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  • GAME

    A game is a structured type of play, usually undertaken for entertainment or fun, and sometimes used as an educational tool.[1] Many games are also considered to be work (such as professional players of spectator sports or games) or art (such as jigsaw puzzles or games involving an artistic layout such as mahjong, solitaire, or some video games).

    Games have a wide range of occasions, reflecting both the generality of its concept and the variety of its play.[2] Games are sometimes played purely for enjoyment, sometimes for achievement or reward as well. They can be played alone, in teams, or online; by amateurs or by professionals. The players may have an audience of non-players, such as when people are entertained by watching a chess championship. On the other hand, players in a game may constitute their own audience as they take their turn to play. Often, part of the entertainment for children playing a game is deciding who is part of their audience and who participates as a player. A toy and a game are not the same. Toys generally allow for unrestricted play, whereas games present rules for the player to follow.

    Key components of games are goals, rules, challenge, and interaction. Games generally involve mental or physical stimulation, and often both. Many games help develop practical skills, serve as a form of exercise, or otherwise perform an educational, simulational, or psychological role.

    Attested as early as 2600 BC,[3][4] games are a universal part of human experience and present in all cultures. The Royal Game of Ur, Senet, and Mancala are some of the oldest known games.[5]

    Definitions

    Look up game in Wiktionary, the free dictionary.

    Ludwig Wittgenstein

    Ludwig Wittgenstein is well known in the history of philosophy for having addressed the definition of the word game. In his Philosophical Investigations,[6] Wittgenstein argued that the elements of games, such as play, rules, and competition, all fail to adequately define what games are. From this, Wittgenstein concluded that people apply the term game to a range of disparate human activities that bear to one another only what one might call family resemblances. As the following game definitions show, this conclusion was not a final one, and today many philosophers, like Thomas Hurka, think that Wittgenstein was wrong and that Bernard Suits’ definition is a good answer to the problem.[7][2]

    Roger Caillois

    French sociologist Roger Caillois, in his book Les jeux et les hommes (Games and Men)(1961),[8] defined a game as an activity that must have the following characteristics:

    • fun: the activity is chosen for its light-hearted character
    • separate: it is circumscribed in time and place
    • uncertain: the outcome of the activity is unforeseeable
    • non-productive: participation does not accomplish anything useful
    • governed by rules: the activity has rules that are different from everyday life
    • fictitious: it is accompanied by the awareness of a different reality

    Chris Crawford

    Game designer Chris Crawford defined the term in the context of computers.[9] Using a series of dichotomies:

    1. Creative expression is art if made for its own beauty and entertainment if made for money.
    2. A piece of entertainment is a plaything if it is interactive. Movies and books are cited as examples of non-interactive entertainment.
    3. If no goals are associated with a plaything, it is a toy. (Crawford notes that by his definition, (a) a toy can become a game element if the player makes up rules, and (b) The Sims and SimCity are toys, not games.) If it has goals, a plaything is a challenge.
    4. If a challenge has no “active agent against whom you compete,” it is a puzzle; if there is one, it is a conflict. (Crawford admits that this is a subjective test. Video games with noticeably algorithmic artificial intelligence can be played as puzzles; these include the patterns used to evade ghosts in Pac-Man.)
    5. Finally, if the player can only outperform the opponent but not attack them to interfere with their performance, the conflict is a competition. (Competitions include racing and figure skating.) However, if attacks are allowed, then the conflict qualifies as a game.

    Crawford’s definition may thus be rendered as an interactive, goal-oriented activity made for money, with active agents to play against, in which players (including active agents) can interfere with each other.

    Other definitions, however, as well as history, show that entertainment and games are not necessarily undertaken for monetary gain.

    Other definitions

    • “My conclusion is that to play a game is to engage in activity directed towards bringing about a specific state of affairs, using only means permitted by rules, where the rules prohibit more efficient in favour of less efficient means, and where such rules are accepted just because they make possible such activity.” Bernard Suits[2]
    • “A game is a form of art in which participants, termed players, make decisions in order to manage resources through game tokens in the pursuit of a goal.” (Greg Costikyan)[10] According to this definition, some “games” that do not involve choices, such as Chutes and Ladders, Candy Land, and War are not technically games any more than a slot machine is.
    • “A game is a form of play with goals and structure.” (Kevin J. Maroney)[11]
    • “A game is a system in which players engage in an artificial conflict, defined by rules, that results in a quantifiable outcome.” (Katie Salen and Eric Zimmerman)[12]
    • “A game is an activity among two or more independent decision-makers seeking to achieve their objectives in some limiting context.” (Clark C. Abt)[13]
    • “At its most elementary level then we can define game as an exercise of voluntary control systems in which there is an opposition between forces, confined by a procedure and rules in order to produce a disequilibrial outcome.” (Elliot Avedon and Brian Sutton-Smith)[14]
    • “To play a game is to engage in activity directed toward bringing about a specific state of affairs, using only means permitted by specific rules, where the means permitted by the rules are more limited in scope than they would be in the absence of the rules, and where the sole reason for accepting such limitation is to make possible such activity.” (Bernard Suits)[15]
    • “When you strip away the genre differences and the technological complexities, all games share four defining traits: a goal, rules, a feedback system, and voluntary participation.” (Jane McGonigal)[16]

    Gameplay elements and classification

    Games can be characterized by “what the player does”.[9] This is often referred to as gameplay. Major key elements identified in this context are tools and rules that define the overall context of game.

    Tools

    A selection of pieces from different games. From top: Chess pawns, marbles, Monopoly tokens, dominoes, Monopoly hotels, jacks and checkers pieces.

    Games are often classified by the components required to play them (e.g., miniatures, a ball, cards, a board and pieces, or a computer). In places where the use of leather is well-established, the ball has been a popular game piece throughout recorded history, resulting in a worldwide popularity of ball games such as rugby, basketball, soccer (football), cricket, tennis, and volleyball. Other tools are more idiosyncratic to a certain region. Many countries in Europe, for instance, have unique standard decks of playing cards. Other games, such as chess, may be traced primarily through the development and evolution of their game pieces.

    Many game tools are tokens, meant to represent other things. A token may be a pawn on a board, play money, or an intangible item such as a point scored.

    Games such as hide-and-seek or tag do not use any obvious tool; rather, their interactivity is defined by the environment. Games with the same or similar rules may have different gameplay if the environment is altered. For example, hide-and-seek in a school building differs from the same game in a park; an auto race can be radically different depending on the track or street course, even with the same cars.

    Rules and aims

    Games are often characterized by their tools and rules. While rules are subject to variations and changes, enough change in the rules usually results in a “new” game. For instance, baseball can be played with “real” baseballs or with wiffleballs. However, if the players decide to play with only three bases, they are arguably playing a different game. There are exceptions to this in that some games deliberately involve the changing of their own rules, but even then there are often immutable meta-rules.

    Rules generally determine the time-keeping system, the rights and responsibilities of the players, scoring techniques, preset boundaries, and each player’s goals.

    The rules of a game may be distinguished from its aims.[17][18] For most competitive games, the ultimate aim is winning: in this sense, checkmate is the aim of chess.[19] Common win conditions are being the first to amass a certain quota of points or tokens (as in Settlers of Catan), having the greatest number of tokens at the end of the game (as in Monopoly), or some relationship of one’s game tokens to those of one’s opponent (as in chess’s checkmate). There may also be intermediate aims, which are tasks that move a player toward winning. For instance, an intermediate aim in football is to score goals, because scoring goals will increase one’s likelihood of winning the game, but is not alone sufficient to win the game.

    An aim identifies a sufficient condition for successful action, whereas the rule identifies a necessary condition for permissible action.[18] For example, the aim of chess is to checkmate, but although it is expected that players will try to checkmate each other, it is not a rule of chess that a player must checkmate the other player whenever possible. Similarly, it is not a rule of football that a player must score a goal on a penalty; while it is expected the player will try, it is not required. While meeting the aims often requires a certain degree of skill and (in some cases) luck, following the rules of a game merely requires knowledge of the rules and some careful attempt to follow them; it rarely (if ever) requires luck or demanding skills.

    Skill, strategy, and chance

    A game’s tools and rules will result in its requiring skill, strategy, luck, or a combination thereof and are classified accordingly.

    Games of skill include games of physical skill, such as wrestling, tug of war, hopscotch, target shooting, and games of mental skill, such as checkers and chess. Games of strategy include checkers, chess, Go, arimaa, and tic-tac-toe, and often require special equipment to play them. Games of chance include gambling games (blackjack, Mahjong, roulette, etc.), as well as snakes and ladders and rock, paper, scissors; most require equipment such as cards or dice. However, most games contain two or all three of these elements. For example, American football and baseball involve both physical skill and strategy, while tiddlywinks, poker, and Monopoly combine strategy and chance. Many card and board games combine all three; most trick-taking games involve mental skill, strategy, and an element of chance, as do many strategic board games such as Risk, Settlers of Catan, and Carcassonne.

    Single-player games

    “Single-player game” redirects here. For single-player video games, see Single-player video game.

    Most games require multiple players. However, single-player games are unique in respect to the type of challenges a player faces. Unlike a game with multiple players competing with or against each other to reach the game’s goal, a one-player game is a battle solely against an element of the environment (an artificial opponent), against one’s own skills, against time, or against chance. Playing with a yo-yo or playing tennis against a wall is not generally recognized as playing a game due to the lack of any formidable opposition. Many games described as “single-player” may be termed actually puzzles or recreations.

    Multiplayer games

    “Multiplayer game” redirects here. For multiplayer video games, see Multiplayer video game.

    The Card Players by Lucas van Leyden (1520) depicting a multiplayer card game

    A multiplayer game is a game of several players who may be independent opponents or teams. Games with many independent players are difficult to analyze formally using game theory as the players may form and switch coalitions.[20] The term “game” in this context may mean either a true game played for entertainment or a competitive activity describable in principle by mathematical game theory.

    Game theory

    Main article: Game theory

    John Nash proved that games with several players have a stable solution provided that coalitions between players are disallowed. Nash won the Nobel prize for economics for this important result which extended von Neumann’s theory of zero-sum games. Nash’s stable solution is known as the Nash equilibrium.[21]

    If cooperation between players is allowed, then the game becomes more complex; many concepts have been developed to analyze such games. While these have had some partial success in the fields of economics, politics and conflict, no good general theory has yet been developed.[21]

    In quantum game theory, it has been found that the introduction of quantum information into multiplayer games allows a new type of equilibrium strategy not found in traditional games. The entanglement of player’s choices can have the effect of a contract by preventing players from profiting from what is known as betrayal.[22]

    Types

    See also: List of game genres

    Tug of war is an easily organized, impromptu game that requires little equipment.

    Games can take a variety of forms, from competitive sports to board games and video games.

    Sports

    Main article: Sport

    Association football is a popular sport worldwide.

    Many sports require special equipment and dedicated playing fields, leading to the involvement of a community much larger than the group of players. A city or town may set aside such resources for the organization of sports leagues.

    Popular sports may have spectators who are entertained just by watching games. A community will often align itself with a local sports team that supposedly represents it (even if the team or most of its players only recently moved in); they often align themselves against their opponents or have traditional rivalries. The concept of fandom began with sports fans.

    Lawn games

    Lawn games are outdoor games that can be played on a lawn, an area of mowed grass (or alternately, on graded soil) generally smaller than a sports field (pitch). Variations of many games that are traditionally played on a sports field are marketed as “lawn games” for home use in a front or back yard. Common lawn games include horseshoes, sholf, croquet, bocce, and lawn bowls.

    Tabletop games

    Main article: Tabletop game

    A tabletop game is a game where the elements of play are confined to a small area and require little physical exertion, usually simply placing, picking up, and moving game pieces. Most of these games are played at a table around which the players are seated and on which the game’s elements are located. However, many games falling into this category, particularly party games, are more free-form in their play and can involve physical activity such as mime. Still, these games do not require a large area in which to play them, large amounts of strength or stamina, or specialized equipment other than what comes in a box.

    Dexterity and coordination games

    This class of games includes any game in which the skill element involved relates to manual dexterity or hand-eye coordination but excludes the class of video games (see below). Games such as jacks, paper football, and Jenga require only very portable or improvised equipment and can be played on any flat level surface, while other examples, such as pinball, billiards, air hockey, foosball, and table hockey, require specialized tables or other self-contained modules on which the game is played. The advent of home video game systems largely replaced some of these, such as table hockey; however, air hockey, billiards, pinball and foosball remain popular fixtures in private and public game rooms. These games and others, as they require reflexes and coordination, are generally performed more poorly by intoxicated persons but are unlikely to result in injury because of this; as such, the games are popular as drinking games. In addition, dedicated drinking games such as quarters and beer pong also involve physical coordination and are popular for similar reasons.

    Board games

    Main article: Board game

    Parcheesi is an American adaptation of a Pachisi, originating in India.

    Board games use as a central tool a board on which the players’ status, resources, and progress are tracked using physical tokens. Many also involve dice or cards. Most games that simulate war are board games (though a large number of video games have been created to simulate strategic combat), and the board may be a map on which the players’ tokens move. Virtually all board games involve “turn-based” play; one player contemplates and then makes a move, then the next player does the same, and a player can only act on their turn. This is opposed to “real-time” play as is found in some card games, most sports and most video games.

    Some games, such as chess and Go, are entirely deterministic, relying only on the strategy element for their interest. Such games are usually described as having “perfect information“; the only unknown is the exact thought processes of one’s opponent, not the outcome of any unknown event inherent in the game (such as a card draw or die roll). Children’s games, on the other hand, tend to be very luck-based, with games such as Candy Land and Chutes and Ladders having virtually no decisions to be made. By some definitions, such as that by Greg Costikyan, they are not games since there are no decisions to make which affect the outcome.[10] Many other games involving a high degree of luck do not allow direct attacks between opponents; the random event simply determines a gain or loss in the standing of the current player within the game, which is independent of any other player; the “game” then is actually a “race” by definitions such as Crawford’s.

    Most other board games combine strategy and luck factors; the game of backgammon requires players to decide the best strategic move based on the roll of two dice. Trivia games have a great deal of randomness based on the questions a person gets. German-style board games are notable for often having rather less of a luck factor than many board games.

    Board game groups include race games, roll-and-move games, abstract strategy games, word games, and wargames, as well as trivia and other elements. Some board games fall into multiple groups or incorporate elements of other genres: Cranium is one popular example, where players must succeed in each of four skills: artistry, live performance, trivia, and language.

    Card games

    Main article: Card game

    Further information: Collectible card game

    Playing Cards, by Theodoor Rombouts, 17th century

    Card games use a deck of cards as their central tool. These cards may be a standard Anglo-American (52-card) deck of playing cards (such as for bridge, poker, Rummy, etc.), a regional deck using 32, 36 or 40 cards and different suit signs (such as for the popular German game skat), a tarot deck of 78 cards (used in Europe to play a variety of trick-taking games collectively known as Tarot, Tarock or Tarocchi games), or a deck specific to the individual game (such as Set or 1000 Blank White Cards). Uno and Rook are examples of games that were originally played with a standard deck and have since been commercialized with customized decks. Some collectible card games such as Magic: The Gathering are played with a small selection of cards that have been collected or purchased individually from large available sets.

    Some board games include a deck of cards as a gameplay element, normally for randomization or to keep track of game progress. Conversely, some card games such as Cribbage use a board with movers, normally to keep score. The differentiation between the two genres in such cases depends on which element of the game is foremost in its play; a board game using cards for random actions can usually use some other method of randomization, while Cribbage can just as easily be scored on paper. These elements as used are simply the traditional and easiest methods to achieve their purpose.

    Dice games

    Main article: Dice game

    Students in Laos using dice to improve numeracy skills. They roll three dice, then use basic math operations to combine those into a new number which they cover on the board. The goal is to cover four squares in the row.

    Dice games use a number of dice as their central element. Board games often use dice for a randomization element, and thus each roll of the dice has a profound impact on the outcome of the game, however dice games are differentiated in that the dice do not determine the success or failure of some other element of the game; they instead are the central indicator of the person’s standing in the game. Popular dice games include Yahtzee, Farkle, Bunco, liar’s dice/Perudo, and poker dice. As dice are, by their very nature, designed to produce random numbers, these games usually involve a high degree of luck, which can be directed to some extent by the player through more strategic elements of play and through tenets of probability theory. Such games are thus popular as gambling games; the game of craps is perhaps the most famous example, though liar’s dice and poker dice were originally conceived of as gambling games.

    Domino and tile games

    Main articles: Tile-based game and Dominoes

    Domino games are similar in many respects to card games, but the generic device is instead a set of tiles called dominoes, which traditionally each have two ends, each with a given number of dots, or “pips”, and each combination of two possible end values as it appears on a tile is unique in the set. The games played with dominoes largely center around playing a domino from the player’s “hand” onto the matching end of another domino, and the overall object could be to always be able to make a play, to make all open endpoints sum to a given number or multiple, or simply to play all dominoes from one’s hand onto the board. Sets vary in the number of possible dots on one end, and thus of the number of combinations and pieces; the most common set historically is double-six, though in more recent times “extended” sets such as double-nine have been introduced to increase the number of dominoes available, which allows larger hands and more players in a game. Muggins, Mexican Train, and Chicken Foot are very popular domino games. Texas 42 is a domino game more similar in its play to a “trick-taking” card game.

    Variations of traditional dominoes abound: Triominoes are similar in theory but are triangular and thus have three values per tile. Similarly, a game known as Quad-Ominos uses four-sided tiles.

    Some other games use tiles in place of cards; Rummikub is a variant of the Rummy card game family that uses tiles numbered in ascending rank among four colors, very similar in makeup to a 2-deck “pack” of Anglo-American playing cards. Mahjong is another game very similar to Rummy that uses a set of tiles with card-like values and art.

    Lastly, some games use graphical tiles to form a board layout, on which other elements of the game are played. Settlers of Catan and Carcassonne are examples. In each, the “board” is made up of a series of tiles; in Settlers of Catan the starting layout is random but static, while in Carcassonne the game is played by “building” the board tile-by-tile. Hive, an abstract strategy game using tiles as moving pieces, has mechanical and strategic elements similar to chess, although it has no board; the pieces themselves both form the layout and can move within it.

    Pencil and paper games

    Main article: Paper-and-pencil game

    Pencil and paper games require little or no specialized equipment other than writing materials, though some such games have been commercialized as board games (Scrabble, for instance, is based on the idea of a crossword puzzle, and tic-tac-toe sets with a boxed grid and pieces are available commercially). These games vary widely, from games centering on a design being drawn such as Pictionary and “connect-the-dots” games like sprouts, to letter and word games such as Boggle and Scattergories, to solitaire and logic puzzle games such as Sudoku and crossword puzzles.

    Guessing games

    Main article: Guessing game

    A guessing game has as its core a piece of information that one player knows, and the object is to coerce others into guessing that piece of information without actually divulging it in text or spoken word. Charades is probably the most well-known game of this type, and has spawned numerous commercial variants that involve differing rules on the type of communication to be given, such as Catch Phrase, Taboo, Pictionary, and similar. The genre also includes many game shows such as Win, Lose or Draw, Password and $25,000 Pyramid.

    Video games

    Main article: Video game

    See also: Electronic game

    Video games are computer- or microprocessor-controlled games. Computers can create virtual spaces for a wide variety of game types. Some video games simulate conventional game objects like cards or dice, while others can simulate environs either grounded in reality or fantastical in design, each with its own set of rules or goals.

    A computer or video game uses one or more input devices, typically a button/joystick combination (on arcade games); a keyboard, mouse or trackball (computer games); or a controller or a motion sensitive tool (console games). More esoteric devices such as paddle controllers have also been used for input.

    There are many genres of video game; the first commercial video game, Pong, was a simple simulation of table tennis. As processing power increased, new genres such as adventure and action games were developed that involved a player guiding a character from a third person perspective through a series of obstacles. This “real-time” element cannot be easily reproduced by a board game, which is generally limited to “turn-based” strategy; this advantage allows video games to simulate situations such as combat more realistically. Additionally, the playing of a video game does not require the same physical skill, strength or danger as a real-world representation of the game, and can provide either very realistic, exaggerated or impossible physics, allowing for elements of a fantastical nature, games involving physical violence, or simulations of sports. Lastly, a computer can, with varying degrees of success, simulate one or more human opponents in traditional table games such as chess, leading to simulations of such games that can be played by a single player.

    In more open-ended video games, such as sandbox games, a virtual environment is provided in which the player may be free to do whatever they like within the confines of a particular game’s universe. Sometimes, there is a lack of goals or opposition, which has stirred some debate on whether these should be considered “games” or “toys”. (Crawford specifically mentions Will Wright‘s SimCity as an example of a toy.)[9]

    Online games

    Main article: Online game

    Online games have been part of culture from the very earliest days of networked and time-shared computers. Early commercial systems such as Plato were at least as widely famous for their games as for their strictly educational value. In 1958, Tennis for Two dominated Visitor’s Day and drew attention to the oscilloscope at the Brookhaven National Laboratory; during the 1980s, Xerox PARC was known mainly for Maze War, which was offered as a hands-on demo to visitors.

    Modern online games are played using an Internet connection; some have dedicated client programs, while others require only a web browser. Some simpler browser games appeal to more casual game-playing demographic groups (notably older audiences) that otherwise play very few video games.[23]

    Role-playing games

    Main article: Role-playing game

    Role-playing games, often abbreviated as RPGs, are a type of game in which the participants (usually) assume the roles of characters acting in a fictional setting. The original role playing games – or at least those explicitly marketed as such – are played with a handful of participants, usually face-to-face, and keep track of the developing fiction with pen and paper. Together, the players may collaborate on a story involving those characters; create, develop, and “explore” the setting; or vicariously experience an adventure outside the bounds of everyday life. Pen-and-paper role-playing games include, for example, Dungeons & Dragons and GURPS.

    The term role-playing game has also been appropriated by the video game industry to describe a genre of video games. These may be single-player games where one player experiences a programmed environment and story, or they may allow players to interact through the internet. The experience is usually quite different from traditional role-playing games. Single-player games include Final Fantasy, Fable, The Elder Scrolls, and Mass Effect. Online multi-player games, often referred to as massively multiplayer online role playing games, or MMORPGs, include RuneScape, EverQuest 2, Guild Wars, MapleStory, Anarchy Online, and Dofus. As of 2009, the most successful MMORPG has been World of Warcraft, which controls the vast majority of the market.[24]

    Business games

    Main article: Team building

    Business games can take a variety of forms, from interactive board games to interactive games involving different props (balls, ropes, hoops, etc.) and different kinds of activities. The purpose of these games is to link to some aspect of organizational performance and to generate discussions about business improvement. Many business games focus on organizational behaviors. Some of these are computer simulations while others are simple designs for play and debriefing. Team building is a common focus of such activities.

    Simulation

    Main article: Simulation game

    The term “game” can include simulation[25][26] or re-enactment of various activities or use in “real life” for various purposes: e.g., training, analysis, prediction. Well-known examples are war games and role-playing. The root of this meaning may originate in the human prehistory of games deduced by anthropology from observing primitive cultures, in which children’s games mimic the activities of adults to a significant degree: hunting, warring, nursing, etc. These kinds of games are preserved in modern times.[original research?]

  • Nim

    Nim

    Nim is a mathematical combinatorial game in which two players take turns removing (or “nimming”) objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object.

    Nim is fundamental to the Sprague–Grundy theorem, which essentially says that every impartial game is equivalent to a nim game with a single pile.

    History

    [edit]

    Variants of nim have been played since ancient times.[1] The game is said to have originated in China—it closely resembles the Chinese game of jiǎn-shízi (捡石子), or “picking stones”[2]—but the origin is uncertain; the earliest European references to nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901,[3] but the origins of the name were never fully explained. The Oxford English Dictionary derives the name from the German verb nimm, meaning “take”.

    At the 1939 New York World’s Fair, Westinghouse displayed a machine, the Nimatron, that played nim.[4] From May 11 to October 27, 1940, only a few people were able to beat the machine in that six-month period; if they did, they were presented with a coin that said “Nim Champ”.[5] It was also one of the first-ever electronic computerized games. Ferranti built a nim-playing computer which was displayed at the Festival of Britain in 1951. In 1952, Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. L. Maxson Corporation, developed a machine weighing 23 kilograms (50 lb) which played nim against a human opponent and regularly won.[6] A nim playing machine has been described made from tinkertoys.[7]

    The game of nim was the subject of Martin Gardner‘s February 1958 Mathematical Games column in Scientific American. A version of nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961).[8]

    Game play and illustration

    [edit]

    Nim is typically played as a misère game, in which the player to take the last object loses. Nim can also be played as a “normal play” game whereby the player taking the last object wins. In either normal play or a misère game, when there is exactly one heap with at least two objects, the player who takes next can easily win. If this removes either all or all but one objects from the heap that has two or more, then no heaps will have more than one object, so the players are forced to alternate removing exactly one object until the game ends. If the player leaves an even number of non-zero heaps (as the player would do in normal play), the player takes last; if the player leaves an odd number of heaps (as the player would do in misère play), then the other player takes last.

    The normal game is between two players and is played with three heaps of any number of objects. The two players alternate taking any number of objects from any one of the heaps. The goal is to be the last to take an object. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object.

    The following example of a normal game is played between fictional players Bob and Alice, who start with heaps of three, four and five objects.

    Heap AHeap BHeap CMove
    345Game begins
    145Bob takes 2 from A
    142Alice takes 3 from C
    132Bob takes 1 from B
    122Alice takes 1 from B
    022Bob takes entire A heap, leaving two 2s
    012Alice takes 1 from B
    011Bob takes 1 from C leaving two 1s. (In misère play he would take 2 from C leaving [0, 1, 0])
    001Alice takes 1 from B
    000Bob takes entire C heap and wins

    Winning positions

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    The practical strategy to win at the game of nim is for a player to get the other into one of the following positions, and every successive turn afterwards they should be able to make one of the smaller positions. Only the last move changes between misère and normal play.

    2 heaps3 heaps4 heaps
    1 1 *1 1 1 **1 1 1 1 *
    2 21 2 31 1 n n
    3 31 4 51 2 4 7
    4 41 6 71 2 5 6
    5 51 8 91 3 4 6
    6 62 4 61 3 5 7
    7 72 5 72 3 4 5
    8 83 4 72 3 6 7
    9 93 5 62 3 8 9
    n n4 8 124 5 6 7
    4 9 134 5 8 9
    5 8 13n n m m
    5 9 12n n n n
    * Only valid for normal play.
    ** Only valid for misère.

    For the generalisations, n and m can be any value > 0, and they may be the same.

    Mathematical theory

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    Normal-play nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum).

    While all normal-play impartial games can be assigned a nim value, that is not the case under the misère convention. Only tame games can be played using the same strategy as misère nim.

    Nim is a special case of a poset game where the poset consists of disjoint chains (the heaps).

    The evolution graph of the game of nim with three heaps is the same as three branches of the evolution graph of the Ulam–Warburton automaton.[9]

    Nim has been mathematically solved for any number of initial heaps and objects, and there is an easily calculated way to determine which player will win and which winning moves are open to that player.

    The key to the theory of the game is the binary digital sum of the heap sizes, i.e., the sum (in binary), neglecting all carries from one digit to another. This operation is also known as “bitwise xor” or “vector addition over GF(2)” (bitwise addition modulo 2). Within combinatorial game theory it is usually called the nim-sum, as it will be called here. The nim-sum of x and y is written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and 5 is as follows:

    Binary Decimal
 
  0112    310    Heap A
  1002    410    Heap B
  1012    510    Heap C
  ---
  0102    210    The nim-sum of heaps A, B, and C, 3 ⊕ 4 ⊕ 5 = 2

    An equivalent procedure, which is often easier to perform mentally, is to express the heap sizes as sums of distinct powers of 2, cancel pairs of equal powers, and then add what is left:

    3 = 0 + 2 + 1 =     2   1      Heap A
4 = 4 + 0 + 0 = 4              Heap B
5 = 4 + 0 + 1 = 4       1      Heap C
--------------------------------------------------------------------
2 =                 2          What is left after canceling 1s and 4s

    In normal play, the winning strategy is to finish every move with a nim-sum of 0. This is always possible if the nim-sum is not zero before the move. If the nim-sum is zero, then the next player will lose if the other player does not make a mistake. To find out which move to make, let X be the nim-sum of all the heap sizes. Find a heap where the nim-sum of X and heap-size is less than the heap-size; the winning strategy is to play in such a heap, reducing that heap to the nim-sum of its original size with X. In the example above, taking the nim-sum of the sizes is X = 3 ⊕ 4 ⊕ 5 = 2. The nim-sums of the heap sizes A=3, B=4, and C=5 with X=2 areA ⊕ X = 3 ⊕ 2 = 1 [Since (011) ⊕ (010) = 001 ]B ⊕ X = 4 ⊕ 2 = 6C ⊕ X = 5 ⊕ 2 = 7

    The only heap that is reduced is heap A, so the winning move is to reduce the size of heap A to 1 (by removing two objects).

    As a particular simple case, if there are only two heaps left, the strategy is to reduce the number of objects in the bigger heap to make the heaps equal. After that, no matter what move the opponent makes, the player can make the same move on the other heap, guaranteeing that they take the last object.

    When played as a misère game, nim strategy is different only when the normal play move would leave only heaps of size one. In that case, the correct move is to leave an odd number of heaps of size one (in normal play, the correct move would be to leave an even number of such heaps).

    These strategies for normal play and a misère game are the same until the number of heaps with at least two objects is exactly equal to one. At that point, the next player removes either all objects (or all but one) from the heap that has two or more, so no heaps will have more than one object (in other words, so all remaining heaps have exactly one object each), so the players are forced to alternate removing exactly one object until the game ends. In normal play, the player leaves an even number of non-zero heaps, so the same player takes last; in misère play, the player leaves an odd number of non-zero heaps, so the other player takes last.

    In a misère game with heaps of sizes three, four and five, the strategy would be applied like this:

    ABCNim-sumMove
    3450102=210I take 2 from A, leaving a sum of 000, so I will win.
    1450002=010You take 2 from C
    1431102=610I take 2 from B
    1230002=010You take 1 from C
    1220012=110I take 1 from A
    0220002=010You take 1 from C
    0210112=310The normal play strategy would be to take 1 from B, leaving an even number (2) heaps of size 1. For misère play, I take the entire B heap, to leave an odd number (1) of heaps of size 1.
    0010012=110You take 1 from C, and lose.

    Proof of the winning formula

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    The soundness of the optimal strategy described above was demonstrated by C. Bouton.

    Theorem. In a normal nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is not zero. Otherwise, the second player has a winning strategy.

    Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+) and also satisfies an additional property, x ⊕ x = 0.

    Let x1, …, xn be the sizes of the heaps before a move, and y1, …, yn the corresponding sizes after a move. Let s = x1 ⊕ … ⊕ xn and t = y1 ⊕ … ⊕ yn. If the move was in heap k, we have xi = yi for all i ≠ k, and xk > yk. By the properties of ⊕ mentioned above, we have

    t=0⊕t=s⊕s⊕t=s⊕(x1⊕⋯⊕xn)⊕(y1⊕⋯⊕yn)=s⊕(x1⊕y1)⊕⋯⊕(xn⊕yn)=s⊕0⊕⋯⊕0⊕(xk⊕yk)⊕0⊕⋯⊕0=s⊕xk⊕yk(∗)t=s⊕xk⊕yk{\displaystyle {\begin{aligned}t&=0\oplus t\\&=s\oplus s\oplus t\\&=s\oplus (x_{1}\oplus \cdots \oplus x_{n})\oplus (y_{1}\oplus \cdots \oplus y_{n})\\&=s\oplus (x_{1}\oplus y_{1})\oplus \cdots \oplus (x_{n}\oplus y_{n})\\&=s\oplus 0\oplus \cdots \oplus 0\oplus (x_{k}\oplus y_{k})\oplus 0\oplus \cdots \oplus 0\\&=s\oplus x_{k}\oplus y_{k}\\[10pt](*)\quad t&=s\oplus x_{k}\oplus y_{k}\end{aligned}}}

    The theorem follows by induction on the length of the game from these two lemmas.

    Lemma 1. If s = 0, then t ≠ 0 no matter what move is made.

    Proof: If there is no possible move, then the lemma is vacuously true (and the first player loses the normal play game by definition). Otherwise, any move in heap k will produce t = xk ⊕ yk from (*). This number is nonzero, since xk ≠ yk.

    Lemma 2. If s ≠ 0, it is possible to make a move so that t = 0.

    Proof: Let d be the position of the leftmost (most significant) nonzero bit in the binary representation of s, and choose k such that the dth bit of xk is also nonzero. (Such a k must exist, since otherwise the dth bit of s would be 0.) Then letting yk = s ⊕ xk, we claim that yk < xk: all bits to the left of d are the same in xk and yk, bit d decreases from 1 to 0 (decreasing the value by 2d), and any change in the remaining bits will amount to at most 2d−1. The first player can thus make a move by taking xk − yk objects from heap k, then

    t = s ⊕ xk ⊕ yk           (by (*))
  = s ⊕ xk ⊕ (s ⊕ xk)
  = 0.

    The modification for misère play is demonstrated by noting that the modification first arises in a position that has only one heap of size 2 or more. Notice that in such a position s ≠ 0, and therefore this situation has to arise when it is the turn of the player following the winning strategy. The normal play strategy is for the player to reduce this to size 0 or 1, leaving an even number of heaps with size 1, and the misère strategy is to do the opposite. From that point on, all moves are forced.

    Variations

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    The subtraction game

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    Interactive subtraction game: Players take turns removing 1, 2 or 3 objects from an initial pool of 21 objects. The player taking the last object wins.

    In another game which is commonly known as nim (but is better called the subtraction game), an upper bound is imposed on the number of objects that can be removed in a turn. Instead of removing arbitrarily many objects, a player can only remove 1 or 2 or … or k at a time. This game is commonly played in practice with only one heap.

    Bouton’s analysis carries over easily to the general multiple-heap version of this game. The only difference is that as a first step, before computing the nim-sums we must reduce the sizes of the heaps modulo k + 1. If this makes all the heaps of size zero (in misère play), the winning move is to take k objects from one of the heaps. In particular, in ideal play from a single heap of n objects, the second player can win if and only if

    • 0 = n (mod k + 1) (in normal play), or
    • 1 = n (mod k + 1) (in misère play).

    This follows from calculating the nim-sequence of S(1, 2, …, k),

    0.123…k0123…k0123…=0˙.123…k˙,{\displaystyle 0.123\ldots k0123\ldots k0123\ldots ={\dot {0}}.123\ldots {\dot {k}},}

    from which the strategy above follows by the Sprague–Grundy theorem.

    The 21 game

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    See also: 21 (drinking game)

    The game “21” is played as a misère game with any number of players who take turns saying a number. The first player says “1” and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say “21” loses. This can be modeled as a subtraction game with a heap of 21 − n objects. The winning strategy for the two-player version of this game is to always say a multiple of 4; it is then guaranteed that the other player will ultimately have to say 21; so in the standard version, wherein the first player opens with “1”, they start with a losing move.

    The 21 game can also be played with different numbers, e.g., “Add at most 5; lose on 34”.

    A sample game of 21 in which the second player follows the winning strategy:

    PlayerNumber
    11
    24
    15, 6 or 7
    28
    19, 10 or 11
    212
    113, 14 or 15
    216
    117, 18 or 19
    220
    121

    The 100 game

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    A similar version is the “100 game”: Two players start from 0 and alternately add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g., 01, 12, 23, 34,…) and control the game by jumping through all the numbers of this sequence. Once a player reaches 89, the opponent can only choose numbers from 90 to 99, and the next answer can in any case be 100.

    A multiple-heap rule

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    See also: Wythoff’s game

    In another variation of nim, besides removing any number of objects from a single heap, one is permitted to remove the same number of objects from each heap.

    Circular nim

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    See also: Kayles

    Yet another variation of nim is “circular nim”, wherein any number of objects are placed in a circle and two players alternately remove one, two or three adjacent objects. For example, starting with a circle of ten objects,

    . . . . . . . . . .

    three objects are taken in the first move

    _ . . . . . . . _ _

    then another three

    _ . _ _ _ . . . _ _

    then one

    _ . _ _ _ . . _ _ _

    but then three objects cannot be taken out in one move.

    Grundy’s game

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    In Grundy’s game, another variation of nim, a number of objects are placed in an initial heap and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of 5+1 or 4+2, but not 3+3. Grundy’s game can be played as either misère or normal play.

    Greedy nim

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    Greedy nim is a variation wherein the players are restricted to choosing stones from only the largest pile.[10] It is a finite impartial game. Greedy nim misère has the same rules as greedy nim, but the last player able to make a move loses.

    Let the largest number of stones in a pile be m and the second largest number of stones in a pile be n. Let pm be the number of piles having m stones and pn be the number of piles having n stones. Then there is a theorem that game positions with pm even are P positions. [11] This theorem can be shown by considering the positions where pm is odd. If pm is larger than 1, all stones may be removed from this pile to reduce pm by 1 and the new pm will be even. If pm = 1 (i.e. the largest heap is unique), there are two cases:

    • If pn is odd, the size of the largest heap is reduced to n (so now the new pm is even).
    • If pn is even, the largest heap is removed entirely, leaving an even number of largest heaps.

    Thus, there exists a move to a state where pm is even. Conversely, if pm is even, if any move is possible (pm ≠ 0), then it must take the game to a state where pm is odd. The final position of the game is even (pm = 0). Hence, each position of the game with pm even must be a P position.

    Index-k nim

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    A generalization of multi-heap nim was called “nimk{\displaystyle {}_{k}}” or “index-k” nim by E. H. Moore,[12] who analyzed it in 1910. In index-k nim, instead of removing objects from only one heap, players can remove objects from at least one but up to k different heaps. The number of elements that may be removed from each heap may be either arbitrary or limited to at most r elements, like in the “subtraction game” above.

    The winning strategy is as follows: Like in ordinary multi-heap nim, one considers the binary representation of the heap sizes (or heap sizes modulo r + 1). In ordinary nim one forms the XOR-sum (or sum modulo 2) of each binary digit, and the winning strategy is to make each XOR sum zero. In the generalization to index-k nim, one forms the sum of each binary digit modulo k + 1.

    Again, the winning strategy is to move such that this sum is zero for every digit. Indeed, the value thus computed is zero for the final position, and given a configuration of heaps for which this value is zero, any change of at most k heaps will make the value non-zero. Conversely, given a configuration with non-zero value, one can always take from at most k heaps, carefully chosen, so that the value will become zero.

    Building nim

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    Building nim is a variant of nim wherein the two players first construct the game of nim. Given n stones and s empty piles, the players, alternating turns, place exactly one stone into a pile of their choice.[13] Once all the stones are placed, a game of Nim begins, starting with the next player that would move. This game is denoted BN(n,s).

    Higher-dimensional nim

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    n-d nim is played on a k1×⋯×kn{\displaystyle k_{1}\times \dots \times k_{n}} board, whereon any number of continuous pieces can be removed from any hyper-row. The starting position is usually the full board, but other options are allowed.[14]

    Graph nim

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    The starting board is a disconnected graph, and players take turns to remove adjacent vertices.[15]

    Candy nim

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    Candy nim is a version of normal-play nim in which players try to achieve two goals at the same time: taking the last object (in this case, candy) and taking the maximum number of candies by the end of the game.[16]