The Math of Chess

"Excelsior" by Sam Loyd

White to move, mate in 5. (Sam Loyd, 1861)

It probably comes as no surprise to anyone that knows me that I enjoy playing chess. Indeed, chess is a common hobby among mathematicians (as are Go, juggling, and magic). But I was still amazed to see that a book I was reading, Chess Endings for the Practical Player by Ludek Pachman, was translated by someone named Hardy, and edited by someone named John E. Littlewood. I thought, “No way.” Hardy and Littlewood do chess? The same two that gave us this?

Well, I was right. The Hardy is Otto Hardy, not G.H. Hardy, and the Littlewood is John Eric Littlewood, not John Edensor Littlewood. But wow, close huh?

Had I been wrong — had the Hardy and Littlewood actually worked together on the book — it still shouldn’t have been shocking. It is fairly common to find mathematicians that study and/or play chess. Former chess world champion Max Euwe had his Ph.D. in mathematics, as do GMs John Nunn and Jon Speelman. (Nunn is also a GM chess problem solver – one of only three in history.) Noam Elkies (the youngest full tenured professor in the history of Harvard – at 26!) has published several papers on the mathematics of chess, and earned his solving GM title in 2001.

Math and chess have long been associated with one another. Some famous problems include the Knight’s Tour (any early version of which appeared in the Sanskrit poem Kâvyâlankâra by Rudrata ca. 900):

An open knight's tour.

the Eight Queens Problem, and the Mutilated Chessboard Problem (posed by Martin Gardner). There’s even a book called Mathematics and Chess.

Not all mathematical study of chess is purely recreational. The so-called rook polynomials have found applications in matrix theory and number theory. There’s even a Rook Reciprocity Theorem!

Check out Wikipedia’s Chess and Mathematics category for even more connections between the two, and also see Mathworld’s pages related to chess and math. Finally, no discussion of chess would be complete without a reference to xkcd’s comic that inspired these amazing photos.


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10 Responses to “The Math of Chess”

  1. Jonathan Says:

    There’s a Martin Gardner math puzzle book giveaway at Cambridge UP’s blog if you haven’t checked it out yet.

    Look for the latest weekly contest in the Categories bar. We’re on week #2 right now.



  2. YourTutorOnline Says:

    An interesting variation of the Knights Tour problem is to do it for a non-standard chessboard (say 25 x 15 squares). It was one of my first assignments in my Computer Science undergrad. I remember that assignment giving me quite a headache.

  3. Any mathematicians on the forum? - Chess Forums | Free Online Chess Game & Forum Says:

    […] (Harvard Uni. Freshman Seminar) Other Sites: Mathematics and chess – The Math of Chess: The Math of Chess 360 Mathematics and Chess: An article- MATHEMATICS AND THE GAMES OF CHESS “Mathematics and […]

  4. pecaturjogja Says:

    Yes..chess is the touchstone of the human intellect 😀

  5. Nate Says:

    Are there specific math equations that would make a person more likely to win a chess game? I don’t see any addition, subtration, division, multiplication, statistics or geometry that create advantage. Only the best move and other worse ones. Maybe I’m missing something obvious but I don’t think that chess uses math, sure correlations can be made but nothing that drives the game.

    • Sakura Kinomoto Says:

      Yeah. As far as I know, the mental operations used in chess are different from those used in math. And, mathematicians generally use theorems to actually prove something with some “facet” of the natural world while true theorems don’t truly do anything except allow one to make the best move in the hopes of getting closer to winning.

  6. terminator Says:

    there is no mathematical law that can solve a chess

  7. hoc Says:

    yes there is

  8. rtt199001 Says:

    hey! guys, i’m working on 2D coordination system transformation on chess board to solve some basic chess endings!!! such as Square rule applications

  9. Albanius Says:

    Jonathan Penrose the chess player is the brother of Roger Penrose the mathematical physicist. Are the chess playing Littlewoods related to the mathematician?

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