Mathematics and Chess Page

Chess teaches many things, including strategic thinking. Though one might think at first that this type of thinking is unrelated to mathematics, in fact, chess also teaches a type of "calculation" (see Soltis's book [4] for the exact idea). To anyone who thinks doing mathematics and playing chess are unrelated, this page is for you.

A paraphrase from the entry under Mathematics and Chess in [5]: In 1893, a Professor Binet (of Stanford-Binet IQ test fame) made a study of the connection between mathematics and chess. After questioning a large number of leading plyers, he discovered that 90% were very good mental calculators. On the other hand, he discovered that although mathematicians are often interested in chess, few become top-class players.... Professor Binet commented that both chess and mathematics have a common direction and the same taste for combinations, abstraction, and precision. One characteristic which was missing from mathematics was the combat, in which two individuals contend for mastery, with all the qualities required of generals in the field of battle.

This page contains information on

• which mathematicians (which we define as someone who has earned a PhD or equivalent in Mathematics) play(ed) chess at the International Master level or above (also included are those who have an IM or above in chess problem solving or composing), and
• how to get papers on mathematical chess problems.

Mathematicians who play, or have played, chess:

• Adolf Anderssen (1818-1879). Pre World Championships but is regarded as the strongest player in the world between 1859 and 1866. He received a degree (probably not a PhD) in mathematics from Breslau University and taught mathematics at the Friedrichs gymnasium from 1847 to 1879. He was promoted to Professor in 1865 and was given an honorary doctorate by Breslau (for his accomplishments in chess) in 1865.
• Robert Coveyou (1915 - 1996) completed an M.S. degree in Mathematics, and joined the Oak Ridge National Laboratory as a research mathematician. He became a recognized expert in pseudo-random number generators. He is known for the quotation "The generation of random numbers is too important to be left to chance," which is based on a title of a paper he wrote. An excellent tournament chess player, he was Tennessee State Champion eight times.
• Noam Elkies (1966-), a Professor of Mathematics at Harvard University specializing in number theory, is a study composer and problem solver (ex-world champion). Prof. Elkies, at age 26, became the youngest scholar ever to have attained a tenured professorship at Harvard. One of his endgame studies is mentioned, for example, in the book Technique for the tournament player, by GM Yusupov and IM Dvoretsky, Henry Holt, 1995. He wrote 11 very interesting columns on Endgame Exporations (posted by permission).

  Some other retrograde chess consructions of his may be found at the Dead Reckoning web site of Andrew Buchanan.

  See also Professor Elkies's very interesting Chess and Mathematics Seminar 2003, 2004, pages and the mathematical papers on his chess page.

• Machgielis (Max) Euwe (1901-1981), World Chess Champion from 1935-1937, President of FIDE (Fédération Internationale des Echecs) from 1970 to 1978, and arbitrator over the turbulent Fischer - Spassky World Championship match in Reykjavik, Iceland in 1972. I don't know as many details of his mathematical career as I'd like. One source gives: PhD (or actually its Dutch equivalent) in Mathematics from Amsterdam University in 1926. Another gives: Doctorate in philosophy in 1923 and taught as a career. Published a paper on the mathematics of chess "Mengentheoretische Betrachtungen uber das Schachspiel".
• Ed Formanek (194?-), International Master. Ph.D. Rice University 1970. Currently on the mathematics faculty at Penn State Univ. Works primarily in matrix theory and representation theory.
• Charles Kalme (Nov 15, 1939-March 22, 2002), earned his master title in chess at 15, was US Junior champ in 1954, 1955, US Intercollegiate champ in 1957, and drew in his game against Bobby Fischer in the 1960 US championship. In 1960, he also was selected to be on the First Team All-Ivy Men's Soccer team, as well as the US Student Olympiad chess team. (Incidently, it is reported that this team, which included William Lombary on board one, did so well against the Soviets in their match that Boris Spassky, board one on the Soviet team, was denied forieng travel for two years as punishment.) In 1961 graduated 1st in his class at the Moore School of Electrical Engineering, The University of Pennsylvania, in Philadelphia. He also received the Cane award (a leadership award) that year. After getting his PhD from NYU (advisor Lipman Bers) in 1967 he to UC Berkeley for 2 years then to USC for 4-5 years. He published 2 papers in mathematics in this period, "A note on the connectivity of components of Kleinian groups", Trans. Amer. Math. Soc. 137 1969 301--307, and "Remarks on a paper by Lipman Bers", Ann. of Math. (2) 91 1970 601--606. He also translated Siegel and Moser, Lectures on celestial mechanics, Springer-Verlag, New York, 1971, from the German original. He was important in the early stages of computer chess programming. In fact, his picture and annotations of a game were featured in the article "An advice-taking chess computer" which appeared in the June 1973 issue of Scientific American. He was an associate editor at Math Reviews from 1975-1977 and then worked in the computer industry. Later in his life he worked on trying to bring computers to elementary schools in his native Latvia A National Strategy for Bringing Computer Literacy to Latvian Schools . His highest rating was acheived later in his life during a "chess comeback": 2458.

  Here is his game against Bobby Fischer referred to above:

  [Event "?"]
  [Site "New York ch-US"]
  [Date "1960.??.??"]
  [Round "3"]
  [White "Fischer, Robert J"]
  [Black "Kalme, Charles"]
  [Result "1/2-1/2"]
  [NIC ""]
  [Eco "C92"]
  [Opening "Ruy Lopez, Closed, Ragozin-Petrosian (Keres) Variation"]
  
  1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.O-O Be7 6.Re1 b5 7.Bb3 O-O 
  8.c3 d6 9.h3 Nd7 10.a4 Nc5 11.Bd5 Bb7 12.axb5 axb5 13.Rxa8 Qxa8 
  14.d4 Nd7 15.Na3 b4 16.Nc4 exd4 17.cxd4 Nf6 18.Bg5 Qd8 19.Qa4 Qa8 
  20.Qxa8 Rxa8 21.Bxf6 Bxf6 22.e5 dxe5 23.Ncxe5 Nxe5 24.Bxb7 Nd3 
  25.Bxa8 Nxe1 26.Be4 b3 27.Nd2 1/2-1/2
  

• Emanuel Lasker (1868-1941), World Chess Champion from 1894-1921, PhD (or actually its German equivalent) in Mathematics from Erlangen Univ in 1902. Author of the influential paper [1], where the well-known Lasker-Noether Primary Ideal Decomposition Theorem in Commutative Algebra was proven . (See [2] for a statement in the modern terminology. For more information, search "Lasker, Emanuel" in the chess encyclopedia, as well as the links provided there.)
• Lev Loshinski (1913-1976) , F.I.D.E. International Grandmaster of Chess Compositions. Taught mathematics (at Moscow State University?). (PhD unknown but considering the reputation of Moscow State University, he may have one.)
• A. Jonathan Mestel, grandmaster in over-the-board play and in chess problem solving, is an applied mathematician specializing in fluid mechanics and is the author of numerous research papers. He is on the mathematics faculty of the Imperial College in London.
• Walter D. Morris (196?-), International Master. Currently on the mathematics faculty at George Mason Univ in Virginia.
• Nick J. Patterson, International Master (?), D. Phil. (from Cambridge Univ.) in 197? in group theory, under Prof. Thompson. Has published several papers in group theory, combinatorics, and the theory of error-correcting codes. For some of his chess games, click here. He was the Irish Chess Champion in 1969.
• John Nunn (1955-), Chess Grandmaster, D. Phil. (from Oxford Univ.) in 1978 at the age of 23 (and the youngest undergraduate at Oxford since Cardinal Wolsey). PhD thesis in Algebraic Topology and author of the paper [3] (Search "Nunn" in the chess encyclopedia for more chess information.)
• Martin Kreuzer (1962-), CC Grandmaster, is rated over 2600 in correspondence chess (ICCF, as of Jan 2000). His OTB rating is over 2300 according to the chessbase encyclopedia. His specialty is computational commutative algebra and applications. Here is a recent game of his:
  Kreuzer, M - Stickler, A
  [Eco "B42"]
  1.e4 c5 2.Nf3 e6 3.d4 cxd4 4.Nxd4 a6 5.Bd3 Nc6 6.c3 Nge7 7.0-0 Ng6 8.Be3 Qc7 9.Nxc6 bxc6 10.f4 Be7 11.Qe2 0-0 12.Nd2 d5 13.g3 c5 14.Nf3 Bb7 15.exd5 exd5 16.Rae1 Rfe8 17.f5 Nf8 18.Qf2 Nd7 19.g4 f6 20.g5 fxg5 21.Nxg5 Bf6 22.Bf4 Qc6 23.Re6 Rxe6 24.fxe6 Bxg5 25.Bxg5 d4 26.Qf7+ Kh8 27.Rf3 Qd5 28.exd7 Qxg5+ 29.Rg3 Qe5 30.d8=Q+ Rxd8 31.Qxb7 Rf8 32.Qe4 Qh5 33.Qe2 Qh6 34.cxd4 cxd4 35.Bxa6 Qc1+ 36.Kg2 Qc6+ 37.Rf3 Re8 38.Qf1 Re3 39.Be2 h6 40.Kf2 Re8 41.Bd3 Qd6 42.Kg1 Kg8 43.a3 Qe7 44.b4 Ra8 45.Qc1 Qd7 46.Qf4 1-0
• Chess problem composer Hans-Peter Rehm (1942-), a Professor of Mathematics at Karlsruhe Univ. He has written several papers in mathematics, such as "Prime factorization of integral Cayley octaves", Ann. Fac. Sci. Toulouse Math (1993), but most in differential algebra, his specialty. Some of his problems can be found on the internet, for example: problem set (in German). A collection of his problems has been published as: Hans+Peter+Rehm=Schach Ausgewählte Schachkompositionen & Aufsätze (= selected chess problems and articles), Aachen 1994.

Some other possible entries for the above list:

• Alexander, Conel Hugh O'Donel (1909-1974), late British chess champion. Alexander may not have been a mathematician but he did mathematical (code and cryptography) work during WWII, as did the famous Soviet chess player David Bronstein (see the book Kahn, Kahn on codes). He was the strongest English player after WWII, until Jonathan Penrose appeared (see below for more on Penrose.) (Search "Alexander" in the chess encyclopedia for more information.)
• Christoph Bandelow teaches mathematics at the Ruhr-University Bochum. He specializes in stochastic processes and has written a number of excellent books on the magic cube (or "Rubik's cube") and related puzzles. Some of his chess problems are (by permission) : problem 1, problem 2, problem 3. (More to come.) Prof Bandelow was also a pioneer in computer problem solving, having written (in 1961) the first German computer program to solve chess problems (this program is described in "Schach und Zahl").
• Magdy Amin Assem (195?-1996) specialized in p-adic representation theory and harmonic analysis on p-adic reductive groups. He published several important papers before a ruptured aneurysm tragically took his life. He was IM strength (rated 2379) in 1996.
• Prof. Vania Mascioni, also IECG Chairperson (IECG is the Internet Email Chess Group), is rated 2326 by IECG (as of 4-99). He is a professor of Mathematics at the University of Texas at Austin (his area is Functional Analysis and Operator Theory).
• Stanislaw Ulam, the famous mathematician and physicist (author of the autobiographical, Adventures of a mathemacian) was a strong chess player. Rating unknown.
• Kenneth S. Rogoff, Professor of Economics at Harvard University, is a Grandmaster. He has a PhD in Economics but has published in statistical journals.
• Kenneth W. Regan, Professor of Computer Science at the State Univ. of New York Buffalo, is currently rated 2453. His research is in computational complexity, a field of computer science which has a significant mathematical component.
• Otto Blathy, who is a very famous many mover problemist, held a doctorate in mathematics from Budapest and Vienna universities at his time. (For a reference, see A.Soltis: Chess to Enjoy. pp.30-34.)
• Canadian grandmaster Duncan Suttles (b.1945 in San Francisco, moved to Vancouver as a child). Suttles studied for though did not (yet anyway) receive a PhD in mathematics. Suttles also has the grandmaster title in correspondence chess.
• Problem composer J. G. Mauldon (deceased, formerly a mathematician at Amherst College) has written several papers in mathematics. One of his retro problems can be found on the internet, for example: problem.
• Problem composer John D. Beasley has also written several books on the mathematics of games. He is secretary of the British Chess Variant Society.

There is some misleading information given either in the literature or on some internet web pages.

• Karl Fabel (1905-1975), F.I.D.E. International Master of Chess Compositions. Not a tournament player but an ingenious problem composer. He received a Doctorate in Chemistry and reportedly worked as a mathematician, civil judge, and patents expert. He was, according to his friend Christoph Bandelow, a chemist not a mathematician. Some Fabel problems: problem 1, problem 2. He was also the co-author of the book Schach und Zahl on mathematics and chess and the problem book Rund um das Schachbrett. Publisher: Walter de Gruyter 1955.
• Rueben Fine was not a mathematician (however, his son Ben is an active research mathematician who teaches at Fairfield University in Connecticut). Reuben Fine was a psychologist.
• GM James Tarjan (a Los Angeles librarian, I'm told) is the brother of the well-known computer scientist (some of his research has been published in mathematical journals) Robert Tarjan.
• World chess champion Kasparov is not a mathematician (as far as I know), though he has made contributions to computer science. (There is a well-know mathematician named Kasparov who works in K-theory and C^*-algebras but they are different people.)
• Jonathan Penrose (mentioned above - one of the strongest chess players in Britain in the 1950's and 1960's) is the brother of the well-known mathematician and physicist Sir Roger Penrose.

1. Eero Bonsdorff, Dr Karl Fabel, Olavi Riihimaa, Schach und Zahl, unterhaltsame schachmathematik, Walter Rau Verlag, Dusseldorf, 1966
2. Lasker, E. "Zur theorie der moduln und ideale," Math. Ann. 60(1905)20-116
3. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser, Boston, 1985
4. Nunn, J. D. M. "The homotopy types of finite H-spaces," Topology 18 (1979), no. 1, 17--28
5. A. Soltis, The Inner Game of Chess, David McKay Co. Inc, (Random House), New York, 1994
6. A. Sunnucks, The Encyclopedia of Chess, 2nd ed, St Martins Press, New York, 1976

Papers about mathematical problems in chess:

I only know of a few sources:

• Timothy Chow, "A Short Proof of the Rook Reciprocity Theorem", in volume 3, 1996, of the Electronic Journal of Combinatorics.
• Noam Elkies, "On numbers and endgames: Combinatorial game theory in chess endgames", in 1996 "Games of No Chance" = Proceedings of the workshop on combinatorial games held July'94 at MSRI. Available from MSRI Publications -- Volume 29 or Noam Elkies' site.
• Noam Elkies and Richard Stanley, "Chess and Mathematics".
• Max Euwe, "Mengentheoretische Betrachtungen uber das Schachspiel", Konin. Akad. Weten. (Proc Acad Sciences, Netherlands), vol 32, 1929, 633-642
• Awani Kumar, "Knight's Tours in 3 Dimensions", in The Games and Puzzles Journal The On-line Journal for Mathematical Recreations, Issue 43, January-April 2006.
• Richard M. Low and Mark Stamp, "King and Rook Vs. King on a Quarter-Infinite Board", in Integers, volume 6(2006).
• Igor Rivin, Ilan Vardi, Paul Zimmermann, "The N-queens problem," American Mathematical Monthly 101 (1994), no. 7, 629-639.
• Lewis Benjamin Stiller, "Exploiting symmetries on parallel architecture", PhD thesis, CS Dept, Johns Hopkins Univ. 1995 Closely related is his Games of No Chance paper, "Multilinear Algebra and Chess Endgames".
• Mario VELUCCHI's NON-Dominating Queens Problem or math chess problems
• Wikipedia's, Eight queens puzzle.
• Herbert S. Wilf, "The Problem of the Kings", and Michael Larsen, "The Problem of Kings", both in volume 2, 1995, of the Electronic Journal of Combinatorics.
• Papers on odd king tours by D. Joyner and M. Fourte (appeared in the J. of Rec. Math., 2003) and even king tours by M. Kidwell and C. Bailey (in Mathematics Magazine, vol 58, 1985).
• Lesson 3 in the chess lessons by Coach Epshteyn at UMBC.
• Wikipedia has an artcle on mathematicians who studied chess.

Created 10-28-97. Thanks to Christoph Bandelow, Max Burkett, Elaine Griffith, Hannu Lehto, John Kalme, Ewart Shaw, Richard Stanley, Will Traves, Steven Dowd, Z. Kornin, and Noam Elkies for help and corrections on this page.

http://www.permutationpuzzles.org/chess/math_chess.html

Last updated 2012-11-12. Any comments or additions to suggest? Please email me at: