Combinatorial Games: Selected Bibliography with a Succinct Gourmet Introduction

Aviezri S. Fraenkel

Department of Applied Mathematics and Computer Science Weizmann Institute of Science Rehovot 76100, Israel

               fraenkel@wisdom.weizmann.ac.il
         http://www.wisdom.weizmann.ac.il/∼fraenkel

1 What are Combinatorial Games?

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is go. The shorter terminology game, games is used below to designate combinatorial games.

2 Why are Games Intriguing and Tempting?

Amusing oneself with games may sound like a frivolous occupation. But the fact is that the bulk of interesting and natural mathematical problems that are hardest in complexity classes beyond NP, such as Pspace, Exptime and Expspace, are two-player games; occasionally even one-player games (puzzles) or even zero-player games (Conway’s “Life”). Some of the reasons for the high complexity of two-player games are outlined in the next section. Before that we note that in addition to a natural appeal of the subject, there are applications or connections to various areas, including complexity, logic, graph and matroid theory, networks, error-correcting codes, surreal numbers, on-line algorithms, biology — and analysis and design of mathematical and commercial games!

But when the chips are down, it is this “natural appeal” that lures both amateurs and professionals to become addicted to the subject. What is the essence of this appeal? Perhaps the urge to play games is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers. A common expression of these vile cravings is found in the passions roused by local, national and international tournaments. An intellectually refined version of these dark desires, well hidden beneath the façade of scientific research, is the consuming drive “to beat them all”, to be more clever than the most clever, in short — to create the tools to Mathter them all in hot combinatorial combat! Reaching this goal is particularly satisfying and sweet in the context of combinatorial games, in view of their inherent high complexity.

With a slant towards artificial intelligence, Pearl wrote that games “offer a perfect laboratory for studying complex problem-solving methodologies. With a few parsimonious rules, one can create complex situations that require no less insight, creativity, and expertise than problems actually encountered in areas such as business, government, scientific, legal, and others. Moreover, unlike these applied areas, games offer an arena in which computerized decisions can be evaluated by absolute standards of performance and in which proven human experts are both available and willing to work towards the goal of seeing their expertise emulated by a machine. Last, but not least, games possess addictive entertaining qualities of a very general appeal. That helps maintain a steady influx of research talents into the field and renders games a convenient media for communicating powerful ideas about general methods of strategic planning.”

To further explore the nature of games, we consider, informally, two subclasses.

(i) Games People Play (playgames): games that are challenging to the point that people will purchase them and play them.

(ii) Games Mathematicians Play (mathgames): games that are challenging to mathematicians or other scientists to play with and ponder about, but not necessarily to “the man in the street”.

Examples of playgames are chess, go, hex, reversi; of mathgames: Nim-type games, Wythoff games, annihilation games, octal games.

Some “rule of thumb” properties, which seem to hold for the majority of playgames and mathgames are listed below.

I. Complexity. Both playgames and mathgames tend to be computationally intractable. There are a few tractable mathgames, such as Nim, but most games still live in Wonderland: we are wondering about their as yet unknown complexity. Roughly speaking, however, NP-hardness is a necessary but not a sufficient condition for being a playgame! (Some games on Boolean formulas are Exptime-complete, yet none of them seems to have the potential of commercial marketability.)

II. Boardfeel. None of us may know an exact strategy from a midgame position of chess, but even a novice, merely by looking at the board, gets some feel who of the two players is in a stronger position – even what a strong or weak next move is. This is what we loosely call boardfeel. Our informal definition of playgames and mathgames suggests that the former do have a boardfeel, whereas the latter don’t. For many mathgames, such as Nim, a player without prior knowledge of the strategy has no inkling whether any given position is “strong” or “weak” for a player. Even when defeat is imminent, only one or two moves away, the player sustaining it may be in the dark about the outcome, which will stump him. The player has no boardfeel. (Even many mathgames, including Nim-type games, can be played, equivalently, on a board.)

Thus, in the boardfeel sense, simple games are complex and complex games are simple! This paradoxical property also doesn’t seem to have an analog in the realm of decision problems. The boardfeel is the main ingredient which makes PlayGames interesting to play.

III. Math Appeal. Playgames, in addition to being interesting to play, also have considerable mathematical appeal. This has been exposed recently by the theory of partizan games established by Conway and applied to endgames of go by Berlekamp, students and associates. On the other hand, mathgames have their own special combinatorial appeal, of a somewhat different flavor. They appeal to and are created by mathematicians of various disciplines, who find special intellectual challenges in analyzing them. As Peter Winkler called a subset of them: “games people don’t play”. We might also call them, in a more positive vein, “games mathematicians play”. Both classes of games have applications to areas outside game theory. Examples: surreal numbers (playgames), error correcting codes (mathgames). Both provide enlightenment through bewilderment, as David Wolfe and Tom Rodgers put it.

IV. Existence. There are relatively few successful playgames around. It seems to be hard to invent a playgame that catches the masses. In contrast, mathgames abound. They appeal to a large subclass of mathematicians and other scientists, who cherish producing them and pondering about them. The large proportion of mathgamespapers in the games bibliography below reflects this phenomenon.

We conclude, inter alia, that for playgames, high complexity is desirable. Whereas in all respectable walks of life we strive towards solutions or at least approximate solutions which are polynomial, there are two less respectable human activities in which high complexity is appreciated. These are cryptography (covert warfare) and games (overt warfare). The desirability of high complexity in cryptography — at least for the encryptor!

— is clear. We claim that it is also desirable for playgames.

It’s no accident that games and cryptography team up: in both there are adversaries, who pit their wits against each other! But games are, in general, considerably harder than cryptography. For the latter, the problem whether the designer of a cryptosystem has a safe system can be expressed with two quantifiers only: ∃ a cryptosystem such that ∀ attacks on it, the cryptosystem remains unbroken? In contrast, the decision problem whether White can win if White moves first in a chess game, has the form: “∃∀∃∀ ⋯ move: White wins?”, expressing the question whether White has an opening winning move — with an unbounded number of alternating quantifiers. This makes games the more challenging and fascinating of the two, besides being fun! See also the next section.

Thus, it’s no surprise that the skill of playing games, such as checkers, chess, or go has long been regarded as a distinctive mark of human intelligence.

3 Why are Combinatorial Games Hard?

Existential decision problems, such as graph hamiltonicity and Traveling Salesperson (Is there a round tour through specified cities of cost ≤ C?), involve a single existential quantifier (“Is there… ?”). In mathematical terms an existential problem boils down to finding a path—sometimes even just verifying its existence—in a large “decision-tree” of all possibilities, that satisfies specified properties. The above two problems, as well as thousands of other interesting and important combinatorial-type problems are NPcomplete. This means that they are conditionally intractable, i.e., the best way to solve them seems to require traversal of most if not all of the decision tree, whose size is exponential in the input size of the problem. No essentially better method is known to date at any rate, and, roughly speaking, if an efficient solution will ever be found for any NP-complete problem, then all NP-complete problems will be solvable efficiently.

The decision problem whether White can win if White moves first in a chess game, on the other hand, has the form: Is there a move of White such that for every move of Black there is a move of White such that for every move of Black there is a move of White … such that White can win? Here we have a large number of alternating existential and universal quantifiers rather than a single existential one. We are looking for an entire subtree rather than just a path in the decision tree. Because of this, most nonpolynomial games are at least Pspace-hard. The problem for generalized chess on an n × n board, and even for a number of seemingly simpler mathgames, is, in fact, Exptime-complete, which is a provable intractability.

Put in simple language, in analyzing an instance of Traveling Salesperson, the problem itself is passive: it does not resist your attempt to attack it, yet it is difficult. In a game, in contrast, there is your opponent, who, at every step, attempts to foil your effort to win. It’s similar to the difference between an autopsy and surgery. Einstein, contemplating the nature of physics said, “Der Allmächtige ist nicht boshaft; Er ist raffiniert” (The Almighty is not mean; He is sophisticated). NP-complete existential problems are perhaps sophisticated. But your opponent in a game can be very mean!

Another manifestation of the high complexity of games is associated with a most basic tool of a game : its game-graph. It is a directed graph G whose vertices are the positions of the game, and (u, v) is an edge if and only if there is a move from position u to position v. Since every combination of tokens in the given game is a single vertex in G, the latter has normally exponential size. This holds, in particular, for both Nim and chess. Analyzing a game means reasoning about its game-graph. We are thus faced with a problem that is a priori exponential, quite unlike many present-day interesting existential problems.

A fundamental notion is the sum (disjunctive compound) of games. A sum is a finite collection of disjoint games; often very basic, simple games. Each of the two players, at every turn, selects one of the games and makes a move in it. If the outcome is not a draw, the sum-game ends when there is no move left in any of the component games. If the outcome is not a tie either, then in normal play, the player first unable to move loses and the opponent wins. The outcome is reversed in misère play.

If a game decomposes into a disjoint sum of its components, either from the beginning (Nim) or after a while (domineering), the potential for its tractability increases despite the exponential size of the game graph. As Elwyn Berlekamp remarked, the situation is similar to that in other scientific endeavors, where we often attempt to decompose a given system into its functional components. This approach may yield improved insights into hardware, software or biological systems, human organizations, and abstract mathematical objects such as groups.

If a game doesn’t decompose into a sum of disjoint components, it is more likely to be intractable (Geography or Poset Games). Intermediate cases happen when the components are not quite fixed (which explains why misère play of sums of games is much harder than normal play) or not quite disjoint (Welter). Thane Plambeck has recently made progress with misère play, and we will be hearing more about this shortly.

The hardness of games is eased somewhat by the efficient freeware package “Combinatorial Game Suite”, courtesy of Aaron Siegel.

4 Breaking the Rules

As the experts know, some of the most exciting games are obtained by breaking some of the rules for combinatorial games, such as permitting a player to pass a bounded or unbounded number of times, i.e., relaxing the requirement that players play alternately; or permitting a number of players other than two.

But permitting a payoff function other than (0,1) for the outcome (lose, win) and a payoff of (\frac{1}{2}, \frac{1}{2}) for either (tie, tie) or (draw, draw) usually, but not always, leads to games that are not considered to be combinatorial games; or to borderline cases.

5 Why Is the Bibliography Vast?

In the realm of existential problems, such as sorting or Traveling Salesperson, most present-day interesting decision problems can be classified into tractable, conditionally intractable, and provably intractable ones. There are exceptions, to be sure, such as graph isomorphism, whose complexity is still unknown. But the exceptions are few. In contrast, most games are still in Wonderland, as pointed out in §2(I) above. Only a few games have been classified into the complexity classes they belong to. Despite recent impressive progress, the tools for reducing Wonderland are still few and inadequate.

To give an example, many interesting games have a very succinct input size, so a polynomial strategy is often more difficult to come by (Richard Guy and Cedric Smith’s octal games; Grundy’s game). Succinctness and non-disjointness of games in a sum may be present simultaneously (Poset games). In general, the alternating quantifiers, and, to a smaller measure, “breaking the rules”, add to the volume of Wonderland. We suspect that the large size of Wonderland, a fact of independent interest, is the main contributing factor to the bulk of the bibliography on games.

6 Why Isn’t it Larger?

The bibliography below is a partial list of books and articles on combinatorial games and related material. It is partial not only because I constantly learn of additional relevant material I did not know about previously, but also because of certain self-imposed restrictions. The most important of these is that only papers with some original and nontrivial mathematical content are considered. This excludes most historical reviews of games and most, but not all, of the work on heuristic or artificial intelligence approaches to games, especially the large literature concerning computer chess. I have, however, included the compendium Levy [1988], which, with its 50 articles and extensive bibliography, can serve as a first guide to this world. Also some papers on chess-endgames and clever exhaustive computer searches of some games have been included.

On the other hand, papers on games that break some of the rules of combinatorial games are included liberally, as long as they are interesting and retain a combinatorial flavor. These are vague and hard to define criteria, yet combinatorialists usually recognize a combinatorial game when they see it. Besides, it is interesting to break also this rule sometimes! We have included some references to one-player games, e.g., towers of Hanoi, n-queen problems, 15-puzzle and peg-solitaire, but only few zero-player games (such as Life and games on “sand piles”). We have also included papers on various applications of games, especially when the connection to games is substantial or the application is interesting or important.

High-class meetings on combinatorial games, such as in Columbus, OH (1990), at MSRI (1994, 2000), at BIRS (2005) resulted in books, or a special issue of a journal – for the Dagstuhl seminar (2002). During 1990–2001, Theoretical Computer Science ran a special Mathematical Games Section whose main purpose was to publish papers on combinatorial games. TCS still solicits papers on games. In 2002, Integers—Electronic J. of Combinatorial Number Theory began publishing a Combinatorial Games Section. The combinatorial games community is growing in quantity and quality!

7 The Dynamics of the Literature

The game bibliography below is very dynamic in nature. Previous versions have been circulated to colleagues, intermittently, since the early 1980’s. Prior to every mailing updates were prepared, and usually also afterwards, as a result of the comments received from several correspondents. The listing can never be “complete”. Thus also the present form of the bibliography is by no means complete.

Because of its dynamic nature, it is natural that the bibliography became a “Dynamic Survey” in the Dynamic Surveys (DS) section of the Electronic Journal of Combinatorics (ElJC) and The World Combinatorics Exchange (WCE). The ElJC and WCE are on the World Wide Web (WWW), and the DS can be accessed at http://www.combinatorics.org/ (click on “Surveys”). The ElJC has mirrors at various locations. Furthermore, the European Mathematical Information Service (EMIS) mirrors this Journal, as do all of its mirror sites (currently over forty of them). See http://www.emis.de/tech/mirrors.html

8 An Appeal

I ask readers to continue sending to me corrections and comments; and inform me of significant omissions, remembering, however, that it is a selected bibliography. I prefer to get reprints, preprints or URLs, rather than only titles — whenever possible.

Material on games is mushrooming on the Web. The URLs can be located using a standard search engine, such as Google.

9 Idiosyncrasies

Most of the bibliographic entries refer to items written in English, though there is a sprinkling of Danish, Dutch, French, German, Japanese, Slovakian and Russian, as well as some English translations from Russian. The predominance of English may be due to certain prejudices, but it also reflects the fact that nowadays the lingua franca of science is English. In any case, I’m soliciting also papers in languages other than English, especially if accompanied by an abstract in English.

On the administrative side, Technical Reports, submitted papers and unpublished theses have normally been excluded; but some exceptions have been made. Abbreviations of book series and journal names usually follow the Math Reviews conventions. Another convention is that de Bruijn appears under D, not B; von Neumann under V, not N, McIntyre under M not I, etc.

Earlier versions of this bibliography have appeared, under the title “Selected bibliography on combinatorial games and some related material”, as the master bibliography for the book Combinatorial Games, AMS Short Course Lecture Notes, Summer 1990, Ohio State University, Columbus, OH, Proc. Symp. Appl. Math. 43 (R. K. Guy, ed.), AMS 1991, pp. 191–226 with 400 items, and in the Dynamic Surveys section of the Electronic J. of Combinatorics in November 1994 with 542 items (updated there at odd times). It also appeared as the master bibliography in Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 1994, Berkeley, CA (R. J. Nowakowski, ed.), MSRI Publ. Vol. 29, Cambridge University Press, Cambridge, 1996, pp. 493–537, under the present title, containing 666 items. The version published in the palindromic year 2002 contained the palindromic number 919 of references. It constituted a growth of 38%. It appeared in ElJC and as the master bibliography in More Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 2000, Berkeley, CA (R. J. Nowakowski, ed.), MSRI Publ. Vol. 42, Cambridge University Press, Cambridge, pp. 475-535. The current update (mid-2003), in ElJC, contains 1001 items, another palindrome.

10 Acknowledgments

Many people have suggested additions to the bibliography, or contributed to it in other ways. Ilan Vardi distilled my Math-master (§2) into Mathter. Among those that contributed more than two or three items are: Akeo Adachi, Ingo Althöfer, Thomas Andreae, Eli Bachmupsky, Adriano Barlotti, József Beck, the late Claude Berge, Gerald E. Bergum, H. S. MacDonald Coxeter, Thomas S. Ferguson, James A. Flanigan, Fred Galvin, Martin Gardner, Alan J. Goldman, Solomon W. Golomb, Richard K. Guy, Shigeki Iwata, David S. Johnson, Victor Klee, Donald E. Knuth, Anton Kotzig, Jeff C. Lagarias, Michel Las Vergnas, Hendrik W. Lenstra, Hermann Loimer, F. Lockwood Morris, Richard J. Nowakowski, Judea Pearl, J. Michael Robson, David Singmaster, Wolfgang Slany, Cedric A. B. Smith, Rastislaw Telgársky, Mark D. Ward, Yōhei Yamasaki and others. Thanks to all and keep up the game! Special thanks to Mark Ward who went through the entire file with a fine comb in late 2005, when it contained 1,151 items, correcting errors and typos. Many thanks also to various anonymous helpers who assisted with the initial TEX file, to Silvio Levy, who has edited and transformed it into L^{A}TEX2e in 1996, and to Wolfgang Slany, who has transformed it into a BIBTeX file at the end of the previous millenium, and solved a “new millenium” problem encountered when the bibliography grew beyond 999 items. Keen users of the bibliography will notice that there is a beginning of MR references, due to Richard Guy’s suggestion, facilitated by his student Alex Fink.

11 The Bibliography

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6. E. W. Adams and D. C. Benson [1956], Nim-Type Games, Technical Report No. 31 , Department of Mathematics, Pittsburgh, PA.

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523. A. S. Fraenkel [1981], Planar kernel and Grundy with d ≤ 3, d_{out} ≤ 2, d_{in} ≤ 2 are NP-complete, Discrete Appl. Math. 3, 257–262.

524. A. S. Fraenkel [1982], How to beat your Wythoff games’ opponent on three fronts, Amer. Math. Monthly 89, 353–361.

525. A. S. Fraenkel [1983], 15 Research problems on games, Discrete Math. in ”Research Problems” section, Vols. 43-46.

526. A. S. Fraenkel [1984], Wythoff games, continued fractions, cedar trees and Fibonacci searches, Theoret. Comput. Sci. 29, 49–73, an earlier version appeared in Proc. 10th Internat. Colloq. on Automata, Languages and Programming (J. Diaz, ed.), Vol. 154, Barcelona, July 1983, Lecture Notes in Computer Science, Springer Verlag, Berlin, 1983, pp. 203–225.

527. A. S. Fraenkel [1988], The complexity of chess, Letter to the Editor, J. Recr. Math. 20, 13–14.

528. A. S. Fraenkel [1991], Complexity of games, in: Combinatorial Games, Proc. Symp. Appl. Math. (R. K. Guy, ed.), Vol. 43, Amer. Math. Soc., Providence, RI, pp. 111–153.

529. A. S. Fraenkel [1994], Even kernels, Electr. J. Combinatorics 1, #R5, 13 pp. http://www.combinatorics.org/

530. A. S. Fraenkel [1994], Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups, Trans. Amer. Math. Soc. 341, 639–664, MR1138949 (94d:11011).

531. A. S. Fraenkel [1994], Recreation and depth in combinatorial games, in: The Lighter Side of Mathematics, Proc. E. Strens Memorial Conf. on Recr. Math. and its History, Calgary, 1986, Spectrum Series (R. K. Guy and R. E. Woodrow, eds.), Math. Assoc. of America, Washington, DC, pp. 176–194.

532. A. S. Fraenkel [1996], Error-correcting codes derived from combinatorial games, in: Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 1994, Berkeley, CA, MSRI Publ. (R. J. Nowakowski, ed.), Vol. 29, Cambridge University Press, Cambridge, pp. 417–431.

533. A. S. Fraenkel [1996], Scenic trails ascending from sea-level Nim to alpine chess, in: Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 1994, Berkeley, CA, MSRI Publ. (R. J. Nowakowski, ed.), Vol. 29, Cambridge University Press, Cambridge, pp. 13–42.

534. A. S. Fraenkel [1997], Combinatorial game theory foundations applied to digraph kernels, Electr. J. Combinatorics 4(2), #R10, 17 pp., Volume in honor of Herbert Wilf. http://www.combinatorics.org/

535. A. S. Fraenkel [1998], Heap games, numeration systems and sequences, Ann. Comb. 2, 197–210, an earlier version appeared in: Fun With Algorithms, Vol. 4 of Proceedings in Informatics (E. Lodi, L. Pagli and N. Santoro, eds.), Carleton Scientific, University of Waterloo, Waterloo, Ont., pp. 99–113, 1999. Conference took place on the island of Elba, June 1998., MR1681514 (2000b:91001).

536. A. S. Fraenkel [1998], Multivision: an intractable impartial game with a linear winning strategy, Amer. Math. Monthly 105, 923–928.

537. A. S. Fraenkel [2000], Recent results and questions in combinatorial game complexities, Theoret. Comput. Sci. 249, 265–288, Conference version in: Proc. AWOCA98 — Ninth Australasian Workshop on Combinatorial Algorithms, C.S. Iliopoulos, ed., Perth, Western Australia, 27–30 July, 1998, special AWOCA98 issue, pp. 124-146, MR1798313 (2001j:91033).

538. A. S. Fraenkel [2001], Virus versus mankind, Proc. 2nd Intern. Conference on Computers and Games CG’2000 (T. Marsland and I. Frank, eds.), Vol. 2063, Hamamatsu, Japan, Oct. 2000, Lecture Notes in Computer Science, Springer, pp. 204– 213.

539. A. S. Fraenkel [2002], Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282, 271–284, special“ Fun With Algorithms” issue, MR1909052 (2003h:91036).

540. A. S. Fraenkel [2002], Mathematical chats between two physicists, in: Puzzler’s Tribute: a Feast for the Mind, honoring Martin Gardner (D. Wolfe and T. Rodgers, eds.), A K Peters, Natick, MA, pp. 383-386.

541. A. S. Fraenkel [2002], Two-player games on cellular automata, in: More Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 2000, Berkeley, CA, MSRI Publ. (R. J. Nowakowski, ed.), Vol. 42, Cambridge University Press, Cambridge, pp. 279–306, MR1973018 (2004b:91004).

542. A. S. Fraenkel [2004], Complexity, appeal and challenges of combinatorial games, Theoret. Comp. Sci. 313, 393–415, Expanded version of a keynote address at Dagstuhl Seminar “Algorithmic Combinatorial Game Theory”, Feb. 2002, special issue on Algorithmic Combinatorial Game Theory, MR2056935.

543. A. S. Fraenkel [2004], New games related to old and new sequences, Integers, Electr. J of Combinat. Number Theory 4, #G6, 18 pp., Comb. Games Sect., 1st version in Proc.10-th Advances in Computer Games (ACG-10 Conf.), H. J. van den Herik, H. Iida and E. A. Heinz eds., Graz, Austria, Nov. 2003, Kluwer, pp. 367-382, MR2042724. http://www.integers-ejcnt.org/vol4.html

544. A. S. Fraenkel [2005], Euclid and Wythoff games, Discrete Math. 304, 65–68, MR2184445 (2006f:91006).

545. A. S. Fraenkel [2006], Nim is easy, chess is hard — but why??, J. Internat. Computer Games Assoc. 29(4), 203–206, earlier version appeared in Plus Mag. (electronic), pluschat section, http://plus.maths.org/issue40/editorial/index.html.

546. A. S. Fraenkel [2007], The Raleigh game, in: Combinatorial Number Theory, de Gruyter, pp. 199–208, Proc. Integers Conference, Carrollton, Georgia, October 27-30,2005, in celebration of the 70th birthday of Ronald Graham. B. Landman, M. Nathanson, J. Nešetřil, R. Nowakowski, C. Pomerance eds., also appeared in Integers, Electr. J. of Combinat. Number Theory 7(2), special volume in honor of Ron Graham, #A13, 11 pp., MR2337047 (2008e:91021). http://www.integers-ejcnt.org/vol7(2).html

547. A. S. Fraenkel [2007], Why are games exciting and stimulating?, Math Horizons pp. 5–7; 32–33, special issue: “Games, Gambling, and Magic” February. German translation by Niek Neuwahl, poster-displayed at traveling exhibition “Games & Science, Science & Games”, opened in Göttingen July 17 – Aug 21, 2005.

548. A. S. Fraenkel [2008], Games played by Boole and Galois, Discrete Appl. Math. 156, 420–427, **MR pending.

549. A. S. Fraenkel [2009], The cyclic Butler University game, in: Mathematical Wizardry for a Gardner, Volume honoring Martin Gardner, A K Peters, Natick, MA, E. Pegg Jr, A. H. Schoen, and T. Rodgers, eds., to appear.

550. A. S. Fraenkel and I. Borosh [1973], A generalization of Wythoff’s game, J. Combin. Theory (Ser. A) 15, 175–191.

551. A. S. Fraenkel, M. R. Garey, D. S. Johnson, T. Schaefer and Y. Yesha [1978], The complexity of checkers on an n × n board — preliminary report, Proc. 19th Ann. Symp. Foundations of Computer Science (Ann Arbor, MI, Oct. 1978), IEEE Computer Soc., Long Beach, CA, pp. 55–64.

552. A. S. Fraenkel and E. Goldschmidt [1987], Pspace-hardness of some combinatorial games, J. Combin. Theory (Ser. A) 46, 21–38.

553. A. S. Fraenkel and F. Harary [1989], Geodetic contraction games on graphs, Internat. J. Game Theory 18, 327–338.

554. A. S. Fraenkel and H. Herda [1980], Never rush to be first in playing Nimbi, Math. Mag. 53, 21–26.

555. A. S. Fraenkel, A. Jaffray, A. Kotzig and G. Sabidussi [1995], Modular Nim, Theoret. Comput. Sci. (Math Games) 143, 319–333.

556. A. S. Fraenkel and C. Kimberling [1994], Generalized Wythoff arrays, shuffles and interspersions, Discrete Math. 126, 137–149, MR1264482 (95c:11028).

557. A. S. Fraenkel and A. Kontorovich [2007], The Sierpiński sieve of Nim-varieties and binomial coefficients, in: Combinatorial Number Theory, de Gruyter, pp. 209–227, Proc. Integers Conference, Carrollton, Georgia, October 27-30,2005, in celebration of the 70th birthday of Ronald Graham. B. Landman, M. Nathanson, J. Nešetřil, R. Nowakowski, C. Pomerance eds., appeared also in Integers, Electr. J. of Combinat. Number Theory 7(2), special volume in honor of Ron Graham, A14, 19 pp., MR2337048. http://www.integers-ejcnt.org/vol7(2).html

558. A. S. Fraenkel and A. Kotzig [1987], Partizan octal games: partizan subtraction games, Internat. J. Game Theory 16, 145–154.

559. A. S. Fraenkel and D. Krieger [2004], The structure of complementary sets of integers: a 3-shift theorem, Internat. J. Pure and Appl. Math. 10, 1–49, MR2020683 (2004h:05012).

560. A. S. Fraenkel and D. Lichtenstein [1981], Computing a perfect strategy for n × n chess requires time exponential in n, J. Combin. Theory (Ser. A) 31, 199–214, preliminary version in Proc. 8th Internat. Colloq. Automata, Languages and Programming (S. Even and O. Kariv, eds.), Vol. 115, Acre, Israel, 1981, Lecture Notes in Computer Science, Springer Verlag, Berlin, pp. 278–293, MR629595 (83b:68044).

561. A. S. Fraenkel, M. Loebl and J. Nešetřil [1988], Epidemiography II. Games with a dozing yet winning player, J. Combin. Theory (Ser. A) 49, 129–144.

562. A. S. Fraenkel and M. Lorberbom [1989], Epidemiography with various growth functions, Discrete Appl. Math. 25, 53–71, special issue on Combinatorics and Complexity.

563. A. S. Fraenkel and M. Lorberbom [1991], Nimhoff games, J. Combin. Theory (Ser. A) 58, 1–25.

564. A. S. Fraenkel and J. Nešetřil [1985], Epidemiography, Pacific J. Math. 118, 369– 381.

565. A. S. Fraenkel and M. Ozery [1998], Adjoining to Wythoff’s game its P -positions as moves, Theoret. Comput. Sci. 205, 283–296.

566. A. S. Fraenkel and Y. Perl [1975], Constructions in combinatorial games with cycles, Coll. Math. Soc. János Bolyai 10, 667–699, Proc. Internat. Colloq. on Infinite and Finite Sets, Vol. 2 (A. Hajnal, R. Rado and V. T. Sós, eds.) Keszthely, Hungary, 1973, North-Holland.

567. A. S. Fraenkel and O. Rahat [2001], Infinite cyclic impartial games, Theoret. Comput. Sci. 252, 13–22, special ”Computers and Games” issue; first version appeared in Proc. 1st Intern. Conf. on Computer Games CG’98, Tsukuba, Japan, Nov. 1998, Lecture Notes in Computer Science, Vol. 1558, Springer, pp. 212-221, 1999., MR1715689 (2000m:91028).

568. A. S. Fraenkel and O. Rahat [2003], Complexity of error-correcting codes derived from combinatorial games, Proc. Intern. Conference on Computers and Games CG’2002, Edmonton, Alberta, Canada, July 2002, (Y. Björnsson, M. Müller and J. Schaeffer, eds.), Vol. LNCS 2883, Lecture Notes in Computer Science, Springer,

pp. 201–21.

569. A. S. Fraenkel and E. Reisner [2008], The game of End-Wythoff, in: Games of No Chance III, Proc. BIRS Workshop on Combinatorial Games, July, 2005, Banff, Alberta, Canada, MSRI Publ. (M. H. Albert and R. J. Nowakowski, eds.), Cambridge University Press, Cambridge.

570. A. S. Fraenkel and E. R. Scheinerman [1991], A deletion game on hypergraphs, Discrete Appl. Math. 30, 155–162.

571. A. S. Fraenkel, E. R. Scheinerman and D. Ullman [1993], Undirected edge geography, Theoret. Comput. Sci. (Math Games) 112, 371–381.

572. A. S. Fraenkel and S. Simonson [1993], Geography, Theoret. Comput. Sci. (Math Games) 110, 197–214.

573. A. S. Fraenkel and U. Tassa [1975], Strategy for a class of games with dynamic ties, Comput. Math. Appl. 1, 237–254.

574. A. S. Fraenkel and U. Tassa [1982], Strategies for compounds of partizan games, Math. Proc. Camb. Phil. Soc. 92, 193–204.

575. A. S. Fraenkel, U. Tassa and Y. Yesha [1978], Three annihilation games, Math. Mag. 51, 13–17, special issue on Recreational Math.

576. A. S. Fraenkel and Y. Yesha [1976], Theory of annihilation games, Bull. Amer. Math. Soc. 82, 775–777.

577. A. S. Fraenkel and Y. Yesha [1979], Complexity of problems in games, graphs and algebraic equations, Discrete Appl. Math. 1, 15–30.

578. A. S. Fraenkel and Y. Yesha [1982], Theory of annihilation games — I, J. Combin. Theory (Ser. B) 33, 60–86.

579. A. S. Fraenkel and Y. Yesha [1986], The generalized Sprague–Grundy function and its invariance under certain mappings, J. Combin. Theory (Ser. A) 43, 165–177.

580. A. S. Fraenkel and D. Zusman [2001], A new heap game, Theoret. Comput. Sci. 252, 5–12, special ”Computers and Games” issue; first version appeared in Proc. 1st Intern. Conf. on Computer Games CG’98, Tsukuba, Japan, Nov. 1998, Lecture Notes in Computer Science, Vol. 1558, Springer, pp. 205-211, 1999., MR1715688 (2000m:91027).

581. C. N. Frangakis [1981], A backtracking algorithm to generate all kernels of a directed graph, Intern. J. Comput. Math. 10, 35–41.

582. P. Frankl [1987], Cops and robbers in graphs with large girth and Cayley graphs, Discrete Appl. Math. 17, 301–305.

583. P. Frankl [1987], On a pursuit game on Cayley graphs, Combinatorica 7, 67–70.

584. P. Frankl and N. Tokushige [2003], The game of n-times nim, Discrete Math. 260, 205–209, MR1948387 (2003m:05017).

585. W. Fraser, S. Hirshberg and D. Wolfe [2005], The structure of the distributive lattice of games born by day n, Integers, Electr. J of Combinat. Number Theory 5(2), #A06, 11 pp., MR2192084 (2006g:91041). http://www.integers-ejcnt.org/vol5(2).html

586. D. Fremlin [1973], Well-founded games, Eureka 36, 33–37.

587. G. H. Fricke, S. M. Hedetniemi, S. T. Hedetniemi, A. A. McRae, C. K. Wallis, M. S. Jacobson, H. W. Martin and W. D. Weakley [1995], Combinatorial problems on chessboards: a brief survey, in: Graph Theory, Combinatorics, and Algorithms: Proc. 7th Quadrennial Internat. Conf. on the Theory and Applications of Graphs (Y. Alavi and A. Schwenk, eds.), Vol. 1, Wiley, pp. 507–528.

588. E. J. Friedman and A. S. Landsberg [2007], Nonlinear dynamics in combinatorial games: renormalizing Chomp, Chaos 17(2), 023117 1–14, MR2340612.

589. E. J. Friedman and A. S. Landsberg [2008], On the geometry of combinatorial games: a renormalization approach, in: Games of No Chance III, Proc. BIRS Workshop on Combinatorial Games, July, 2005, Banff, Alberta, Canada, MSRI Publ. (M. H. Albert and R. J. Nowakowski, eds.), Cambridge University Press, Cambridge.

590. A. Frieze, M. Krivelevich, O. Pikhurko and T. Szabó [2005], The game of JumbleG, Combin. Probab. Comput. 14, 783–793, MR2174656 (2006k:05207).

591. M. Fukuyama [2003], A Nim game played on graphs, Theoret. Comput. Sci. 304, 387–399, MR1992342 (2004f:91041).

592. M. Fukuyama [2003], A Nim game played on graphs II, Theoret. Comput. Sci. 304, 401–419, MR1992343 (2004g:91036).

593. W. W. Funkenbusch [1971], SIM as a game of chance, J. Recr. Math. 4(4), 297–298.

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595. H. N. Gabow and H. H. Westermann [1992], Forests, frames, and games: algorithms for matroid sums and applications, Algorithmica 7, 465–497.

596. D. Gale [1974], A curious Nim-type game, Amer. Math. Monthly 81, 876–879.

597. D. Gale [1979], The game of Hex and the Brouwer fixed-point theorem, Amer. Math. Monthly 86, 818–827.

598. D. Gale [1986], Problem 1237 (line-drawing game), Math. Mag. 59, 111, solution by J. Hutchinson and S. Wagon, ibid. 60 (1987) 116.

599. D. Gale [1991 – 1996], Mathematical Entertainments, Math. Intelligencer 13 - 18, column on mathematical games and gems, Winter 1991 – Fall 1996.

600. D. Gale [1998], Tracking the Automatic Ant and Other Mathematical Explorations, Springer-Verlag, New York, A collection of Mathematical Entertainments columns from The Mathematical Intelligencer.

601. D. Gale and A. Neyman [1982], Nim-type games, Internat. J. Game Theory 11, 17–20.

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604. H. Galeana-Sánchez [1982], A counterexample to a conjecture of Meyniel on kernelperfect graphs, Discrete Math. 41, 105–107.

605. H. Galeana-Sánchez [1986], A theorem about a conjecture of Meyniel on kernelperfect graphs, Discrete Math. 59, 35–41.

606. H. Galeana-Sánchez [1988], A new method to extend kernel-perfect graphs to kernelperfect critical graphs, Discrete Math. 69, 207–209, MR937787 (89d:05086).

607. H. Galeana-Sánchez [1990], On the existence of (k, l)-kernels in digraphs, Discrete Math. 85, 99–102, MR1078316 (91h:05056).

608. H. Galeana-Sánchez [1992], On the existence of kernels and h-kernels in directed graphs, Discrete Math. 110, 251–255.

609. H. Galeana-Sánchez [1995], B_{1} and B_{2}-orientable graphs in kernel theory, Discrete Math. 143, 269–274.

610. H. Galeana-Sánchez [1996], On claw-free M-oriented critical kernel-imperfect digraphs, J. Graph Theory 21, 33–39, MR1363686 (96h:05089).

611. H. Galeana-Sánchez [1998], Kernels in edge-colored digraphs, Discrete Math. 184, 87–99, MR1609359 (99a:05055).

612. H. Galeana-Sánchez [2000], Semikernels modulo F and kernels in digraphs, Discrete Math. 218, 61–71, MR1754327 (2001d:05068).

613. H. Galeana-Sánchez [2002], Kernels in digraphs with covering number at most 3, Discrete Math. 259, 121–135, MR1948776 (2003m:05090).

614. H. Galeana-Sánchez [2004], Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel, Discuss. Math. Graph Theory 24, 171–182, MR2120561 (2005m:05104).

615. H. Galeana-Sánchez [2006], Kernels and perfectness in arc-local tournament digraphs, Discrete Math. 306(19-20), 2473–2480, special issue: Creation and Recreation: A Tribute to the Memory of Claude Berge, MR2261913 (2007g:05071).

616. H. Galeana-Sánchez [2008], Kernels in edge-coloured orientations of nearly complete graphs, Discrete Math. 308, 4599–4607, **MR pending.

617. H. Galeana-Sánchez and J. de J. García-Ruvalcaba [2000], Kernels in the closure of coloured digraphs, Discuss. Math. Graph Theory 20, 243–254, MR1817495 (2001m:05123).

618. H. Galeana-Sánchez and J. d. J. García-Ruvalcaba [2001], On graphs all of whose {C_{3}, T_{3}}-free arc colorations are kernel-perfect, Discuss. Math. Graph Theory 21(1), 77–93, MR1867488 (2002j:05075).

619. H. Galeana-Sánchez and M.-k. Guevara [2007], Kernel perfect and critical kernel imperfect digraphs structure, in: 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Vol. 28 of Electron. Notes Discrete Math., Elsevier, Amsterdam, pp. 401–408, MR2324045.

620. H. Galeana-Sánchez and X. Li [1998], Kernels in a special class of digraphs, Discrete Math. 178, 73–80, MR1483740 (98f:05073).

621. H. Galeana-Sánchez and X. Li [1998], Semikernels and (k, l)-kernels in digraphs, SIAM J. Discrete Math. 11, 340–346 (electronic), MR1617163 (99c:05078).

622. H. Galeana-Sánchez and V. Neumann-Lara [1984], On kernels and semikernels of digraphs, Discrete Math. 48, 67–76.

623. H. Galeana-Sánchez and V. Neumann Lara [1986], On kernel-perfect critical digraphs, Discrete Math. 59, 257–265, MR842278 (88b:05069).

624. H. Galeana-Sánchez and V. Neumann-Lara [1991], Extending kernel perfect digraphs to kernel perfect critical digraphs, Discrete Math. 94, 181–187.

625. H. Galeana-Sánchez and V. Neumann Lara [1991], Orientations of graphs in kernel theory, Discrete Math. 87, 271–280, MR1095472 (92d:05134).

626. H. Galeana-Sánchez and V. Neumann-Lara [1994], New extensions of kernel perfect digraphs to kernel imperfect critical digraphs, Graphs Combin. 10, 329–336.

627. H. Galeana-Sánchez and V. Neumann-Lara [1998], New classes of critical kernel-imperfect digraphs, Discuss. Math. Graph Theory 18, 85–89, MR1646233 (99g:05087).

628. H. Galeana-Sánchez and V. Neumann-Lara [1998], On the dichromatic number in kernel theory, Math. Slovaca 48, 213–219, MR1647678 (99k:05080).

629. H. Galeana-Sánchez and L. Pastrana Ramírez [1998], Kernels in edge coloured line digraph, Discuss. Math. Graph Theory 18, 91–98, MR1646234 (99g:05088).

630. H. Galeana-Sánchez, L. Pastrana Ramírez and H. A. Rincón-Mejía [1991], Semikernels, quasi kernels, and Grundy functions in the line digraph, SIAM J. Discrete Math. 4, 80–83, MR1090291 (92c:05068).

631. H. Galeana-Sánchez, L. P. Ramírez and H. A. Rincón Mejía [2005], Kernels in monochromatic path digraphs, Discuss. Math. Graph Theory 25, 407–417, MR2233005.

632. H. Galeana-Sánchez and H. A. Rincón-Mejía [1998], A sufficient condition for the existence of k-kernels in digraphs, Discuss. Math. Graph Theory 18, 197–204, MR1687843 (2000c:05074).

633. H. Galeana-Sánchez and R. Rojas-Monroy [2004], Kernels in pretransitive digraphs, Discrete Math. 275, 129–136, MR2026280 (2004j:05059).

634. H. Galeana-Sánchez and R. Rojas-Monroy [2006], Kernels in quasi-transitive digraphs, Discrete Math. 306(16), 1969–1974, MR2251578 (2007b:05092).

635. H. Galeana-Sánchez and R. Rojas-Monroy [2007], Extensions and kernels in digraphs and in edge-coloured digraphs, WSEAS Trans. Math. 6(2), 334–341, MR2273872.

636. H. Galeana-Sánchez and R. Rojas-Monroy [2007], Kernels in orientations of pretransitive orientable graphs, in: Graph theory in Paris, Trends Math., Birkhäuser, Basel, pp. 197–208, MR2279176.

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