mate, but otherwise reporting its best candidate after that time. This might not be repeatable. – Douglas Zare Sep 10 '15 at 22:17

Blood Bowl! All about managing probability. http://en.wikipedia.org/wiki/Blood_bowl

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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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What is mathematical about Dominion? – Timothy Chow Jan 30 '13 at 18:58
@TimothyChow: I added a bit of explanation. – Felix Goldberg Jan 31 '13 at 9:26

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

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What gives a mathematician an advantage when playing or studying Othello over an equally smart engineer or philosopher? I learned to play Othello at a decent level to win a multi-game tournament, and I don't recall anything standing out. The moves don't commute, unlike in Lights Out, and the endgame seems to require a brute-force search. If you look at the opening heuristic of trying to minimize your exposed pieces, this could resemble games in combinatorial game theory. – Douglas Zare Apr 29 '15 at 13:57

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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Yes, and few mathematicians rise above B class. Sad. – Igor Rivin Jan 14 '11 at 21:40
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, en.wikipedia.org/wiki/Mornington_Crescent_(game). "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
"...on the London Underground itself. A Facebook application has also been produced." The game has also been compared to the game, World War Three (see, e.g., www1.maths.leeds.ac.uk/~pmt6jrp/personal/quintics.html). – Gerry Myerson Apr 30 '15 at 1:52
Although I do like Mornington Crescent, it really doesn't count because it doesn't actually have any rules - there is no mathematics involved. You might as well suggest Numberwang. – Nate Eldredge Nov 22 '15 at 19:15
@Nate, I'm not familiar with Numberwang, but here are the rules for Finchley Central: "Players take turns to name London Underground stations: the first one to say Finchley Central'' wins." – Gerry Myerson Nov 22 '15 at 22:06

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
@GerhardPaseman: Pattern recognition and the theory of images is hugely mathematical. One of my favorite books to refer to on the subject in college was Monique Pavel's "Fundamentals of Pattern Recognition" which started with the topological theory of images and the group theory of mappings and generalised this to a categorial setting. Once you have object recognition, then you analyse semantic interpretations and similarity mappings to understand the goals. Are you suggesting these are things that can't be mathematically modeled, or that they shouldn't be, or? – ex0du5 Jan 31 '13 at 19:40
I think my point is that some view games mathematically, for the goal of practicing mathematics. While I understand and often sympathize with such an endeavour, I remind you that some games are for pure social entertainment, and mathematical analysis is often counter to that goal. I would consider a semiotic approach to Charades analysis, and I am not clear what goals your suggested approach are trying to reach. Gerhard "Likes Board Over Party Games" Paseman, 2013.02.01 – Gerhard Paseman Feb 2 '13 at 6:27

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