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A Combinatorial Game Theoretic Analysis of Chess Endgames
A COMBINATORIAL GAME THEORETIC ANALYSIS OF CHESS ENDGAMES QINGYUN WU, FRANK YU,¨ MICHAEL LANDRY 1. Abstract In this paper, we attempt to analyze Chess endgames using combinatorial game theory. This is a challenge, because much of combinatorial game theory applies only to games under normal play, in which players move according to a set of rules that define the game, and the last player to move wins. A game of Chess ends either in a draw (as in the game above) or when one of the players achieves checkmate. As such, the game of chess does not immediately lend itself to this type of analysis. However, we found that when we redefined certain aspects of chess, there were useful applications of the theory. (Note: We assume the reader has a knowledge of the rules of Chess prior to reading. Also, we will associate Left with white and Right with black). We first look at positions of chess involving only pawns and no kings. We treat these as combinatorial games under normal play, but with the modification that creating a passed pawn is also a win; the assumption is that promoting a pawn will ultimately lead to checkmate. Just using pawns, we have found chess positions that are equal to the games 0, 1, 2, ?, ", #, and Tiny 1. Next, we bring kings onto the chessboard and construct positions that act as game sums of the numbers and infinitesimals we found. The point is that these carefully constructed positions are games of chess played according to the rules of chess that act like sums of combinatorial games under normal play. -
Touring Problems – a Mathematical Approach
TOURING PROBLEMS – A MATHEMATICAL APPROACH VIPUL NAIK Abstract. The article surveys “touring”, a problem of “fun” math, and employs serious techniques to attack it. The text is suitable for high school students interested in Olympiad mathematics, as well as undergraduate students studying combinatorics and discrete math- ematics. It can be used to introduce the concept of a graph. A sequel to this article, on a similar analysis for “tiling” problems, shall also appear soon. 1. Prebeginnings 1.1. The knight’s tour. Chess pieces, with their variously defined moves, are a rich source of puzzles in recreational mathematics. Chess puzzles include the following types : • Nonattacking pieces puzzle : Given a piece with some moves defined, place as many copies of this piece as possible so that no two can kill each other. The proto- typical example of this is the eight queens problem – placing eight queens on the chessboard so that no two can kill each other. • Board domination puzzle : Given a piece with some moves defined, place as few copies of the piece as possible so that every square is either covered by some copy or is under direct attack by a copy of the piece. • Touring puzzle : Given a piece with some moves defined, construct a tour of the entire chessboard with that piece. The only nontrivial case of this for a standard chess piece is the knight’s tour. Each of these problems revolves around a piece, with certain moves defined for that piece. Instead of standard chess pieces, we may sometimes be interested in defining chess pieces with somewhat different moves. -
UIL Text 111212
UIL Chess Puzzle Solvin g— Fall/Winter District 2016-2017 —Grades 4 and 5 IMPORTANT INSTRUCTIONS: [Test-administrators, please read text in this box aloud.] This is the UIL Chess Puzzle Solving Fall/Winter District Test for grades four and five. There are 20 questions on this test. You have 30 minutes to complete it. All questions are multiple choice. Use the answer sheet to mark your answers. Multiple choice answers pur - posely do not indicate check, checkmate, or e.p. symbols. You will be awarded one point for each correct answer. No deductions will be made for incorrect answers on this test. Finishing early is not rewarded, even to break ties. So use all of your time. Some of the questions may be hard, but all of the puzzles are interesting! Good luck and have fun! If you don’t already know chess notation, reading and referring to the section below on this page will help you. How to read and answer questions on this test Piece Names Each chessman can • To answer the questions on this test, you’ll also be represented need to know how to read chess moves. It’s by a symbol, except for the pawn. simple to do. (Figurine Notation) K King Q • Every square on the board has an “address” Queen R made up of a letter and a number. Rook B Bishop N Knight Pawn a-h (We write the file it’s on.) • To make them easy to read, the questions on this test use the figurine piece symbols on the right, above. -
Fundamental Endings CYRUS LAKDAWALA
First Steps : Fundamental Endings CYRUS LAKDAWALA www.everymanchess.com About the Author Cyrus Lakdawala is an International Master, a former National Open and American Open Cham- pion, and a six-time State Champion. He has been teaching chess for over 30 years, and coaches some of the top junior players in the U.S. Also by the Author: Play the London System A Ferocious Opening Repertoire The Slav: Move by Move 1...d6: Move by Move The Caro-Kann: Move by Move The Four Knights: Move by Move Capablanca: Move by Move The Modern Defence: Move by Move Kramnik: Move by Move The Colle: Move by Move The Scandinavian: Move by Move Botvinnik: Move by Move The Nimzo-Larsen Attack: Move by Move Korchnoi: Move by Move The Alekhine Defence: Move by Move The Trompowsky Attack: Move by Move Carlsen: Move by Move The Classical French: Move by Move Larsen: Move by Move 1...b6: Move by Move Bird’s Opening: Move by Move Petroff Defence: Move by Move Fischer: Move by Move Anti-Sicilians: Move by Move Opening Repertoire ... c6 First Steps: the Modern 3 Contents About the Author 3 Bibliography 5 Introduction 7 1 Essential Knowledge 9 2 Pawn Endings 23 3 Rook Endings 63 4 Queen Endings 119 5 Bishop Endings 144 6 Knight Endings 172 7 Minor Piece Endings 184 8 Rooks and Minor Pieces 206 9 Queen and Other Pieces 243 4 Introduction Why Study Chess at its Cellular Level? A chess battle is no less intense for its lack of brevity. Because my messianic mission in life is to make the chess board a safer place for students and readers, I break the seal of confessional and tell you that some students consider the idea of enjoyable endgame study an oxymoron. -
Chess Endgame News
Chess Endgame News Article Published Version Haworth, G. (2014) Chess Endgame News. ICGA Journal, 37 (3). pp. 166-168. ISSN 1389-6911 Available at http://centaur.reading.ac.uk/38987/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Publisher: The International Computer Games Association All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online 166 ICGA Journal September 2014 CHESS ENDGAME NEWS G.McC. Haworth1 Reading, UK This note investigates the recently revived proposal that the stalemated side should lose, and comments further on the information provided by the FRITZ14 interface to Ronald de Man’s DTZ50 endgame tables (EGTs). Tables 1 and 2 list relevant positions: data files (Haworth, 2014b) provide chess-line sources and annotation. Pos.w-b Endgame FEN Notes g1 3-2 KBPKP 8/5KBk/8/8/p7/P7/8/8 b - - 34 124 Korchnoi - Karpov, WCC.5 (1978) g2 3-3 KPPKPP 8/6p1/5p2/5P1K/4k2P/8/8/8 b - - 2 65 Anand - Kramnik, WCC.5 (2007) 65. … Kxf5 g3 3-2 KRKRB 5r2/8/8/8/8/3kb3/3R4/3K4 b - - 94 109 Carlsen - van Wely, Corus (2007) 109. … Bxd2 == g4 7-7 KQR..KQR.. 2Q5/5Rpk/8/1p2p2p/1P2Pn1P/5Pq1/4r3/7K w Evans - Reshevsky, USC (1963), 49. -
Multilinear Algebra and Chess Endgames
Games of No Chance MSRI Publications Volume 29, 1996 Multilinear Algebra and Chess Endgames LEWIS STILLER Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for high-performance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute using earlier techniques, including a win requiring a record 243 moves. (3) To contribute to the study of the history of chess endgames, by focusing on the work of Friedrich Amelung (in particular his apparently lost analysis of certain six-piece endgames) and that of Theodor Molien, one of the founders of modern group representation theory and the first person to have systematically numerically analyzed a pawnless endgame. 1. Introduction Parallel and vector architectures can achieve high peak bandwidth, but it can be difficult for the programmer to design algorithms that exploit this bandwidth efficiently. Application performance can depend heavily on unique architecture features that complicate the design of portable code [Szymanski et al. 1994; Stone 1993]. The work reported here is part of a project to explore the extent to which the techniques of multilinear algebra can be used to simplify the design of high- performance parallel and vector algorithms [Johnson et al. 1991]. The approach is this: Define a set of fixed, structured matrices that encode architectural primitives • of the machine, in the sense that left-multiplication of a vector by this matrix is efficient on the target architecture. Formulate the application problem as a matrix multiplication. -
Chess-Training-Guide.Pdf
Q Chess Training Guide K for Teachers and Parents Created by Grandmaster Susan Polgar U.S. Chess Hall of Fame Inductee President and Founder of the Susan Polgar Foundation Director of SPICE (Susan Polgar Institute for Chess Excellence) at Webster University FIDE Senior Chess Trainer 2006 Women’s World Chess Cup Champion Winner of 4 Women’s World Chess Championships The only World Champion in history to win the Triple-Crown (Blitz, Rapid and Classical) 12 Olympic Medals (5 Gold, 4 Silver, 3 Bronze) 3-time US Open Blitz Champion #1 ranked woman player in the United States Ranked #1 in the world at age 15 and in the top 3 for about 25 consecutive years 1st woman in history to qualify for the Men’s World Championship 1st woman in history to earn the Grandmaster title 1st woman in history to coach a Men's Division I team to 7 consecutive Final Four Championships 1st woman in history to coach the #1 ranked Men's Division I team in the nation pnlrqk KQRLNP Get Smart! Play Chess! www.ChessDailyNews.com www.twitter.com/SusanPolgar www.facebook.com/SusanPolgarChess www.instagram.com/SusanPolgarChess www.SusanPolgar.com www.SusanPolgarFoundation.org SPF Chess Training Program for Teachers © Page 1 7/2/2019 Lesson 1 Lesson goals: Excite kids about the fun game of chess Relate the cool history of chess Incorporate chess with education: Learning about India and Persia Incorporate chess with education: Learning about the chess board and its coordinates Who invented chess and why? Talk about India / Persia – connects to Geography Tell the story of “seed”. -
CHESS INFORMANT Contain
Drawing by Bob Brandreth , The , E"..-y ail: mono. the Yuqoalav ChI .. Federation brings out a Dew book of the tin.. 1 gom.. plared dwinq the preceding baH y.ar. A unique. Dewly-deviled aystem of annotating gwu_ by coded ligna moida all languuge obetcd... 1'Ju. malt. possible a univeraally usable and yet V'Osonably-priced book which brings the neweat ideaa in the opening,; and throughout the game to every ch.. enthusiast more quickly U"m ever before. Book 6 confaina 821 gam.. played between July 1 and D.cember 31, 1968. A qreat aelectiOD of theoretically important gam_ from 28 toumcnrumta and match.. , inc1uding the Lugano Olympiad. World Student Team Cbmnpionsbip (Ybb.), Mar del Plata. Netanya, Amaterdam. Skopje, Debrecen, Sombot. Havana. Vinkovci, Belgrade, Palma d. Majorca, and Athens, S.pacial New Featurel Beginning with Book 6. each CHESS INFORMANT contain. a aection for FIDE communicati0D8, re placing the former official publication FIDE REVIEW. The FIDE section in this iau. contains comple'e Regu1ationa for the Toumamenta and Match BII for the Men'. and l.cdl·,' World CbampiC'Dlhipa. Pr.. crih n the entire competition .,atem from Zonal cmd Interzonal Toummnenta throuqb the Ccmdidatea Matches to the World Championship Match. Book 6 has aections leaturing 51 brilliant Combinations and 45 Endings from actual play during the preceding six months. Another interesting feature ia a table listing in Older the Ten Beat Gam,ea from Book 5 and showing how each of the eight Grandmastem on the jury voted. Contains an Engliah·lanquage introduction. esplanation of the annotation cod•• indez of play em and comm._tcrton. -
Finding Checkmate in N Moves in Chess Using Backtracking And
Finding Checkmate in N moves in Chess using Backtracking and Depth Limited Search Algorithm Moch. Nafkhan Alzamzami 13518132 Informatics Engineering School of Electrical Engineering and Informatics Institute Technology of Bandung, Jalan Ganesha 10 Bandung [email protected] [email protected] Abstract—There have been many checkmate potentials that has pieces of a dark color. There are rules about how pieces players seem to miss. Players have been training for finding move, and about taking the opponent's pieces off the board. checkmates through chess checkmate puzzles. Such problems can The player with white pieces always makes the first move.[4] be solved using a simple depth-limited search and backtracking Because of this, White has a small advantage, and wins more algorithm with legal moves as the searching paths. often than Black in tournament games.[5][6] Keywords—chess; checkmate; depth-limited search; Chess is played on a square board divided into eight rows backtracking; graph; of squares called ranks and eight columns called files, with a dark square in each player's lower left corner.[8] This is I. INTRODUCTION altogether 64 squares. The colors of the squares are laid out in There have been many checkmate potentials that players a checker (chequer) pattern in light and dark squares. To make seem to miss. Players have been training for finding speaking and writing about chess easy, each square has a checkmates through chess checkmate puzzles. Such problems name. Each rank has a number from 1 to 8, and each file a can be solved using a simple depth-limited search and letter from a to h. -
British Endgame Study News Volume 15 Number 3 Septernber 2010
British Endgame Study News Volume 15 Number 3 Septernber 2010 Edited and. published by John Beasley, 7 St James Road, Harpenden, Herts AL5 4NX ISSN 1363-0318 E-mdil: [email protected] Contents of this issue Editorial 465 Variations on a theme 466 From the world at large 468 News and notices 4'72 This issue- We have a series of related studies from Paul Michelet, the special number looks at the studies of Jirdfich Fritz, and do try the litde trifle atongside before looking inside, by Richard Becker Index 1996-2010. Next time's final issue will be White to play and win accompanied by a composite index of studies by author covering the whole of BESN. I have prepare.d a draft up to and including the present issue which I like to think is conect, but if some kind reader with a complete run of lhe magazine and time to spare is willing to check it for me I shall be most grateful_ Special number 63. It appears that the modetn Mahi encyklopedie.im,a is wrong, and fhat "Jan" Vaniura was in truth Josef Vaniura. Casopis iesbjch iachisti 1911, page 95, "Jos. Vandura" (from Emil Vlasdk and Jaroslav pol6iek, forwarding information from ZdenEk Zdvodnj); chess column in ieskl s/ovo,2Z.i.l922,..los:' (sent to me by Bedrich Formdnek)l obituary in Casopis ieskoslovenskjch iachisttit 1922, pnge 2l, "losef' (ciied by caige, drawn ro my anenrion by Timorhy Whitworrh, and sent to me by the library in Den Haag), The incorrect rlamc "Jan,' appears to derive from an article by FrantiSek Dedrle in Ceskoslovensky iach 1947. -
Chess & Bridge
2013 Catalogue Chess & Bridge Plus Backgammon Poker and other traditional games cbcat2013_p02_contents_Layout 1 02/11/2012 09:18 Page 1 Contents CONTENTS WAYS TO ORDER Chess Section Call our Order Line 3-9 Wooden Chess Sets 10-11 Wooden Chess Boards 020 7288 1305 or 12 Chess Boxes 13 Chess Tables 020 7486 7015 14-17 Wooden Chess Combinations 9.30am-6pm Monday - Saturday 18 Miscellaneous Sets 11am - 5pm Sundays 19 Decorative & Themed Chess Sets 20-21 Travel Sets 22 Giant Chess Sets Shop online 23-25 Chess Clocks www.chess.co.uk/shop 26-28 Plastic Chess Sets & Combinations or 29 Demonstration Chess Boards www.bridgeshop.com 30-31 Stationery, Medals & Trophies 32 Chess T-Shirts 33-37 Chess DVDs Post the order form to: 38-39 Chess Software: Playing Programs 40 Chess Software: ChessBase 12` Chess & Bridge 41-43 Chess Software: Fritz Media System 44 Baker Street 44-45 Chess Software: from Chess Assistant 46 Recommendations for Junior Players London, W1U 7RT 47 Subscribe to Chess Magazine 48-49 Order Form 50 Subscribe to BRIDGE Magazine REASONS TO SHOP ONLINE 51 Recommendations for Junior Players - New items added each and every week 52-55 Chess Computers - Many more items online 56-60 Bargain Chess Books 61-66 Chess Books - Larger and alternative images for most items - Full descriptions of each item Bridge Section - Exclusive website offers on selected items 68 Bridge Tables & Cloths 69-70 Bridge Equipment - Pay securely via Debit/Credit Card or PayPal 71-72 Bridge Software: Playing Programs 73 Bridge Software: Instructional 74-77 Decorative Playing Cards 78-83 Gift Ideas & Bridge DVDs 84-86 Bargain Bridge Books 87 Recommended Bridge Books 88-89 Bridge Books by Subject 90-91 Backgammon 92 Go 93 Poker 94 Other Games 95 Website Information 96 Retail shop information page 2 TO ORDER 020 7288 1305 or 020 7486 7015 cbcat2013_p03to5_woodsets_Layout 1 02/11/2012 09:53 Page 1 Wooden Chess Sets A LITTLE MORE INFORMATION ABOUT OUR CHESS SETS.. -
Usborne Chess Puzzle Answers – Level 2 to Print out This Answer Sheet, Click on ‘File’ and Then ‘Print’ in the Menu at the Top of Your Browser
Usborne Chess Puzzle Answers – level 2 To print out this answer sheet, click on ‘File’ and then ‘Print’ in the menu at the top of your browser. 1. Black should play 1…Bc8, so the Bishop covers the crucial a6–c8 diagonal and stops any of White’s Pawns from advancing. 2. 1…Be2+ forks the King and the Knight.When the King moves out of check, Black can take the Knight. 3.The best move is 1.Ra5+, which will capture the Rook on h5 after the King moves out of check. 1.Rxa6 wins only a Pawn and 1.Re2+ captures the e8 Knight with a skewer attack. 4. 1.Qg7++. Note that if White plays 1.Qa8+, Black blocks with 1…Bf8 and the game continues. 5. 1.Bf6++. 6. 1…Nd3++.The Pawn on e2 is pinned by the black Queen so the white King is trapped in a smothered mate. 7. 1…Nh3+ forces White to play 2.Kh1. Black can then play 2…Bb7++. 8. Black can play 1…Nf3+, shielding the King from check, and at the same time, checking White so that White’s next move, 2.Kh1, is forced. (White’s Queen is pinned by the Rook on a1 and cannot capture the Knight.) Black can now play 2…Rxh2++. 9. If Black plays 1…Bh3,White cannot prevent 2…Qg2++. 10. 1…Nf3+, 2.Kh1 Qxh2++ or 1…Nh3+, 2.Kh1 Bxg2++. 11. 1.Ba7+ Ka8, 2.Qc8++ or 1.Ra8+ Kxa8 (the Rook is sacrificed), 2.Qc8++. 12. 1.Rxa7+ Kxa7, 2.Qa5++.