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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. -
CONTENTS Contents
CONTENTS Contents Symbols 5 Preface 6 Introduction 9 1 Glossary of Attacking and Strategic Terms 11 2 Double Attack 23 2.1: Double Attacks with Queens and Rooks 24 2.2: Bishop Forks 31 2.3: Knight Forks 34 2.4: The Í+Ì Connection 44 2.5: Pawn Forks 45 2.6: The Discovered Double Attack 46 2.7: Another Type of Double Attack 53 Exercises 55 Solutions 61 3 The Role of the Pawns 65 3.1: Pawn Promotion 65 3.2: The Far-Advanced Passed Pawn 71 3.3: Connected Passed Pawns 85 3.4: The Pawn-Wedge 89 3.5: Passive Sacrifices 91 3.6: The Kamikaze Pawn 92 Exercises 99 Solutions 103 4 Attacking the Castled Position 106 4.1: Weakness in the Castled Position 106 4.2: Rooks and Files 112 4.3: The Greek Gift 128 4.4: Other Bishop Sacrifices 133 4.5: Panic on the Long Diagonal 143 4.6: The Knight Sacrifice 150 4.7: The Exchange Sacrifice 162 4.8: The Queen Sacrifice 172 Exercises 176 Solutions 181 5 Drawing Combinations 186 5.1: Perpetual Check 186 5.2: Repetition of Position 194 5.3: Stalemate 197 5.4: Fortress and Blockade 202 5.5: Positional Draws 204 Exercises 207 Solutions 210 6 Combined Tactical Themes 213 6.1: Material, Endings, Zugzwang 214 6.2: One Sacrifice after Another 232 6.3: Extraordinary Combinations 242 6.4: A Diabolical Position 257 Exercises 260 Solutions 264 7 Opening Disasters 268 7.1: Open Games 268 7.2: Semi-Open Games 274 7.3: Closed Games 288 8 Tactical Examination 304 Test 1 306 Test 2 308 Test 3 310 Test 4 312 Test 5 314 Test 6 316 Hints 318 Solutions 320 Index of Names 331 Index of Openings 335 THE ROLE OF THE PAWNS 3 The Role of the Pawns Ever since the distant days of the 18th century 3.1: Pawn Promotion (let us call it the time of the French Revolution, or of François-André Danican Philidor) we have known that “pawns are the soul of chess”. -
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. -
Do First Mover Advantages Exist in Competitive Board Games: the Importance of Zugzwang
DO FIRST MOVER ADVANTAGES EXIST IN COMPETITIVE BOARD GAMES: THE IMPORTANCE OF ZUGZWANG Douglas L. Micklich Illinois State University [email protected] ABSTRACT to the other player(s) in the game (Zagal, et.al., 2006) Examples of such games are chess and Connect-Four. The players try to The ability to move first in competitive games is thought to be secure some sort of first-mover advantage in trying to attain the sole determinant on who wins the game. This study attempts some advantage of position from which a lethal attack can be to show other factors which contribute and have a non-linear mounted. The ability to move first in competitive board games effect on the game’s outcome. These factors, although shown to has thought to have resulted more often in a situation where that be not statistically significant, because of their non-linear player, the one moving first, being victorious. The person relationship have some positive correlations to helping moving first will normally try to take control from the outset and determine the winner of the game. force their opponent into making moves that they would not otherwise have made. This is a strategy which Allis refers to as “Zugzwang”, which is the principle of having to play a move INTRODUCTION one would rather not. To be able to ensure that victory through a gained advantage In Allis’s paper “A Knowledge-Based Approach of is attained, a position must first be determined. SunTzu in the Connect-Four: The Game is Solved: White Wins”, the author “Art of War” described position in this manner: “this position, a states that the player of the black pieces can follow strategic strategic position (hsing), is defined as ‘one that creates a rules by which they can at least draw the game provided that the situation where we can use ‘the individual whole to attack our player of the red pieces does not start in the middle column (the rival’s) one, and many to strike a few’ – that is, to win the (Allis, 1992). -
Electronic Funds Transfer Agreement and Disclosure VISA Checkmate and Electronic Services
Electronic Funds Transfer Agreement and Disclosure VISA CheckMate and Electronic Services This Electronic Fund Transfers Agreement and Disclosure is the contract which covers your and our rights and responsibilities concerning the electronic fund transfers (EFT) services offered to you by University of Virginia Community Credit Union (“Credit Union”). In this Agreement, the words “you,” “your,” and “yours” mean those who sign the application or account card as applicants, joint owners, or any authorized users. The words “we,” “us,” and “our” mean the Credit Union. The word “account” means any one (1) or more share and checking accounts you have with the Credit Union. Electronic fund transfers are electronically initiated transfers of money from your account through the EFT services described below. By signing an application or account card for EFT services, signing your card, or using any service, each of you, jointly and severally, agree to the terms and conditions in this Agreement and any amendments for the EFT services offered. Furthermore, electronic fund transfers that meet the definition of remittance transfers are governed by 12 C.F.R. part 1005, subpart B—Requirements for remittance transfers, and consequently, terms of this agreement may vary for those types of transactions. A “remittance transfer” is an electronic transfer of funds of more than $15.00 which is requested by a sender and sent to a designated recipient in a foreign country by a remittance transfer provider. Terms applicable to such transactions may vary from those disclosed herein and will be disclosed to you at the time such services are requested and rendered in accordance with applicable law. -
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. -
Dvoretsky Lessons 12
The Instructor Tragicomedies in Pawn Endgames “Pawn endgames are rare birds in practice. Players avoid them, because they do not like them, because they do not understand them. It’s certainly no secret that pawn endings are ‘terra incognita’ - even for many masters, right up to the level of grandmasters and world champions.” N. Grigoriev Herewith, I offer proof that these words, spoken by a famous expert on pawn endings, are true. Without commentary, I give below the final moves of some actual games, and offer the readers the chance to comment on them, to uncover all the mistakes committed by both players. The endgames you will be dealing with here are not all that difficult; but still, the players on both sides have The provided you with plenty of opportunities for critical commentary. 1...Kf8 2. Qf5+ Qxf5 3. gf Kg7 4. c4 f3 5. h6+ Kxh6 6. c5 dc 7. f6 Kg6 Instructor White resigned. Mark Dvoretsky 1...g5 2. Kf3 Kd5 3. c6 Kd6 4. Ke4 a6 5. ba Kxc6 6. Kf3 Kb6 7. h4 gh 8. Kg4 Kxa6 9. Kxh4 Kb6 10. Kg4 Kc6 11. h4 Kd6. White resigned. file:///C|/Cafe/Dvoretsky/dvoretsky.htm (1 of 9) [9/11/2001 7:09:02 AM] The Instructor 1. Kh7 Kf7 2. Kh8 Kf8 3. g5. Black resigned. 1. Kg5 Kf8 2. Kxf5 Kf7 3. Kg4 Kf6 4. Kf4 Kf7 5. Kf5 Ke7 6. Ke5 Kf7 7. Kd6 Kf6 8. Kd7 Kf7 9. h6 Kg6 10. f4 Kf7 11. f5 Kf6 Drawn Gazic - Petursson European Junior Championship, Groningen 1978/79 The draw is obvious after 1...Kh8! Black mistakenly allowed the trade of queens. -
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. -
Virginia Chess Federation 2008 - #6
VIRGINIA CHESS Newsletter The bimonthly publication of the Virginia Chess Federation 2008 - #6 Grandmaster Larry Kaufman See page 1 VIRGINIA CHESS Newsletter 2008 - Issue #6 Editor: Circulation: Macon Shibut Ernie Schlich 8234 Citadel Place 1370 South Braden Crescent Vienna VA 22180 Norfolk VA 23502 [email protected] [email protected] k w r Virginia Chess is published six times per year by the Virginia Chess Federation. Membership benefits (dues: $10/yr adult; $5/yr junior under 18) include a subscription to Virginia Chess. Send material for publication to the editor. Send dues, address changes, etc to Circulation. The Virginia Chess Federation (VCF) is a non-profit organization for the use of its members. Dues for regular adult membership are $10/yr. Junior memberships are $5/yr. President: Mike Hoffpauir, 405 Hounds Chase, Yorktown VA 23693, mhoffpauir@ aol.com Treasurer: Ernie Schlich, 1370 South Braden Crescent, Norfolk VA 23502, [email protected] Secretary: Helen Hinshaw, 3430 Musket Dr, Midlothian VA 23113, jallenhinshaw@comcast. net Scholastics Coordinator: Mike Hoffpauir, 405 Hounds Chase, Yorktown VA 23693, [email protected] VCF Inc. Directors: Helen Hinshaw (Chairman), Rob Getty, John Farrell, Mike Hoffpauir, Ernie Schlich. otjnwlkqbhrp 2008 - #6 1 otjnwlkqbhrp Larry Kaufman, of Maryland, is a familiar face at Virginia tournaments. Among others he won the Virginia Open in 1969, 1998, 2000, 2006 and 2007! Recently Larry achieved a lifelong goal by attaining the title of International Grandmaster, and agreed to tell VIRGINIA CHESS readers how it happened. -ed World Senior Chess Championship by Larry Kaufman URING THE LAST FIVE YEARS OR SO, whenever someone asked me Dif I still hoped to become a GM, I would reply something like this: “I’m too old now to try to do it the normal way, but perhaps when I reach 60 I will try to win the World Senior, which carries an automatic GM title. -
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.. -
CHECKMATE PHARMACEUTICALS, INC. (Exact Name of Registrant As Specified in Its Charter)
UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event reported): March 29, 2021 CHECKMATE PHARMACEUTICALS, INC. (Exact name of registrant as specified in its charter) Delaware 001-39425 36-4813934 (State or other jurisdiction (Commission (I.R.S. Employer of incorporation) File Number) Identification No.) Checkmate Pharmaceuticals, Inc. 245 Main Street, 2nd Floor Cambridge, MA 02142 (Address of principal executive offices, including zip code) (617) 682-3625 (Registrant’s telephone number, including area code) Not Applicable (Former Name or Former Address, if Changed Since Last Report) Check the appropriate box below if the Form 8-K filing is intended to simultaneously satisfy the filing obligation of the registrant under any of the following provisions: ☐ Written communications pursuant to Rule 425 under the Securities Act (17 CFR 230.425) ☐ Soliciting material pursuant to Rule 14a-12 under the Exchange Act (17 CFR 240.14a-12) ☐ Pre-commencement communications pursuant to Rule 14d-2(b) under the Exchange Act (17 CFR 240.14d-2(b)) ☐ Pre-commencement communications pursuant to Rule 13e-4(c) under the Exchange Act (17 CFR 240.13e-4(c)) Securities registered pursuant to Section 12(b) of the Act: Trade Name of each exchange Title of each class Symbol(s) on which registered Common Stock, $0.0001 par value per share CMPI The Nasdaq Global Market Indicate by check mark whether the registrant is an emerging growth company as defined in Rule 405 of the Securities Act of 1933 (§ 230.405 of this chapter) or Rule 12b-2 of the Securities Exchange Act of 1934 (§ 240.12b-2 of this chapter).