Combinatorial Game Theory
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On Variations of Nim and Chomp Arxiv:1705.06774V1 [Math.CO] 18
On Variations of Nim and Chomp June Ahn Benjamin Chen Richard Chen Ezra Erives Jeremy Fleming Michael Gerovitch Tejas Gopalakrishna Tanya Khovanova Neil Malur Nastia Polina Poonam Sahoo Abstract We study two variations of Nim and Chomp which we call Mono- tonic Nim and Diet Chomp. In Monotonic Nim the moves are the same as in Nim, but the positions are non-decreasing numbers as in Chomp. Diet-Chomp is a variation of Chomp, where the total number of squares removed is limited. 1 Introduction We study finite impartial games with two players where the same moves are available to both players. Players alternate moves. In a normal play, the person who does not have a move loses. In a misère play, the person who makes the last move loses. A P-position is a position from which the previous player wins, assuming perfect play. We can observe that all terminal positions are P-positions. An N-position is a position from which the next player wins given perfect play. When we play we want to end our move with a P-position and want to see arXiv:1705.06774v1 [math.CO] 18 May 2017 an N-position before our move. Every impartial game is equivalent to a Nim heap of a certain size. Thus, every game can be assigned a non-negative integer, called a nimber, nim- value, or a Grundy number. The game of Nim is played on several heaps of tokens. A move consists of taking some tokens from one of the heaps. The game of Chomp is played on a rectangular m by n chocolate bar with grid lines dividing the bar into mn squares. -
An Introduction to Conway's Games and Numbers
AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS DIERK SCHLEICHER AND MICHAEL STOLL 1. Combinatorial Game Theory Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic struc- ture: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition gen- eralizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is under- stood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Win- ning Ways by Berlekamp, Conway and Guy [WW]; they have recently appeared in their second editions. -
Solving 8 × 8 Domineering
Theoretical Computer Science 230 (2000) 195–206 www.elsevier.com/locate/tcs View metadata, citation and similar papers at core.ac.uk brought to you by CORE Mathematical Games provided by Elsevier - Publisher Connector Solving 8 × 8 Domineering D.M. Breuker, J.W.H.M. Uiterwijk ∗, H.J. van den Herik Department of Computer Science, MATRIKS Research Institute, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, Netherlands Received May 1998; revised October 1998 Communicated by A.S. Fraenkel Abstract So far the game of Domineering has mainly been investigated by combinatorial-games re- searchers. Yet, it is a genuine two-player zero-sum game with perfect information, of which the general formulation is a topic of AI research. In that domain, many techniques have been developed for two-person games, especially for chess. In this article we show that one such technique, i.e., transposition tables, is ÿt for solving standard Domineering (i.e., on an 8×8 board). The game turns out to be a win for the player ÿrst to move. This result coincides with a result obtained independently by Morita Kazuro. Moreover, the technique of transposition tables is also applied to di erently sized m × n boards, m ranging from 2 to 8, and n from m to 9. The results are given in tabular form. Finally, some conclusions on replacement schemes are drawn. In an appendix an analysis of four tournament games is provided. c 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Domineering; Solving games; Transposition tables; Replacement schemes 1. Introduction Domineering is a two-player zero-sum game with perfect information. -
By Glenn A. Emelko
A NEW ALGORITHM FOR EFFICIENT SOFTWARE IMPLEMENTATION OF REED-SOLOMON ENCODERS FOR WIRELESS SENSOR NETWORKS by Glenn A. Emelko Submitted to the Office of Graduate Studies at Case Western Reserve University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in ELECTRICAL ENGINEERING Department of Electrical Engineering and Computer Science Case Western Reserve University Glennan 321, 10900 Euclid Ave. Cleveland, Ohio 44106 May 2009 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of _Glenn A. Emelko____________________________________ candidate for the _Doctor of Philosophy_ degree *. (signed)_Francis L. Merat______________________________ (chair of the committee) _Wyatt S. Newman______________________________ _H. Andy Podgurski_____________________________ _William L. Schultz______________________________ _David A. Singer________________________________ ________________________________________________ (date) _March 2, 2009__________ * We also certify that written approval has been obtained for any proprietary material contained therein. i Dedication For my loving wife Liz, and for my children Tom and Leigh Anne. I thank you for giving me love and support and for believing in me every step along my journey. ii Table of Contents Dedication........................................................................................................................... ii List of Figures......................................................................................................................4 -
Claude Elwood Shannon (1916–2001) Solomon W
Claude Elwood Shannon (1916–2001) Solomon W. Golomb, Elwyn Berlekamp, Thomas M. Cover, Robert G. Gallager, James L. Massey, and Andrew J. Viterbi Solomon W. Golomb Done in complete isolation from the community of population geneticists, this work went unpublished While his incredibly inventive mind enriched until it appeared in 1993 in Shannon’s Collected many fields, Claude Shannon’s enduring fame will Papers [5], by which time its results were known surely rest on his 1948 work “A mathematical independently and genetics had become a very theory of communication” [7] and the ongoing rev- different subject. After his Ph.D. thesis Shannon olution in information technology it engendered. wrote nothing further about genetics, and he Shannon, born April 30, 1916, in Petoskey, Michi- expressed skepticism about attempts to expand gan, obtained bachelor’s degrees in both mathe- the domain of information theory beyond the matics and electrical engineering at the University communications area for which he created it. of Michigan in 1936. He then went to M.I.T., and Starting in 1938 Shannon worked at M.I.T. with after spending the summer of 1937 at Bell Tele- Vannevar Bush’s “differential analyzer”, the an- phone Laboratories, he wrote one of the greatest cestral analog computer. After another summer master’s theses ever, published in 1938 as “A sym- (1940) at Bell Labs, he spent the academic year bolic analysis of relay and switching circuits” [8], 1940–41 working under the famous mathemati- in which he showed that the symbolic logic of cian Hermann Weyl at the Institute for Advanced George Boole’s nineteenth century Laws of Thought Study in Princeton, where he also began thinking provided the perfect mathematical model for about recasting communications on a proper switching theory (and indeed for the subsequent mathematical foundation. -
Arxiv:2101.07237V2 [Cs.CC] 19 Jan 2021 Ohtedsg N Oprtv Nlsso Obntra Games
TRANSVERSE WAVE: AN IMPARTIAL COLOR-PROPAGATION GAME INSPRIED BY SOCIAL INFLUENCE AND QUANTUM NIM Kyle Burke Department of Computer Science, Plymouth State University, Plymouth, New Hampshire, https://turing.plymouth.edu/~kgb1013/ Matthew Ferland Department of Computer Science, University of Southern California, Los Angeles, California, United States Shang-Hua Teng1 Department of Computer Science, University of Southern California, Los Angeles, California, United States https://viterbi-web.usc.edu/~shanghua/ Abstract In this paper, we study a colorful, impartial combinatorial game played on a two- dimensional grid, Transverse Wave. We are drawn to this game because of its apparent simplicity, contrasting intractability, and intrinsic connection to two other combinatorial games, one inspired by social influence and another inspired by quan- tum superpositions. More precisely, we show that Transverse Wave is at the intersection of social-influence-inspired Friend Circle and superposition-based Demi-Quantum Nim. Transverse Wave is also connected with Schaefer’s logic game Avoid True. In addition to analyzing the mathematical structures and com- putational complexity of Transverse Wave, we provide a web-based version of the game, playable at https://turing.plymouth.edu/~kgb1013/DB/combGames/transverseWave.html. Furthermore, we formulate a basic network-influence inspired game, called Demo- graphic Influence, which simultaneously generalizes Node-Kyles and Demi- arXiv:2101.07237v2 [cs.CC] 19 Jan 2021 Quantum Nim (which in turn contains as special cases Nim, Avoid True, and Transverse Wave.). These connections illuminate the lattice order, induced by special-case/generalization relationships over mathematical games, fundamental to both the design and comparative analyses of combinatorial games. 1Supported by the Simons Investigator Award for fundamental & curiosity-driven research and NSF grant CCF1815254. -
Alpha-Beta Pruning
Alpha-Beta Pruning Carl Felstiner May 9, 2019 Abstract This paper serves as an introduction to the ways computers are built to play games. We implement the basic minimax algorithm and expand on it by finding ways to reduce the portion of the game tree that must be generated to find the best move. We tested our algorithms on ordinary Tic-Tac-Toe, Hex, and 3-D Tic-Tac-Toe. With our algorithms, we were able to find the best opening move in Tic-Tac-Toe by only generating 0.34% of the nodes in the game tree. We also explored some mathematical features of Hex and provided proofs of them. 1 Introduction Building computers to play board games has been a focus for mathematicians ever since computers were invented. The first computer to beat a human opponent in chess was built in 1956, and towards the late 1960s, computers were already beating chess players of low-medium skill.[1] Now, it is generally recognized that computers can beat even the most accomplished grandmaster. Computers build a tree of different possibilities for the game and then work backwards to find the move that will give the computer the best outcome. Al- though computers can evaluate board positions very quickly, in a game like chess where there are over 10120 possible board configurations it is impossible for a computer to search through the entire tree. The challenge for the computer is then to find ways to avoid searching in certain parts of the tree. Humans also look ahead a certain number of moves when playing a game, but experienced players already know of certain theories and strategies that tell them which parts of the tree to look. -
Impartial Games
Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society. -
Combinatorial Game Theory Théorie Des Jeux Combinatoires (Org
Combinatorial Game Theory Théorie des jeux combinatoires (Org: Melissa Huggan (Ryerson), Svenja Huntemann (Concordia University of Edmonton) and/et Richard Nowakowski (Dalhousie)) ALEXANDER CLOW, St. Francis Xavier University Red, Blue, Green Poset Games This talk examines Red, Blue, Green (partizan) poset games under normal play. Poset games are played on a poset where players take turns choosing an element of the partial order and removing every element greater than or equal to it in the ordering. The Left player can choose Blue elements (Right cannot) and the Right player can choose Red elements (while the Left cannot) and both players can choose Green elements. Red, Blue and Red, Blue, Green poset games have not seen much attention in the literature, do to most questions about Green poset games (such as CHOMP) remaining open. We focus on results that are true of all Poset games, but time allowing, FENCES, the poset game played on fences (or zig-zag posets) will be considered. This is joint work with Dr.Neil McKay. MATTHIEU DUFOUR AND SILVIA HEUBACH, UQAM & California State University Los Angeles Circular Nim CN(7,4) Circular Nim CN(n; k) is a variation on Nim. A move consists of selecting k consecutive stacks from n stacks arranged in a circle, and then to remove at least one token (and as many as all tokens) from the selected stacks. We will briefly review known results on Circular Nim CN(n; k) for small values of n and k and for some families, and then discuss new features that have arisen in the set of the P-positions of CN(7,4). -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
HEXAFLEXAGONS, PROBABILITY PARADOXES, AND THE TOWER OF HANOI For 25 of his 90 years, Martin Gard- ner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of math- ematics. He has also made signifi- cant contributions to magic, philos- ophy, debunking pseudoscience, and children’s literature. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than a million copies. He continues to write a regular column for the Skeptical Inquirer magazine. (The photograph is of the author at the time of the first edition.) THE NEW MARTIN GARDNER MATHEMATICAL LIBRARY Editorial Board Donald J. Albers, Menlo College Gerald L. Alexanderson, Santa Clara University John H. Conway, F.R. S., Princeton University Richard K. Guy, University of Calgary Harold R. Jacobs Donald E. Knuth, Stanford University Peter L. Renz From 1957 through 1986 Martin Gardner wrote the “Mathematical Games” columns for Scientific American that are the basis for these books. Scientific American editor Dennis Flanagan noted that this column contributed substantially to the success of the magazine. The exchanges between Martin Gardner and his readers gave life to these columns and books. These exchanges have continued and the impact of the columns and books has grown. These new editions give Martin Gardner the chance to bring readers up to date on newer twists on old puzzles and games, on new explanations and proofs, and on links to recent developments and discoveries. -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first.