Board Games and mathematics

Summary: While some games are explicitly mathematical, others are implicitly governed by math.

Humans have been playing games for as long as they have been around. Johan Huizinga was the first to call the attention to the fact that play precedes culture. Board games, a very organized form of play, are part of human social nature. Human communities may differ in many ways, but they all play games. From the ancient Mancala, practiced for millennia in Africa, to our Monopoly, we find board games in many societies. Besides their cultural relevance—they are studied by anthropologists, historians, and others—board games are characterized by their sets of rules, which show mathematical structures and connections that are at times very surprising.

Game Classifications

Chess and Go come to mind as examples of traditional board games, and Monopoly and Scrabble are examples of proprietary games. The distinction between the two types of games is not always easy to identify. In chess, the movements of the pieces and the other rules are the main considerations. Chess is an abstract game, not considering the fact that it originally emulated a battle between two armies. Chess does have similarities with other games. When playing representational games like Monopoly or Diplomacy, players find themselves focusing on the possibilities and strategic choices, forgetting the particular settings. Accoring to David Parlett, positional games refer to games where pieces are played in a board or any other set of markings, as chess, checkers, and Go, and “theme” games are generally representational and commercial, like Monopoly and Diplomacy.

Board game classification has been inspired in the fact, first noted by H. J. R. Murray, that games are typical of early activities of man—the battle, the siege, the race, the hunt, alignment, arrangement, and counting. Parlett’s classification, which evolved from Murray’s and others, is as follows. In race games, the board is a linear track where each player tries to be the first to reach a particular cell or remove a set of pieces from the board. Most of the games under this category use dice or other randomizing devices, like Chutes & Ladders, Ludo, and Backgammon, but not all, such as Hare & Tortoise.

Space games, typically two-dimensional and free placing, comprise the alignment games, as Nine Men Morris; connection games, as Hex and Twixt; traversal games, in which a player tries to have one or several pieces cross the board, as Breakthrough, Halma, and Chinese Checkers; configuration games, where players try to achieve certain displays with their pieces, as Agon; restriction games, where the aim is to try to block the adversary, like Pentominoes; and occupation games, in which the winner is the player who achieves more space in the board, as in Go and Othello. Chase games are asymmetrical, one player having several pieces while the other has only one or two. Their goals are also distinct, as in Fox & Geese. “Displace games” include chess and checkers, where a player aims at capturing most of his opponent’s pieces (as in checkers) or a particular one (as in chess), and other war games; the family of Mancala games belongs also to this class.

History

The Royal Game of Ur, also known as the Game of Twenty Squares, was found in the south of Iraq in the 1920s and is about 4500 years old. The board shows twenty squares, 12 in a three-by-four rectangular array, six in two rows of three, and two connecting cells. The reverse of the board corresponding to the 12 cells showed a zodiac, illustrating that in the past, the same object could be a board game and a divinatory device. Two cuneiform clay tablets give the exact rules for this game. Each player had seven pieces, which moved across the board according to the toss of three tetrahedral dice.

A similar game is found in Ancient Egypt, Senet or the Game of Thirty Squares. It was a race game as well, but it was more than a simple toy. In funerary monuments that date from 4000 years ago, images are shown of the deceased playing Senet against an invisible adversary. Osiris, which is present but not shown, decides on matters of life after death.

The Royal Game of Ur and Senet can be viewed as the oldest relatives of the modern Backgammon, a game in which the moves are decided by the players upon tossing two cubic dice. The player who better understands the probability laws that rule the dice is most often the winner.

The Chinese game Go is four millennia old. Nowadays, it remains one of the most complex games, despite the simplicity of its rules. Go is played on the intersections of a 19-by-19 grid, and each player fights to control the largest area.

Pure strategy games could also be found in Ancient Greece, like Petteia. This game, and the Roman Ludus Latrunculorum, shared the shape of the board, checkered, and the orthogonal movement of the pieces.

Chess, which originated in India about 1400 years ago, traveled to the West with the Arabs, and saw its rules evolve in the process. It was originally created as a war game between two armies, and its pieces represented the actors of the battle. However, the abstract shapes that reached Europe gave way to the symbolic representation of the European medieval society.

The Arabs introduced several other games in Europe. One game they introduced, Alquerque, was played on the intersections of a five-by-five lined board. The adaptation of this game to the chessboard originated the game of Checkers.

Board Games and Mathematics

The oldest known pedagogical game is Rithmomachia, also known as Philosopher’s Game. It was invented in the eleventh century as a didactical device to teach mathematics. It was practiced wherever Boethius’s arithmetic was taught. Pythagorean in nature, this tradition of mathematics dominated teaching at churches and universities for more than 500 years. In an eight-by-16 board, two armies fought each other. Pieces carried numbers and could have one of three shapes: circular, triangular, or square.

The movements depended on the shape of the piece played; the captures depended on the numbers and on arithmetical calculations. Victory was attained by means of a configuration of pieces holding numbers in progression (arithmetic, geometric, harmonic, or combinations of the three). This game spread throughout Europe, and only when the mathematical curriculum at universities changed in the sixteenth century did it vanish. Losing its pedagogical goal turned out to be fatal, as Rithmomachia lacked the qualities to survive as a purely recreational activity. Chinese scholars of the eleventh century also published work on permutations based on the Go board. John H. Conway’s twentieth-century research on the game contributed to the invention of surreal numbers and the development of combinatorial game theory.

Ludus Astronomorum was a board game for seven players based on Ptolemaic astrological principles. In the sixteenth century, William Fulke, a professor at Cambridge who had written a manual of the Philosopher’s Game, created two other games. One, intended to improve on the astronomy game, was Ouranomachia, the other, created to teach geometry, was Metromachia. Fulke published one book on each.

In the eighteenth century, George Berkeley invented a game to help teach algebra, a subject Berkeley had in very high consideration. The game was Ludus Algebraicus and essentially functioned as a randomizing device to generate algebraic equations.

Charles Dodgson invented a game in the nineteenth century to practice logical deduction and wrote a book about it, The Game of Logic, under his pen name, Lewis Carroll.

In Ireland, mathematician William Hamilton created in 1857 the Icosian Game and soon after Traveller’s Dodecahedron. This comprised a dodecahedron and a piece of thread that should touch every vertex according to some rules. It was this game that gave rise to the concept of Hamiltonean Graph.

The familiar game of Nim in which a move consists of choosing from one of a pile of beans and reducing its cardinality, was first solved mathematically at the beginning of the previous century. In its normal form, where the winner is the one who takes the last bean, is the paradigm of a class of games studied in Combinatorial Game Theory. The familiar children’s game Dots & Boxes was also treated mathematically with the same techniques. Some traditional games, like Konane, can be approached the same way.

The game Hex was invented independently by both Piet Hein and John Nash in the 1940s. It is a connection game played on a diamond-shaped board of hexagonal cells. David Gale noted that a game of Hex can never end in a tie, and that this fact is logically equivalent to a deep theorem in topology.

Abstract games with complete information and no chance devices are also called mathematical games. The mental processes present in their practice and in a typical mathematical activity, like problem solving, are far from disjointed.

Bibliography

Avedon, Elliot M., and Brian Sutton-Smith. Study of Games. New York: John Wiley & Sons, 1971.

Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy. Winning Ways for Your Mathematical Plays. Natick, MA: Ak Peters, 2001.

Huizinga, Johan. Homo Ludens. New York: Routledge, 2008.

Murray, Harold James Ruthven. A History of Board-Games Other Than Chess. New York: Hacker Art Books, 1952.

Parlett, David. Oxford History of Board Games. New York: Oxford University Press, 1999.