Answers.Body: <p>Here are some starting spots in reading up on Game theory which is the mathematical tool that would be most appropriate for making claims about chess.</p>
  
  <p><a href="http://www.math.harvard.edu/~elkies/FS23j.03/zermelo.pdf" rel="noreferrer">This</a> is a light history of early game theory. Chess is a "perfect information game" and there are some interesting things one can claim about this category of games. See <a href="https://en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)" rel="noreferrer">this</a> for example. 
  I would also read up on Claude Shannon. <a href="https://en.wikipedia.org/wiki/Shannon_number" rel="noreferrer">Shannon's number</a> gives a bound on game complexity tree. For standard chess we have an intractable puzzle but you could make a smaller variants and solve these smaller games. Here's an example of <a href="https://arxiv.org/pdf/1307.7118v1.pdf" rel="noreferrer">this</a>. The authors claim to have solved a smaller 5x5 variant of chess. </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T13:05:48.227Z
Answers.Text: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game
Answers.UserDisplayName: BCLC
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T05:09:39.020Z
Answers.LastActivityDate: 2019-12-10T05:09:39.020Z
Answers.UserDisplayName: Mason
Answers.Body: <p>Not really.   At least not serious mathematicians. </p>
  
  <p>It is covered somewhat in Game Theory but other  approaches like Group Theory do not seem to fit;  although there might some higher dimensional non-linearish type description that could be found.  </p>
  
  <p>So Algebra which includes Groups,  and Topology too,  seem to be out but perhaps it could fit in Analysis if somebody cared enough to try it.  But that would an advanced mathematician who would be more likely to attempt a better problem that would lead to their PhD and a job teaching.  </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-09T22:30:19.740Z
Answers.Text: Thanks for the insight. If such an attempt succeeds, it ought to lend a lot of recognition. Just think of how many people have struggled to solve it so far.
Answers.UserDisplayName: postoronnim
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-09T23:42:27.873Z
Answers.Text: @postoronnim -  It would be worth recognition.  I am not sure how many qualified people have seriously tried to work on that problem.
Answers.UserDisplayName: yobamamama
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T12:31:28.623Z
Answers.Text: [Noam Elkies](http://www.math.harvard.edu/~elkies/chess.html) springs to mind.
Answers.UserDisplayName: Rebecca J. Stones
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-09T22:09:53.287Z
Answers.LastActivityDate: 2019-12-09T22:09:53.287Z
Answers.UserDisplayName: yobamamama
Answers.Body: <p>I'd like to add some reflections to the answers. First, we probably have to distinguish how chess is played by a chessplayer, computer, or mathematician. The main point here is that these three may use different criteria when making moves. The mathematician tries to optimize his or her game in general while a chessplayer or neural network may use local criteria. In its turn, the chessplayer may wish to make the game interesting rather than just to win. Thus, analyzing the criteria, we may apply mathematics that includes not only the pure optimization but also symmetries, i.e. group theory. To my mind, mathematics is more about such kind of beauty and that's why it can be applied to chess in a non-trivial way.</p>
  <p>Now, I'll try to describe mathematically one of the &quot;symmetrical&quot; criteria which is well-known for a chessplayer (though I'm not a professional player). At the beginning of the game, white has 20 moves, and one could not use all of them according to his or her preferences. But it seems rather natural that the number of moves satisfying some strategy decreases in time. Thus, if two players built the decision tree they would find out that the outdegrees of vertices become smaller and smaller. Thus, the player may wish to <em>terminate</em> the game finding such a move that leads to a <em>path</em>, that is a part of the decision tree s.t. each its vertex is followed exactly by one vertex. This corresponds to the situation when two players both don't know the result and it's the game that shows it in a clear way. To generalize the task, one may wish to find the moves that lead to an abstract subgraph of the decision tree (say, with a well-known group of symmetries).</p>
  <p>P.S. Yet another way to think how group theory can be applied in chess is to note that the moves of <strong>K</strong>, <strong>Q</strong>, <strong>R</strong>, <strong>B</strong>, <strong>N</strong> form natural groups with at most 4 generators.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T07:28:00.113Z
Answers.LastActivityDate: 2021-01-20T07:33:17.900Z
Answers.LastEditDate: 2021-01-20T07:33:17.900Z
Answers.UserDisplayName: Rebel Yell
Answers.Body: <p>As an applied mathematician, I feel that mathematics very seldom solves any problem in a practical sense, apart from those that reduce to some form of counting. There are many problems that lead to differential equations, for example, that present points of interest in their own right, but to apply them to any real engineering situation assumptions must be made that can never be completely reliable. An engineer must understand both their power and their limitations.</p>
  <p>The relationship that exists is that some aspect of a &quot;real world&quot; problem may have an interesting abstract structure and if this captures the interest of enough people it becomes part of mathematics. It goes in directions determined by the possibilities for rigorous thought. After a while it no longer matters if there is any visible connection to the origin.</p>
  <p>In the case of chess, there are a very few abstract aspects with practical value, mostly to do with Pawn endgames. Rigorous statements like &quot;either somebody should win or it's a draw&quot; are too general to be useful and are not really interesting either. Useful statements would apply to specific positions and there are too great a diversity of these for any &quot;mathematical&quot; statement to apply to all of them. A strong player sees the exceptions to the rule</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-07-20T03:25:36.153Z
Answers.LastActivityDate: 2021-07-20T03:25:36.153Z
Answers.UserDisplayName: Philip Roe
Answers.Body: <p>Anyone looked at the mathematics behind alpha-zero describing chess completely even without having a engine.  One might see the engine implementations as measure instruments for such mathematical model of chess.</p>
  <p>It does consider a big space that is bound to contain the solutions, which is far from from assuming impractical complete tree knowledge as in most game theory about chess (that i am aware of).</p>
  <p>It handles uncertainty, in the approximation process.  Allowing access to the solution via approximation (at least in theory).  We just have to be confortable with asymptotic behavior perhaps.</p>
  <p>It also use probability spaces in order to be compatible with the type of data coming in from its measurements (RL).  I would say, as long as you included the data relationship with the model, you might have something of a mathematical characterization, be it of statistical nature.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-12-31T05:15:34.073Z
Answers.LastActivityDate: 2022-12-31T05:23:39.053Z
Answers.LastEditDate: 2022-12-31T05:23:39.053Z
Answers.UserDisplayName: dbdb
Answers.Body: <p>I'd hate to say it, but mathematically speaking chess as we play it is very boring. It is a perfect information game, without any non-determinism, for two alternating players. This means that chess is either a win for white, a win for black, or a draw. The optimal strategy for chess is trivial and known: minimax. As far as maths is concerned, chess is solved.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T17:43:14.413Z
Answers.Text: @emory - because actually _executing_ the optimal strategy is computationally infeasible?  (due to combinatorial explosion)
Answers.UserDisplayName: davidbak
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T18:10:45.737Z
Answers.Text: @emery - perhaps you're confusing the way mathematics is done with the way other work is done? Consider: An engineer, a physicist, and a mathematician are working, when a small fire breaks out in front of their offices.
  The engineer panics and grabs the fire extinguisher, spraying it everywhere, putting out the fire, but causing extra damage in the process.
  The physicist runs some quick calculations, and uses just enough to put out the fire.
  The mathematician sees the fire, looks over to the fire extinguisher and says “a solution exists!” then returns to his office.
Answers.UserDisplayName: davidbak
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T15:09:30.847Z
Answers.Text: @emory You don't understand. Thanks to minimax we don't only know *what to do* (that is to score more points in basketball or cause checkmate of the opponent in chess), but we also know *how exactly to do it*. We simply know that if before every possible chess move we play all the possible combinations of how game could develop and then choose the best one, we are performing the **best possible** strategy. That is minimax algorithm in nutshell. And that is very different to your basketball example.
Answers.UserDisplayName: Firzen
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T15:22:16.400Z
Answers.Text: @Firzen, but if you don't care about such things then yes, chess is solved.
Answers.UserDisplayName: emory
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T15:22:42.930Z
Answers.Text: @emory *"If I recall correctly such a computer would be many times bigger than our solar system and would take many million human lifetimes to run. Is it not the case that such a machine would be so massive that it would form a black hole and the results of its computation could not escape its event horizon."*  I wouldn't be so sure about that. Anyway.. construction of computers is rather physical discipline. We might not have *efficient* best possible chess algorithm, but we have at least some. So, as mentioned above, mathematically, chess is solved.
Answers.UserDisplayName: Firzen
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T15:23:25.057Z
Answers.Text: @emory No, minimax doesn't require a lot of memory at all. Your mobile phone has the computational power to optimally solve chess - it just requires a long, long, long time. And even if you'd need a huge computer until the heat death of the universe, that's no concern of mathematics.
Answers.UserDisplayName: orlp
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T06:15:11.103Z
Answers.Text: Also, given alpha beta pruning, with good move heuristics and a transposition table, you only need to look at about 10^30 ish different positions. That's only about 2^100, which while intractable, isn't that intractable. It's roughly on the order of breaking 256 bit encryption.
Answers.UserDisplayName: Oscar Smith
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T13:06:56.900Z
Answers.Text: LOLwut? as far as maths is concerned, chess CAN be solved. so could checkers. but this didn't stop people from solving checkers
Answers.UserDisplayName: BCLC
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-03T09:15:06.983Z
Answers.Text: Combinatorial games are also perfect information games, and there is not even a draw. That doesn't mean they are completely solved (e.g.: chomp). Very disappointing answer.
Answers.UserDisplayName: Cyriac Antony
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-03T09:38:03.043Z
Answers.Text: @CyriacAntony There doesn't exist a player that can defeat minimax for chomp. You're using a different definition for 'solved' than this answer is. The optimal strategy for chomp is known: minimax.
Answers.UserDisplayName: orlp
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-04T04:06:08.800Z
Answers.Text: @orlp Minimax is an algorithm, not a strategy. If minimax counts as solving, then in the same way we can say that every problem can be solved (except halting problem, etc.) by exhaustive search. How about that?
Answers.UserDisplayName: Cyriac Antony
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-04T11:25:21.100Z
Answers.Text: @CyriacAntony Then please give a definition of a 'strategy'.  _"then in the same way we can say that every problem can be solved (except halting problem, etc.) by exhaustive search"_ Yes? There's nothing wrong with that.
Answers.UserDisplayName: orlp
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T14:15:53.410Z
Answers.LastActivityDate: 2019-12-10T14:15:53.410Z
Answers.UserDisplayName: orlp
Answers.Body: <p>Chess isn't the sort of object that mathematicians are interested in studying. Mathematicians care about problems that are in some way infinite: either infinite objects or finite objects of which there are infinitely many examples. Chess is completely finite. There is only one chess, it uses a finite board, there are finitely many possible positions. If chess is played rationally (and we can't hope to characterize it without that assumption) then games are finite in length. "Rational" basically means that either player will claim a draw as soon as they have the right to, and that means that games are less than about 5900 moves, which means there are only finitely many games.  A mathematician might study the prime numbers, but they're not going to study the single number 1827368429.</p>
  
  <p>To a mathematician, "solve chess" is just a calculation, and mathematicians aren't interested in calculations. The fact that it's a culturally important calculation might encourage somebody to try, but it's hard to see anybody having the time to do something so complicated. It's the sort of thing that would take decades of full-time work and, if you failed, you'd have basically nothing.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T21:34:08.600Z
Answers.Text: Big math does not have an issue with finite things, there are plenty of examples if you look around. The question is not about calculation or computation, it's about giving chess a theoretical footing in the math realm.
Answers.UserDisplayName: postoronnim
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T21:39:35.830Z
Answers.Text: @postoronnim Examples such as what? And I've given you the theoretical footing: it's a single finite object.
Answers.UserDisplayName: David Richerby
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T23:36:27.337Z
Answers.Text: Finite probability spaces, matrices, models, finite graphs, the list goes on.
Answers.UserDisplayName: postoronnim
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T23:58:31.017Z
Answers.Text: @postoronnim There are infinitely many of all of those things. There is only one chess. Just like there are infinitely many prime numbers (each of which is finite) but only one 1827368429. There are thousands of mathematicians who study prime numbers, and none who study 1827368429 in isolation.
Answers.UserDisplayName: David Richerby
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T07:38:19.723Z
Answers.Text: @YiFan I clarified it in the part of the sentence that you didn't quote... If it's horribly wrong, give me a list of examples of mathematicians studying a single finite object.
Answers.UserDisplayName: David Richerby
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T11:38:31.687Z
Answers.Text: Stanislav Ulam described himself as: “a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points”
Answers.UserDisplayName: Martin Bonner supports Monica
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T15:44:43.660Z
Answers.Text: @DavidRicherby `If chess is played rationally (and we can't hope to characterize it without that assumption) then games are finite in length`.  I think we can weaken that assumption based on the rule `Either player can declare a draw if 50 chess moves (50 white and 50 black) have been made and not a single pawn move or a piece has been captured.` If we modify that rule from "can declare" to "must declare" then players can otherwise be irrational and the game will still be finite.
Answers.UserDisplayName: emory
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T23:47:00.710Z
Answers.Text: Oh. In that case I am sure I misread your writing.
Answers.UserDisplayName: Mason
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T23:47:11.363Z
Answers.Text: It is common to place a finite object in a convenient generalization and study this. When game theorists make a claim about the computational complexity of a game like chess, go, checkers, etc. They are talking about the game played on an n by n board.
Answers.UserDisplayName: Mason
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T09:41:21.553Z
Answers.Text: @Mason Right. But those analyses of the asymptotic resource requirements of solving chess on an n-by-n board tell us nothing whatsoever about chess on an 8-by-8 board, which is what the question is about.
Answers.UserDisplayName: David Richerby
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T19:56:04.713Z
Answers.Text: You are right to challenge me to find the fundamental component of my critique: 
  
  Let me answer in perhaps a strange way. Mathematics can be framed like this (kinda/ only sometimes). We pick the assumptions that we want because it builds the type of structures that we want...
  
  If we assume that the OP is asking about "(A) Only chess in it's standard form" and "(B) mathematicians aren't interested in calculations." Then the answer that we write should be of the type you've written. 
  
  But taking those assumptions doesn't seem to make for a question I want to answer.
Answers.UserDisplayName: Mason
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T20:04:01.550Z
Answers.Text: I know this is maybe a strange critique but when I read your writing it seems to come across that you have taking some assumptions/restrictions that the OP didn't necessarily place into their question. So why are we not assuming that the OP asked a meritorious question that might have a "Yes! But the natural math move would be study a generalization..."- answer. As opposed to "No. Because you seem to be misunderstanding the type of thing math gets at..." answer. 
  
  One seems to be more encouraging.
Answers.UserDisplayName: Mason
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T21:04:26.867Z
Answers.Text: @Mason It's a reasonable critique. I don't think I really am adding restrictions but, rather, I'm answering the question that's been asked. You're suggesting generalizing the question to get what I agree is a more interesting answer, and that's also a perfectly reasonable approach.
Answers.UserDisplayName: David Richerby
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-12T21:10:45.360Z
Answers.Text: I appreciate you taking the time to read a critique. And I respect the desire to answer exactly what was asked rather than making too many unjustified guesses as to what they were getting at.
Answers.UserDisplayName: Mason
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-08-15T19:12:21.123Z
Answers.Text: Ok, chess is finite but the count of the possible states (10^120 or so) are far over any reasonable prediction (like: who has a winning algoithm). In all practical sense, chess is infinite. I think it is CS territory.
Answers.UserDisplayName: Gray Sheep
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-10-19T21:08:53.053Z
Answers.Text: I upvoted, as I like discussion, more than helpdesk interaction structure.  I also do not agree with the point offered as debate point.  I think given the amount of comment, it is likely to be informative for its discussion value.  I personally think that big finite is better approximated by infinite and some uncertainty ambient formalism to catch the eventual finiteness made discretization, in game odds given known player chess skill set.  but I think the "answer" is a well presented point of view that is actually tenable, if one thinks chess rules being small finite limits the ambient math.
Answers.UserDisplayName: dbdb
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T20:56:47.213Z
Answers.LastActivityDate: 2019-12-10T20:56:47.213Z
Answers.UserDisplayName: David Richerby
Answers.Body: <p>There are many different aspects of chess which can be formalized mathematically. Since the 19th century at least, chess has been mined as a resource to drive mathematical innovation. So when talking about a mathematical <strong>characterization</strong> of chess, it's not a single modeling that we are talking about, which grabs every feature, but rather a number of models, in which the power of each is paradoxically in the <strong>letting go</strong> of some aspect of chess which is not relevant for that analysis.</p>
  <ul>
  <li><p>Perhaps the earliest success (not counting the counting :-) of <strong>grains
  of rice</strong> on a chessboard) was <strong>Zermelo's Theorem</strong> (yes by one of the
  founders of ZF Set Theory) which states that in chess &quot;either White
  can force a win, or Black can force a win, or both sides can force at
  least a draw&quot;.</p>
  </li>
  <li><p><strong>Combinatorial Game Theory</strong> (developed by celebrated mathematicians
  John H. Conway, Elwyn Berlekamp and Richard Guy) has been
  successfully applied to chess. A couple of assumptions in this theory
  go against its general applicability to chess though. One is that a
  win in CGT is solely if your opponent can’t move which does not
  address stalemate. Secondly there is the notion of “entailing move”
  (e.g. check) where if one person plays, the other player must respond
  rather than the first player playing again. But Noam Elkies has
  derived some non-trivial results in chess endgames by computing their
  CGT value - references given in his chess page
  <a href="http://www.math.harvard.edu/%7Eelkies/chess.html" rel="nofollow noreferrer">http://www.math.harvard.edu/~elkies/chess.html</a>: &quot;On Numbers and
  Endgames&quot; and &quot;Higher Nimbers [sic] in Pawn Endgames on Large
  Chessboards&quot;.</p>
  </li>
  <li><p>You mentioned <strong>group theory</strong> - well curiously the set of combinatorial games under composition does form an abelian group! To be fair, the two obstacles mentioned above do prevent chess itself from behaving in this way.</p>
  </li>
  <li><p>There is a massive amount of combinatorial work which has been done around the chessboard. Vaclav Kotesovec has published online an 800 page book <em>just</em> on the subject of non-attacking chess pieces (<strong>generalizing the 8 queens problem</strong> massively). See <a href="http://www.kotesovec.cz/" rel="nofollow noreferrer">http://www.kotesovec.cz/</a> for the link. This is related to Magic Squares, Experimental Design, etc.</p>
  </li>
  <li><p>There is also, starting in Finland with Eero Bonsdorff, a long lineage in path enumeration chess problems, counting the number of ways that a position can be reached. This often involves the analysis of <strong>Standard Young Tableaux</strong>, which also underpin representation theory of symmetric groups. Fibonacci numbers, Catalan numbers and Euler numbers are all frequently found here, together with other combinatorial identities whose realization can be found lurking in ingenious chess compositions. See <a href="https://pdb.dieschwalbe.de/search.jsp" rel="nofollow noreferrer">https://pdb.dieschwalbe.de/search.jsp</a>, and type g='mathematics' in the search box.</p>
  </li>
  <li><p>The <strong>Knight's Tour</strong> problem is also famous for helping to push the study of Hamiltonian graphs, and particularly the challenge of counting the number of such objects. Nikolai Beluhov and Ed Pegg of Mathematica fame have contributed made some major recent theoretical improvements. See <a href="https://www.mayhematics.com/t/t.htm" rel="nofollow noreferrer">https://www.mayhematics.com/t/t.htm</a>.</p>
  </li>
  <li><p>The theory of <strong>computation</strong> also asks about the complexity of chess. Generalizations of chess to larger nxn boards are for example EXPTIME-complete: <em>A. S. Fraenkel and D. Lichtenstein, &quot;Computing a perfect strategy for nxn chess requires time exponential in n&quot;, Proc. 8th Int. Coll. Automata, Languages, and Programming, Springer LNCS 115 (1981) 278-293 and J. Comb. Th. A 31 (1981) 199-214.</em></p>
  </li>
  <li><p>Determination of <strong>legality &amp; dead position</strong> are both PSPACE-complete on nxn boards: <em>Josh Brunner, Erik D. Demaine, Dylan Hendrickson, Julian Wellman &quot;Even Cooperative Chess is Hard&quot;, arXiv:2010.09271v1 [cs.CC] 19 Oct 2020</em></p>
  </li>
  <li><p><strong>Full axiomatization of generalized chess</strong> I think there is also research value in a mathematical description of how chess conventions apply consistently to chess problems including fairy chess. Guus Rol has a very ambitious program which reduces each turn to “micro-phases” and claims to be able to determine with great accuracy how fairy conditions interact in complex “retro-active” situations, where both retro and forward logic is necessary. I don’t know if he will ever complete his theory.</p>
  </li>
  <li><p><strong>Limited axiomatization</strong>: I personally would like to see a more modest theory, which treats each move as atomic and although it doesn’t cover fairy chess so well, at least can cover retro-active aspects of problem conventions such as castling and e.p. Even that has not been done yet.</p>
  </li>
  <li><p>The mathematical physicist Roger Penrose published a chess position about 2 years ago which was intended to argue his long-held position that there is a fundamentally different kind of reasoning displayed by humans than an AI grounded in <strong>computable functions</strong> can demonstrate. See <a href="https://en.chessbase.com/post/a-chess-problem-holds-the-key-to-human-consciousness" rel="nofollow noreferrer">https://en.chessbase.com/post/a-chess-problem-holds-the-key-to-human-consciousness</a></p>
  </li>
  <li><p>Even though the theory of random graphs, and <strong>Monte Carlo analysis</strong> has been applied very successfully to engines, particularly the most recent generation, I don't think this disqualifies it from being considered a mathematical theory.</p>
  </li>
  <li><p>There is also a linear algebra approach to counting moves and positions. “If chess is a <strong>graph</strong>, what is its <strong>maximum eigenvalue</strong>?” is a very real and interesting question. See Francois Labelle's site at <a href="http://wismuth.com/chess/statistics-games.html" rel="nofollow noreferrer">http://wismuth.com/chess/statistics-games.html</a></p>
  </li>
  <li><p><strong>Linear algebra</strong> can be applied successfully to other chess-related questions e.g. how many ways are there for a rook to move from a1 to h8 in exactly n moves? How about a king?</p>
  </li>
  <li><p>For many years, the well-known mathematicians and chess problemists, <strong>Noam Elkies &amp; Richard Stanley</strong>, have been collaborating on a book on chess and mathematics. I don't know when it will finally emerge, or whether they have given up on this. But &quot;The Mathematical Knight&quot; given on Noam's chess page linked earlier perhaps gives a foretaste of that book. Richard Stanley is one of the leading lights in Combinatorics: see <a href="http://www-math.mit.edu/%7Erstan/chess/queue.pdf" rel="nofollow noreferrer">http://www-math.mit.edu/~rstan/chess/queue.pdf</a>, and Noam's birthday article <a href="https://arxiv.org/pdf/math/0508645.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/math/0508645.pdf</a></p>
  </li>
  <li><p>Chess can be played on boards with <strong>different geometries</strong>: cylinders, tori, mobius strips, even projective planes has been explored but much more can be done.</p>
  </li>
  <li><p>There is also perhaps a role for some kind of &quot;probabilistic mechanics&quot; or some such, to explain large-scale <strong>behaviours in tablebases</strong>. Why, for example, does there not appear to be an increase in the maximum length of winning lines when one scales from 7 pieces to 8 in the board?</p>
  </li>
  </ul>
  <p><strong>A few books</strong></p>
  <ul>
  <li><p>There is also already a book &quot;Mathematics and Chess&quot; by Miodrag S. Petkovic, reprinted by Dover, that if you can excuse the typos contains a raft of different math puzzles inspired by chess. Particularly I liked the geometrical ones, many of which were new to me.</p>
  </li>
  <li><p>A comment below mentioned &quot;Games &amp; Mathematics&quot; by David Wells, which gives chess as an example of an abstract game, and also devotes a chapter to how chess is not mathematics.</p>
  </li>
  <li><p>Another excellent Dover reprint is the late John Beasley's &quot;The Mathematics of Games&quot;, which chess examples throughout. He was an endgame theorist involved with early tablebases, but had a sense of whimsy which led to many other problems. Particularly Chapter 10 &quot;When the counting has to stop&quot; examines what makes chess <em>hard</em>, to play and to analyze with mathematics.</p>
  </li>
  </ul>
  <p><strong>Other links</strong></p>
  <p>Some of these ideas are listed in a Wikipedia page:
  <a href="https://en.wikipedia.org/wiki/Category:Mathematical_chess_problems" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Category:Mathematical_chess_problems</a>, and there's a more specific page <a href="https://en.wikipedia.org/wiki/Mathematical_chess_problem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Mathematical_chess_problem</a>.</p>
  <p>If anyone else has other examples of chess and mathematics, please mention them in comments and I will try to incorporate them in this response. If anyone has links that illustrate the topics I am mentioning, please add them, thanks.</p>
  <p>Finally: math.stackexchange.com has 1300 entries related to chess!?!</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T16:05:21.717Z
Answers.Text: I have been looking for these mathematical- and combinatorial-game-theory-analyses which really focus chess (and which not focus combinatorial-games in general) for several years. I have not find many results. Do you know some concrete books/papers for this topic?
Answers.UserDisplayName: anion
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-11T16:56:05.277Z
Answers.Text: Hi anion, the best are Noam Elkies' two endgame papers - I have added the link to his chess page which gives the specific details.
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T13:04:40.143Z
Answers.Text: the moore-penrose guy. nice.
Answers.UserDisplayName: BCLC
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T13:05:45.190Z
Answers.Text: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game
Answers.UserDisplayName: BCLC
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-20T15:20:34.040Z
Answers.Text: Thank you, BCLC. I'm not sure the term "mathematical games" is well-defined in the wikipedia page, which reads very much like an enthusiastic first draft. It was certainly the title of Martin Gardner's famous column in Scientific American, but he had a broader meaning than game theory. I wondered, is there any game where the rules are defined by "clear mathematical parameters" (whatever they are) but the strategy and outcomes are not? But even Wittgenstein used the word "game" as a poster-child for a word difficult to define...
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-07-20T02:49:20.230Z
Answers.Text: There is another nice book, Mathematics and Games, by David Wells.
Answers.UserDisplayName: Philip Roe
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-03-11T08:44:48.323Z
Answers.Text: @BCLC I mention that chess can be modeled with CGT, together with the two main limitations of this approach.
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-12-10T07:38:12.863Z
Answers.LastActivityDate: 2024-11-15T19:05:15.990Z
Answers.LastEditDate: 2024-11-15T19:05:15.990Z
Answers.UserDisplayName: Laska
Body: <p>From years of being student, researcher, and practitioner of chess, I feel like it possesses completeness and elegance that are found in mathematical proofs and theories. Thus, it seems only natural to imagine that it should be possible to describe chess in mathematical terms. By that I mean an elegant description, like a compact set of formulas or a proof that chess belongs to such and such field of mathematics. Intuitively, it feels like it should be able to fit in group theory. Does anyone know what attempts have been made in this regard? I would like to specifically exclude engines, neural networks, and the like from the discussion because, despite the obvious success in increasing the quality of play, those attempts do not constitute theoretical proof.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-09T20:22:37.847Z
Comments.Text: If we had anything close to a mathematical proof about chess, don't you think that the game would be solved? If there was a complete description about the game analogously to a complete description about a group, then we would be able to form much better engines than those based purely on heuristics.
Comments.UserDisplayName: NoseKnowsAll
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-09T20:44:06.613Z
Comments.Text: @ NoseKnowsAll I know that the game is not solved. My question is about the attempts to do that.
Comments.UserDisplayName: postoronnim
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-09T21:07:02.170Z
Comments.Text: @Rewan Demontay That's definitely something. However, I am curious about a complete solution, or at least attempts. I just have a hunch that chess, or elements thereof like pieces and positions, having symmetries like groups in group theory do, could potentially be classified as a set of those groups. Moreover, group theory being complete, meaning it incorporates any mathematical object, could uncork some previously unseen properties of chess.
Comments.UserDisplayName: postoronnim
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-09T21:54:01.520Z
Comments.Text: It looks more like graph theory, with (directed) arrows indicating which positions can be reached in one move from another position. I do not think there is any kind of relation between group theory and chess; they have quite different characteristics (actions in chess, for example, are in general not invertible).
Comments.UserDisplayName: Alexander Geldhof
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-10T12:51:55.800Z
Comments.Text: Has no one mentioned the [wikipedia page on this subject](https://en.wikipedia.org/wiki/Solving_chess) yet?
Comments.UserDisplayName: OliverRadini
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-10T15:13:03.360Z
Comments.Text: @NoseKnowsAll: "mathematical formulation" is far removed from "mathematical proof". There are lots of things mathematicians talk about in detail without having proofs.
Comments.UserDisplayName: Martin Argerami
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-10T16:41:12.313Z
Comments.Text: Have you seen the Rook Polynomial?
Comments.UserDisplayName: Owl
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-01-20T11:47:58.920Z
Comments.Text: You can describe chess in a mathematical way. It wouldn't really be too hard. The issue is that you won't be able to proof many theorems that have any practical value
Comments.UserDisplayName: David
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-01-20T13:05:34.113Z
Comments.Text: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game
Comments.UserDisplayName: BCLC
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2023-11-22T21:49:39.933Z
Comments.Text: https://mathoverflow.net/questions/287917/what-is-a-chess-piece-mathematically/453847#453847
Comments.UserDisplayName: dbdb
Tags: <theory><mathematics>
Title: Has anyone attempted to characterize chess mathematically?
ThreadText: # Has anyone attempted to characterize chess mathematically?
  
  (Asked by: username_0 on 2019-12-09)
  
  From years of being student, researcher, and practitioner of chess, I feel like it possesses completeness and elegance that are found in mathematical proofs and theories. Thus, it seems only natural to imagine that it should be possible to describe chess in mathematical terms. By that I mean an elegant description, like a compact set of formulas or a proof that chess belongs to such and such field of mathematics. Intuitively, it feels like it should be able to fit in group theory. Does anyone know what attempts have been made in this regard? I would like to specifically exclude engines, neural networks, and the like from the discussion because, despite the obvious success in increasing the quality of play, those attempts do not constitute theoretical proof.
  
  **Tags:** theory, mathematics
  
  > **Comments:**
  > **username_1**: If we had anything close to a mathematical proof about chess, don't you think that the game would be solved? If there was a complete description about the game analogously to a complete description about a group, then we would be able to form much better engines than those based purely on heuristics.  
  > **username_0**: @ username_1 I know that the game is not solved. My question is about the attempts to do that.  
  > **username_0**: @Rewan Demontay That's definitely something. However, I am curious about a complete solution, or at least attempts. I just have a hunch that chess, or elements thereof like pieces and positions, having symmetries like groups in group theory do, could potentially be classified as a set of those groups. Moreover, group theory being complete, meaning it incorporates any mathematical object, could uncork some previously unseen properties of chess.  
  > **username_2**: It looks more like graph theory, with (directed) arrows indicating which positions can be reached in one move from another position. I do not think there is any kind of relation between group theory and chess; they have quite different characteristics (actions in chess, for example, are in general not invertible).  
  > **username_3**: Has no one mentioned the [wikipedia page on this subject](https://en.wikipedia.org/wiki/Solving_chess) yet?  
  > **username_4**: @username_1: "mathematical formulation" is far removed from "mathematical proof". There are lots of things mathematicians talk about in detail without having proofs.  
  > **username_5**: Have you seen the Rook Polynomial?  
  > **username_6**: You can describe chess in a mathematical way. It wouldn't really be too hard. The issue is that you won't be able to proof many theorems that have any practical value  
  > **username_7**: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game  
  > **username_8**: https://mathoverflow.net/questions/287917/what-is-a-chess-piece-mathematically/453847#453847  
  
  ---
  
  ## 8 answers
  
  ### Answer by username_23 on 2019-12-10. Score: 41 | Accepted answer
  
  There are many different aspects of chess which can be formalized mathematically. Since the 19th century at least, chess has been mined as a resource to drive mathematical innovation. So when talking about a mathematical **characterization** of chess, it's not a single modeling that we are talking about, which grabs every feature, but rather a number of models, in which the power of each is paradoxically in the **letting go** of some aspect of chess which is not relevant for that analysis.
  
  * Perhaps the earliest success (not counting the counting :-) of **grains
    of rice** on a chessboard) was **Zermelo's Theorem** (yes by one of the
    founders of ZF Set Theory) which states that in chess "either White
    can force a win, or Black can force a win, or both sides can force at
    least a draw".
  * **Combinatorial Game Theory** (developed by celebrated mathematicians
    John H. Conway, Elwyn Berlekamp and Richard Guy) has been
    successfully applied to chess. A couple of assumptions in this theory
    go against its general applicability to chess though. One is that a
    win in CGT is solely if your opponent can’t move which does not
    address stalemate. Secondly there is the notion of “entailing move”
    (e.g. check) where if one person plays, the other player must respond
    rather than the first player playing again. But Noam Elkies has
    derived some non-trivial results in chess endgames by computing their
    CGT value - references given in his chess page
    [http://www.math.harvard.edu/~elkies/chess.html](http://www.math.harvard.edu/%7Eelkies/chess.html): "On Numbers and
    Endgames" and "Higher Nimbers [sic] in Pawn Endgames on Large
    Chessboards".
  * You mentioned **group theory** - well curiously the set of combinatorial games under composition does form an abelian group! To be fair, the two obstacles mentioned above do prevent chess itself from behaving in this way.
  * There is a massive amount of combinatorial work which has been done around the chessboard. Vaclav Kotesovec has published online an 800 page book *just* on the subject of non-attacking chess pieces (**generalizing the 8 queens problem** massively). See <http://www.kotesovec.cz/> for the link. This is related to Magic Squares, Experimental Design, etc.
  * There is also, starting in Finland with Eero Bonsdorff, a long lineage in path enumeration chess problems, counting the number of ways that a position can be reached. This often involves the analysis of **Standard Young Tableaux**, which also underpin representation theory of symmetric groups. Fibonacci numbers, Catalan numbers and Euler numbers are all frequently found here, together with other combinatorial identities whose realization can be found lurking in ingenious chess compositions. See <https://pdb.dieschwalbe.de/search.jsp>, and type g='mathematics' in the search box.
  * The **Knight's Tour** problem is also famous for helping to push the study of Hamiltonian graphs, and particularly the challenge of counting the number of such objects. Nikolai Beluhov and Ed Pegg of Mathematica fame have contributed made some major recent theoretical improvements. See <https://www.mayhematics.com/t/t.htm>.
  * The theory of **computation** also asks about the complexity of chess. Generalizations of chess to larger nxn boards are for example EXPTIME-complete: *A. S. Fraenkel and D. Lichtenstein, "Computing a perfect strategy for nxn chess requires time exponential in n", Proc. 8th Int. Coll. Automata, Languages, and Programming, Springer LNCS 115 (1981) 278-293 and J. Comb. Th. A 31 (1981) 199-214.*
  * Determination of **legality & dead position** are both PSPACE-complete on nxn boards: *Josh Brunner, Erik D. Demaine, Dylan Hendrickson, Julian Wellman "Even Cooperative Chess is Hard", arXiv:2010.09271v1 [cs.CC] 19 Oct 2020*
  * **Full axiomatization of generalized chess** I think there is also research value in a mathematical description of how chess conventions apply consistently to chess problems including fairy chess. Guus Rol has a very ambitious program which reduces each turn to “micro-phases” and claims to be able to determine with great accuracy how fairy conditions interact in complex “retro-active” situations, where both retro and forward logic is necessary. I don’t know if he will ever complete his theory.
  * **Limited axiomatization**: I personally would like to see a more modest theory, which treats each move as atomic and although it doesn’t cover fairy chess so well, at least can cover retro-active aspects of problem conventions such as castling and e.p. Even that has not been done yet.
  * The mathematical physicist Roger Penrose published a chess position about 2 years ago which was intended to argue his long-held position that there is a fundamentally different kind of reasoning displayed by humans than an AI grounded in **computable functions** can demonstrate. See <https://en.chessbase.com/post/a-chess-problem-holds-the-key-to-human-consciousness>
  * Even though the theory of random graphs, and **Monte Carlo analysis** has been applied very successfully to engines, particularly the most recent generation, I don't think this disqualifies it from being considered a mathematical theory.
  * There is also a linear algebra approach to counting moves and positions. “If chess is a **graph**, what is its **maximum eigenvalue**?” is a very real and interesting question. See Francois Labelle's site at <http://wismuth.com/chess/statistics-games.html>
  * **Linear algebra** can be applied successfully to other chess-related questions e.g. how many ways are there for a rook to move from a1 to h8 in exactly n moves? How about a king?
  * For many years, the well-known mathematicians and chess problemists, **Noam Elkies & Richard Stanley**, have been collaborating on a book on chess and mathematics. I don't know when it will finally emerge, or whether they have given up on this. But "The Mathematical Knight" given on Noam's chess page linked earlier perhaps gives a foretaste of that book. Richard Stanley is one of the leading lights in Combinatorics: see [http://www-math.mit.edu/~rstan/chess/queue.pdf](http://www-math.mit.edu/%7Erstan/chess/queue.pdf), and Noam's birthday article <https://arxiv.org/pdf/math/0508645.pdf>
  * Chess can be played on boards with **different geometries**: cylinders, tori, mobius strips, even projective planes has been explored but much more can be done.
  * There is also perhaps a role for some kind of "probabilistic mechanics" or some such, to explain large-scale **behaviours in tablebases**. Why, for example, does there not appear to be an increase in the maximum length of winning lines when one scales from 7 pieces to 8 in the board?
  
  **A few books**
  
  * There is also already a book "Mathematics and Chess" by Miodrag S. Petkovic, reprinted by Dover, that if you can excuse the typos contains a raft of different math puzzles inspired by chess. Particularly I liked the geometrical ones, many of which were new to me.
  * A comment below mentioned "Games & Mathematics" by username_6 Wells, which gives chess as an example of an abstract game, and also devotes a chapter to how chess is not mathematics.
  * Another excellent Dover reprint is the late John Beasley's "The Mathematics of Games", which chess examples throughout. He was an endgame theorist involved with early tablebases, but had a sense of whimsy which led to many other problems. Particularly Chapter 10 "When the counting has to stop" examines what makes chess *hard*, to play and to analyze with mathematics.
  
  **Other links**
  
  Some of these ideas are listed in a Wikipedia page:
  <https://en.wikipedia.org/wiki/Category:Mathematical_chess_problems>, and there's a more specific page <https://en.wikipedia.org/wiki/Mathematical_chess_problem>.
  
  If anyone else has other examples of chess and mathematics, please mention them in comments and I will try to incorporate them in this response. If anyone has links that illustrate the topics I am mentioning, please add them, thanks.
  
  Finally: math.stackexchange.com has 1300 entries related to chess!?!
  
  > **Comments:**
  > **username_24**: I have been looking for these mathematical- and combinatorial-game-theory-analyses which really focus chess (and which not focus combinatorial-games in general) for several years. I have not find many results. Do you know some concrete books/papers for this topic?  
  > **username_23**: Hi username_24, the best are Noam Elkies' two endgame papers - I have added the link to his chess page which gives the specific details.  
  > **username_7**: the moore-penrose guy. nice.  
  > **username_7**: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game  
  > **username_23**: Thank you, username_7. I'm not sure the term "mathematical games" is well-defined in the wikipedia page, which reads very much like an enthusiastic first draft. It was certainly the title of username_4 Gardner's famous column in Scientific American, but he had a broader meaning than game theory. I wondered, is there any game where the rules are defined by "clear mathematical parameters" (whatever they are) but the strategy and outcomes are not? But even Wittgenstein used the word "game" as a poster-child for a word difficult to define...  
  > **username_13**: There is another nice book, Mathematics and Games, by username_6 Wells.  
  > **username_23**: @username_7 I mention that chess can be modeled with CGT, together with the two main limitations of this approach.  
  
  ---
  
  ### Answer by username_9 on 2019-12-10. Score: 9
  
  Here are some starting spots in reading up on Game theory which is the mathematical tool that would be most appropriate for making claims about chess.
  
  [This](http://www.math.harvard.edu/~elkies/FS23j.03/zermelo.pdf) is a light history of early game theory. Chess is a "perfect information game" and there are some interesting things one can claim about this category of games. See [this](https://en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)) for example.
  I would also read up on Claude Shannon. [Shannon's number](https://en.wikipedia.org/wiki/Shannon_number) gives a bound on game complexity tree. For standard chess we have an intractable puzzle but you could make a smaller variants and solve these smaller games. Here's an example of [this](https://arxiv.org/pdf/1307.7118v1.pdf). The authors claim to have solved a smaller 5x5 variant of chess.
  
  > **Comments:**
  > **username_7**: has no one mentioned that chess is a mathematical game? https://en.wikipedia.org/wiki/Mathematical_game  
  
  ---
  
  ### Answer by username_10 on 2019-12-09. Score: 3
  
  Not really. At least not serious mathematicians.
  
  It is covered somewhat in Game Theory but other approaches like Group Theory do not seem to fit; although there might some higher dimensional non-linearish type description that could be found.
  
  So Algebra which includes Groups, and Topology too, seem to be out but perhaps it could fit in Analysis if somebody cared enough to try it. But that would an advanced mathematician who would be more likely to attempt a better problem that would lead to their PhD and a job teaching.
  
  > **Comments:**
  > **username_0**: Thanks for the insight. If such an attempt succeeds, it ought to lend a lot of recognition. Just think of how many people have struggled to solve it so far.  
  > **username_10**: @username_0 -  It would be worth recognition.  I am not sure how many qualified people have seriously tried to work on that problem.  
  > **username_11**: [Noam Elkies](http://www.math.harvard.edu/~elkies/chess.html) springs to mind.  
  
  ---
  
  ### Answer by username_12 on 2021-01-20. Score: 2
  
  I'd like to add some reflections to the answers. First, we probably have to distinguish how chess is played by a chessplayer, computer, or mathematician. The main point here is that these three may use different criteria when making moves. The mathematician tries to optimize his or her game in general while a chessplayer or neural network may use local criteria. In its turn, the chessplayer may wish to make the game interesting rather than just to win. Thus, analyzing the criteria, we may apply mathematics that includes not only the pure optimization but also symmetries, i.e. group theory. To my mind, mathematics is more about such kind of beauty and that's why it can be applied to chess in a non-trivial way.
  
  Now, I'll try to describe mathematically one of the "symmetrical" criteria which is well-known for a chessplayer (though I'm not a professional player). At the beginning of the game, white has 20 moves, and one could not use all of them according to his or her preferences. But it seems rather natural that the number of moves satisfying some strategy decreases in time. Thus, if two players built the decision tree they would find out that the outdegrees of vertices become smaller and smaller. Thus, the player may wish to *terminate* the game finding such a move that leads to a *path*, that is a part of the decision tree s.t. each its vertex is followed exactly by one vertex. This corresponds to the situation when two players both don't know the result and it's the game that shows it in a clear way. To generalize the task, one may wish to find the moves that lead to an abstract subgraph of the decision tree (say, with a well-known group of symmetries).
  
  P.S. Yet another way to think how group theory can be applied in chess is to note that the moves of **K**, **Q**, **R**, **B**, **N** form natural groups with at most 4 generators.
  
  
  ---
  
  ### Answer by username_13 on 2021-07-20. Score: 1
  
  As an applied mathematician, I feel that mathematics very seldom solves any problem in a practical sense, apart from those that reduce to some form of counting. There are many problems that lead to differential equations, for example, that present points of interest in their own right, but to apply them to any real engineering situation assumptions must be made that can never be completely reliable. An engineer must understand both their power and their limitations.
  
  The relationship that exists is that some aspect of a "real world" problem may have an interesting abstract structure and if this captures the interest of enough people it becomes part of mathematics. It goes in directions determined by the possibilities for rigorous thought. After a while it no longer matters if there is any visible connection to the origin.
  
  In the case of chess, there are a very few abstract aspects with practical value, mostly to do with Pawn endgames. Rigorous statements like "either somebody should win or it's a draw" are too general to be useful and are not really interesting either. Useful statements would apply to specific positions and there are too great a diversity of these for any "mathematical" statement to apply to all of them. A strong player sees the exceptions to the rule
  
  
  ---
  
  ### Answer by username_8 on 2022-12-31. Score: 1
  
  Anyone looked at the mathematics behind alpha-zero describing chess completely even without having a engine. One might see the engine implementations as measure instruments for such mathematical model of chess.
  
  It does consider a big space that is bound to contain the solutions, which is far from from assuming impractical complete tree knowledge as in most game theory about chess (that i am aware of).
  
  It handles uncertainty, in the approximation process. Allowing access to the solution via approximation (at least in theory). We just have to be confortable with asymptotic behavior perhaps.
  
  It also use probability spaces in order to be compatible with the type of data coming in from its measurements (RL). I would say, as long as you included the data relationship with the model, you might have something of a mathematical characterization, be it of statistical nature.
  
  
  ---
  
  ### Answer by username_14 on 2019-12-10. Score: 0
  
  I'd hate to say it, but mathematically speaking chess as we play it is very boring. It is a perfect information game, without any non-determinism, for two alternating players. This means that chess is either a win for white, a win for black, or a draw. The optimal strategy for chess is trivial and known: minimax. As far as maths is concerned, chess is solved.
  
  > **Comments:**
  > **username_15**: @username_17 - because actually _executing_ the optimal strategy is computationally infeasible?  (due to combinatorial explosion)  
  > **username_15**: @emery - perhaps you're confusing the way mathematics is done with the way other work is done? Consider: An engineer, a physicist, and a mathematician are working, when a small fire breaks out in front of their offices. The engineer panics and grabs the fire extinguisher, spraying it everywhere, putting out the fire, but causing extra damage in the process. The physicist runs some quick calculations, and uses just enough to put out the fire. The mathematician sees the fire, looks over to the fire extinguisher and says “a solution exists!” then returns to his office.  
  > **username_16**: @username_17 You don't understand. Thanks to minimax we don't only know *what to do* (that is to score more points in basketball or cause checkmate of the opponent in chess), but we also know *how exactly to do it*. We simply know that if before every possible chess move we play all the possible combinations of how game could develop and then choose the best one, we are performing the **best possible** strategy. That is minimax algorithm in nutshell. And that is very different to your basketball example.  
  > **username_17**: @username_16, but if you don't care about such things then yes, chess is solved.  
  > **username_16**: @username_17 *"If I recall correctly such a computer would be many times bigger than our solar system and would take many million human lifetimes to run. Is it not the case that such a machine would be so massive that it would form a black hole and the results of its computation could not escape its event horizon."*  I wouldn't be so sure about that. Anyway.. construction of computers is rather physical discipline. We might not have *efficient* best possible chess algorithm, but we have at least some. So, as mentioned above, mathematically, chess is solved.  
  > **username_14**: @username_17 No, minimax doesn't require a lot of memory at all. Your mobile phone has the computational power to optimally solve chess - it just requires a long, long, long time. And even if you'd need a huge computer until the heat death of the universe, that's no concern of mathematics.  
  > **username_18**: Also, given alpha beta pruning, with good move heuristics and a transposition table, you only need to look at about 10^30 ish different positions. That's only about 2^100, which while intractable, isn't that intractable. It's roughly on the order of breaking 256 bit encryption.  
  > **username_7**: LOLwut? as far as maths is concerned, chess CAN be solved. so could checkers. but this didn't stop people from solving checkers  
  > **username_19**: Combinatorial games are also perfect information games, and there is not even a draw. That doesn't mean they are completely solved (e.g.: chomp). Very disappointing answer.  
  > **username_14**: @username_19 There doesn't exist a player that can defeat minimax for chomp. You're using a different definition for 'solved' than this answer is. The optimal strategy for chomp is known: minimax.  
  > **username_19**: @username_14 Minimax is an algorithm, not a strategy. If minimax counts as solving, then in the same way we can say that every problem can be solved (except halting problem, etc.) by exhaustive search. How about that?  
  > **username_14**: @username_19 Then please give a definition of a 'strategy'.  _"then in the same way we can say that every problem can be solved (except halting problem, etc.) by exhaustive search"_ Yes? There's nothing wrong with that.  
  
  ---
  
  ### Answer by username_20 on 2019-12-10. Score: -3
  
  Chess isn't the sort of object that mathematicians are interested in studying. Mathematicians care about problems that are in some way infinite: either infinite objects or finite objects of which there are infinitely many examples. Chess is completely finite. There is only one chess, it uses a finite board, there are finitely many possible positions. If chess is played rationally (and we can't hope to characterize it without that assumption) then games are finite in length. "Rational" basically means that either player will claim a draw as soon as they have the right to, and that means that games are less than about 5900 moves, which means there are only finitely many games. A mathematician might study the prime numbers, but they're not going to study the single number 1827368429.
  
  To a mathematician, "solve chess" is just a calculation, and mathematicians aren't interested in calculations. The fact that it's a culturally important calculation might encourage somebody to try, but it's hard to see anybody having the time to do something so complicated. It's the sort of thing that would take decades of full-time work and, if you failed, you'd have basically nothing.
  
  > **Comments:**
  > **username_0**: Big math does not have an issue with finite things, there are plenty of examples if you look around. The question is not about calculation or computation, it's about giving chess a theoretical footing in the math realm.  
  > **username_20**: @username_0 Examples such as what? And I've given you the theoretical footing: it's a single finite object.  
  > **username_0**: Finite probability spaces, matrices, models, finite graphs, the list goes on.  
  > **username_20**: @username_0 There are infinitely many of all of those things. There is only one chess. Just like there are infinitely many prime numbers (each of which is finite) but only one 1827368429. There are thousands of mathematicians who study prime numbers, and none who study 1827368429 in isolation.  
  > **username_20**: @YiFan I clarified it in the part of the sentence that you didn't quote... If it's horribly wrong, give me a list of examples of mathematicians studying a single finite object.  
  > **username_21**: Stanislav Ulam described himself as: “a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points”  
  > **username_17**: @username_20 `If chess is played rationally (and we can't hope to characterize it without that assumption) then games are finite in length`.  I think we can weaken that assumption based on the rule `Either player can declare a draw if 50 chess moves (50 white and 50 black) have been made and not a single pawn move or a piece has been captured.` If we modify that rule from "can declare" to "must declare" then players can otherwise be irrational and the game will still be finite.  
  > **username_9**: Oh. In that case I am sure I misread your writing.  
  > **username_9**: It is common to place a finite object in a convenient generalization and study this. When game theorists make a claim about the computational complexity of a game like chess, go, checkers, etc. They are talking about the game played on an n by n board.  
  > **username_20**: @username_9 Right. But those analyses of the asymptotic resource requirements of solving chess on an n-by-n board tell us nothing whatsoever about chess on an 8-by-8 board, which is what the question is about.  
  > **username_9**: You are right to challenge me to find the fundamental component of my critique:   Let me answer in perhaps a strange way. Mathematics can be framed like this (kinda/ only sometimes). We pick the assumptions that we want because it builds the type of structures that we want...  If we assume that the OP is asking about "(A) Only chess in it's standard form" and "(B) mathematicians aren't interested in calculations." Then the answer that we write should be of the type you've written.   But taking those assumptions doesn't seem to make for a question I want to answer.  
  > **username_9**: I know this is maybe a strange critique but when I read your writing it seems to come across that you have taking some assumptions/restrictions that the OP didn't necessarily place into their question. So why are we not assuming that the OP asked a meritorious question that might have a "Yes! But the natural math move would be study a generalization..."- answer. As opposed to "No. Because you seem to be misunderstanding the type of thing math gets at..." answer.   One seems to be more encouraging.  
  > **username_20**: @username_9 It's a reasonable critique. I don't think I really am adding restrictions but, rather, I'm answering the question that's been asked. You're suggesting generalizing the question to get what I agree is a more interesting answer, and that's also a perfectly reasonable approach.  
  > **username_9**: I appreciate you taking the time to read a critique. And I respect the desire to answer exactly what was asked rather than making too many unjustified guesses as to what they were getting at.  
  > **username_22**: Ok, chess is finite but the count of the possible states (10^120 or so) are far over any reasonable prediction (like: who has a winning algoithm). In all practical sense, chess is infinite. I think it is CS territory.  
  > **username_8**: I upvoted, as I like discussion, more than helpdesk interaction structure.  I also do not agree with the point offered as debate point.  I think given the amount of comment, it is likely to be informative for its discussion value.  I personally think that big finite is better approximated by infinite and some uncertainty ambient formalism to catch the eventual finiteness made discretization, in game odds given known player chess skill set.  but I think the "answer" is a well presented point of view that is actually tenable, if one thinks chess rules being small finite limits the ambient math.  
  
  ---
  
  