Rithmomachia (also known as Rithmomachy, Arithmomachia, Rythmomachy, Rhythmomachy, The Philosophers' Game, and other variants) is an early European mathematical board game. Its earliest known description dates from the eleventh century. The name comes loosely from Greek and means "The Battle of the Numbers." The game is somewhat like chess except that most methods of capture depend on the numbers inscribed on each piece.
The game was used as an educational tool that teachers could introduce while teaching arithmetic as part of the quadrivium to those in Western Europe who received a classical education during the medieval period. David Sepkoski wrote that between the twelfth and sixteenth centuries, "rithmomachia served as a practical exemplar for teaching the contemplative values of Boethian mathematical philosophy, which emphasized the natural harmony and perfection of number and proportion, that it was used both as a mnemonic drill for the study of Boethian number theory and, more importantly, as a vehicle for moral education, by reminding players of the mathematical harmony of creation."[1] The game declined sharply in popularity in the 17th century, as it was no longer used in education, and potential players were not introduced to it during their schooling.
Little is known about the origin of the game. Medieval writers attributed it to Pythagoras, but no trace of it has been discovered in Greek literature. The earliest surviving mentions of it is are from the early 11th century, suggesting it was created in the late 10th or early 11th century. The name and its many variations are from Greek; it is unclear whether this was due to being created by a rare Western European with a classical education that involved learning Greek, or if the game had a genuine origin in Greece and the Greek-speaking Byzantine Empire of the period.
The first written evidence of Rithmomachia dates to around 1030, when a monk named Asilo created a game that illustrated the number theory of Boethius' De institutione arithmetica, for the students of monastery schools. De institutione arithmetica was the standard textbook for instruction in arithmetics in the period for those lucky enough to receive a medieval education. The rules of the game were improved and spelled out more shortly thereafter by another monk, Hermannus Contractus (1013–1054) from Reichenau, and in the school of Liège. In the following centuries, Rithmomachia spread through schools and monasteries in the southern parts of Germany and France. It was used mainly as a teaching aid, but gradually intellectuals started to play it for pleasure. In the 13th century Rithmomachia came to England, where famous mathematician Thomas Bradwardine wrote a text about it. Even Roger Bacon recommended Rithmomachia to his students, while Sir Thomas More let the inhabitants described in the book Utopia play it for recreation. The game was known well enough to justify printed treatises in Latin, French, Italian, and German, in the sixteenth century. Two notices advertising a game set for sale have been found, one in Paris (1556) and the other in London (1563). Nevertheless, no archaeological evidence of the game (such as playing boards) has survived from the medieval and early modern periods, in contrast to many other board games.
Rithmomachia was at its most popular in the 16th century. The Tudor polymath, statesman and publisher of the Geneva Bible, Sir Rowland Hill, published on the game under the title The most ancient and learned Playe, called the Philosopher's Game invented for the honest recreation of Students and other sober persons, in passing the tedious of tyme to the release of their labours, and the exercise of their Wittes in 1562;[2] his house at Soulton Hall also contains a board for the game on a basement parlour or prayer room.
In 1572 Francesco Barozzi published a version in Venice which was translated into German by Augustus II, Duke of Brunswick.[3] It lost its popularity sharply in the 17th century, virtually disappearing as the style of teaching mathematics changed, and Boethius's mathematics was considered old-fashioned and obsolete. One issue was that the rules were never standardized, with major variations from teacher to teacher. The game partially survived by clinging to the now far more popular chess. Gustavus Selenus included the rules of rithmomachia as an appendix in one of his books on chess, and the game persisted as "arithmetic chess" or "numerical checkers" as a side mention in German chess books for some time afterward, a curiosity but a rarely-played one. The game was rediscovered in the 20th century by historians of board games such as Arno Borst.[4]
Rithmomachia is played on a board resembling the one used for chess or checkers with eight squares on the shorter side, but with sixteen on the longer side. The forms used for the pieces were triangles, squares, and rounds. Pyramids could be formed by stacking pieces. The game was noteworthy in that the black and white forces were not symmetrical. Although each side had the same array of pieces, the numbers on them differed, allowing different possible captures and winning configurations to the two players.
The rules below describe the most common version of the game, played through much of the Middle Ages and Renaissance. There was also a variant propounded by Fulke in the 16th century, with significantly different (and somewhat more consistent) capture rules.[5]
There are four types of pieces: Rounds, Triangles, Squares, and Pyramids.
There were a variety of capture methods. Pieces do not land on another piece to capture it, but instead remain in their square and remove the other. If a piece is captured, it changes sides.[6]
There were also a variety of victory conditions for determining when a game would end and who the winner was. There were common victories, and proper victories, which were recommended for more skilled players. Proper victories required placing pieces in linear arrangements in the opponent's side of the board, with the numbers formed by the arrangement following various types of numerical progression. The types of progression required — arithmetic, geometric and harmonic — fit with the mathematical and numerological teachings of Boethius.