Mathematical game
Encyclopedia
This article is about using mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 to study the inner-workings of multiplayer game
Multiplayer game
A multiplayer video game is one which more than one person can play in the same game environment at the same time. Unlike most other games, computer and video games are often single-player activities that put the player against preprogrammed challenges and/or AI-controlled opponents, which often...

s which, on the surface, may not appear mathematical at all. For games that directly involve mathematics in their play, see mathematical puzzle
Mathematical puzzle
Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules as do multiplayer games, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions....

.

Mathematical Games was a column written by Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...

 that appeared in the
Scientific American
Scientific American
Scientific American is a popular science magazine. It is notable for its long history of presenting science monthly to an educated but not necessarily scientific public, through its careful attention to the clarity of its text as well as the quality of its specially commissioned color graphics...

. Information on his column and other recreational mathematics publications can be found in the recreational mathematics article.


A mathematical game is a multiplayer game
Multiplayer game
A multiplayer video game is one which more than one person can play in the same game environment at the same time. Unlike most other games, computer and video games are often single-player activities that put the player against preprogrammed challenges and/or AI-controlled opponents, which often...

 whose rules, strategies, and outcomes can be studied and explained by mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

. Examples of such games are Tic-tac-toe
Tic-tac-toe
Tic-tac-toe, also called wick wack woe and noughts and crosses , is a pencil-and-paper game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The X player usually goes first...

 and Dots and Boxes
Dots and Boxes
Dots and Boxes is a pencil and paper game for two players first published in 1889 by Édouard Lucas.Starting with an empty grid of dots, players take turns, adding a single...

, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game. For example, even though the rules of Mancala
Mancala
Mancala is a family of board games played around the world, sometimes called "sowing" games, or "count-and-capture" games, which describes the game-play. Mancala games play a role in many African and some Asian societies comparable to that of chess in the West, or the game of Go in Eastern Asia...

 are straightforward, mathematicians analyze the game using combinatorial game theory
Combinatorial game theory
Combinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...

.

Mathematical games differ from mathematical puzzles in that all mathematical puzzles require math to solve them whereas mathematical games may not require a knowledge of mathematics to play them or even to win them. Thus the actual mathematics of mathematical games may not be apparent to the average player.

Some mathematical games are topics of interest in recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...

.

When studying the mathematics of games, the mathematical analysis of the game is more important than actually playing the game. To analyze a game mathematically, the mathematician studies the rules of the game in order to understand the inner-workings of the game, to determine winning strategies, and possibly to determine if a game has a solution
Solved game
A solved game is a game whose outcome can be correctly predicted from any position when each side plays optimally. Games which have not been solved are said to be "unsolved"...

.

Abstract Strategy Games (No chance involved)

Sometimes it is not immediately obvious that a particular game involves chance. Often a card game is described as "pure strategy" and such, but a game with any sort of random shuffling or face-down dealing of cards should not be considered to be "no chance". Several abstract strategy games
Abstract strategy game
An abstract strategy game is a strategy game, aiming to minimise luck, and without a theme. Almost all abstract strategy games will conform to the strictest definition of: a board or card game, in which there is no hidden information, no non-deterministic elements , in which two players or teams...

 are listed below:

Lattice board

  • Angels and Devils
    Angel problem
    The angel problem is a question in game theory proposed by John Horton Conway. The game is commonly referred to as the Angels and Devils game. The game is played by two players called the angel and the devil. It is played on an infinite chessboard...

  • Checkers (English draughts
    English draughts
    English draughts or checkers , also called American checkers or straight checkers or in Israel damka, is a form of draughts board game. Unlike international draughts, it is played on an eight by eight squared board with twelve pieces on each side...

    )
    • Checkers variants
      Draughts
      Draughts is a group of abstract strategy board games between two players which involve diagonal moves of uniform pieces and mandatory captures by jumping over the enemy's pieces. Draughts developed from alquerque...

  • Chess
    Chess
    Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...

    • Chess variants
  • Chomp
    Chomp
    Chomp is a 2-player game of strategy played on a rectangular "chocolate bar" made up of smaller square blocks . The players take it in turns to choose one block and "eat it" , together with those that are below it and to its right...

  • Domineering
    Domineering
    Domineering is a mathematical game played on a sheet of graph paper, with any set of designs traced out. For example, it can be played on a 6×6 square, a checkerboard, an entirely irregular polygon, or any combination thereof. Two players have a collection of dominoes which they place on the grid...

  • Dots and boxes
    Dots and Boxes
    Dots and Boxes is a pencil and paper game for two players first published in 1889 by Édouard Lucas.Starting with an empty grid of dots, players take turns, adding a single...

  • Go
    Go (board game)
    Go , is an ancient board game for two players that originated in China more than 2,000 years ago...

    • Go variants
      Go variants
      There are many variations on the basic game of Go. Some are ancient digressions, while other are modern deviations. They are often to be found as side events at tournaments, for instance the U.S...

  • Hex
    Hex (board game)
    Hex is a board game played on a hexagonal grid, theoretically of any size and several possible shapes, but traditionally as an 11x11 rhombus. Other popular dimensions are 13x13 and 19x19 as a result of the game's relationship to the older game of Go...

  • Hexapawn
    Hexapawn
    Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a chessboard. On a board of size n×m, each player begins with m pawns, one for each square in the row closest to them...

  • L game
    L game
    The L game is a simple strategic game invented by Edward de Bono.The L game is a two-player turn-based game played on a board of 4×4 squares. Each player has a 3×2 L-shaped piece, and there are two 1×1 neutral pieces. On each turn, a player first must move their L piece, and then...

  • Pawn duel
    Pawn duel
    Pawn duel is a logical chess game. Two players are taking part, each one has three pawns, placed in opposite to each other on controversial ending lanes. First move belongs to the player controlling the white pawns. Each move means changing the position of a single pawn of your color...

  • Philosopher's football
    Phutball
    Phutball is a two-player board game described in Elwyn Berlekamp, John Horton Conway, and Richard Guy's Winning Ways for your Mathematical Plays.- Rules :...

  • Rhythmomachy

Non-lattice boards and other games

  • Graph pebbling
    Graph pebbling
    Graph pebbling is a mathematical game and area of interest played on a graph with pebbles on the vertices. 'Game play' is composed of a series of pebbling moves. A pebbling move on a graph consists of taking two pebbles off one vertex and placing one on an adjacent vertex...

  • Hackenbush
    Hackenbush
    Hackenbush is a two-player mathematical game that may be played on any configuration of colored line segments connected to one another by their endpoints and to the ground...

  • Chopsticks (Hand game)
  • Nim
    Nim
    Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. 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....

  • Sim
    Sim (pencil game)
    The game of Sim is played by two players on a board consisting of six dots . Each dot is connected to every other dot by a line.Two players take turns coloring any uncolored lines...

  • Sprouts
    Sprouts (game)
    Sprouts is a pencil-and-paper game with interesting mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967.- Rules :...


External links

  • Historical Math Problems/Puzzles at Mathematical Association of America
    Mathematical Association of America
    The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...

    Convergence
  • The Travelling Salesman a famous advanced mathematics game problem.
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