The Philosophers’ Game

Rhythmomachy board from Boissiere 1554 work on the game

  

The Philosophers’ Game

Rithmomachia (also Arithmomachia, known as Ludus philosophorum) is an early European mathematical board game.

A literal translation of the name is “The Battle of the Numbers”.

The first written evidence of Rythmomachia dates back to around 1030, when a monk, named Asilo, created a game that illustrated the number theory of Boëthius’ De institutione arithmetica, for the students of monastery schools. It was used mainly as a teaching aid, but, gradually, intellectuals started to play it for pleasure.

The rules of the game were improved shortly thereafter by the respected monk Hermannus Contractus (1013–1054), from Reichenau, and in the school of Liège.

In the following centuries, Rythmomachia spread quickly through schools and monasteries in the southern parts of Germany and France.

In the 13th century, Rythmomachia came to England, where famous mathematician Thomas Bradwardine wrote a text about it.

Even Roger Bacon recommended Rythmomachia to his students, while Sir Thomas More let the inhabitants of the fictitious Utopia play it for recreation.

The game was well enough known as to justify printed treatises in Latin, French, Italian, and German, in the sixteenth century, and to have public advertisements of the sale of the board and pieces under the shadow of the old Sorbonne.

From the seventeenth century onwards the game, which at its peak rivaled chess for popularity in Europe, virtually disappeared until the late 19th century when rediscovered by historians.

 The rules given here are those established in 1556 by Claude de Boissière.

Pieces and movements

The game was played on a board resembling the one used for chess or checkers, with 8 squares on the shorter side, but with 16 on the longer side.

There are four types of pieces:
Rounds – move one square in any of the four diagonals;
Triangles – can move exactly two squares vertically or horizontally, but not diagonally;
Squares – can move exactly three squares vertically or horizontally, but not diagonally;
Pyramids – are not actually one piece, but more than one piece put together. The White Pyramid is made of a “36” Square, a “25” Square, a “16” Triangle, a “9” Triangle, a ” 4″ Round, and a “1” Round, which totals up to the Pyramid’s value of 91. The Black Pyramid is made up of a “64” Square, a “49” Square, a “36” Triangle, a “25” Triangle, and a “16” Round, which adds up to the Pyramid’s value of 190. These irregular values make it hard for them to be captured by most of the capturing methods listed below, except for Siege. Pyramids can move like a Round, a Triangle, or a Square, as long as they still contain the respective piece, which makes them very valuable.

The game was noteworthy in that the black and white forces were not symmetrical. There are 57 pawns, 29 black ones and 28 white ones, broken down as follows:

White Pieces

  • 8 round pieces, of values
    • 3 / 5 / 7 / 9 / 9 / 25 / 49 / 81
    • (odd numbers 3 to 9 and their squares)
  • 8 triangular pieces, of values
    • 12 / 30 / 56 / 90 / 16 / 36 / 64 / 100
    • (3*4 / 5*6 / 7*8 / 9*10 and the squares of 4 / 6 / 8 / 10)
  • 7 square pieces, of values
    • 28 / 66 / 120 / [190] / 49 / 121 / 225 / 361
    • (4*7 / 6*11 / 8*15 / [10*19] and the squares of 7 / 11 / 15 / 19)
  • the preceding 190 pieces are broken down into a pyramid of:
    • round 16 (4*4)
    • triangular 25 (5*5)
    • triangular 36 (6*6)
    • square 49 (7*7)
    • square 64 (8*8)

Black Pieces

  • 8 round pieces, of values
    • 2 / 4 / 6 / 8 / 4 / 16 / 36 / 64
    • (even numbers 2 to 8 and their squares)
  • 8 triangular pieces, of values
    • 6 / 20 / 42 / 72 / 9 / 25 / 49 / 81
    • (2*3 / 4*5 / 6*7 / 8*9 and the squares of 3 / 5 / 7 / 9)
  • 7 square pieces, of values
    • 15 / 45 / 153 / 25 / 81 / 169 / 289
    • (3*5 / 5*9 / [7*13] / 9*17 and the squares of 5 / 9 / 13 / 17)
    • (note that 3+2=5 / 5+4=9 / 7+6=13 / 9+8=17)
  • the preceding 91 piece is broken down into a pyramid of:
    • round 1 (1*1)
    • round 4 (2*2)
    • triangular 9 (3*3)
    • triangular 16 (4*4)
    • square 25 (5*5)
    • square 36 (6*6)

BLACK SIDE

25   

81   

        169   

289   

15   

45   

25   

20   

42   

49   

91   

153   

9   

6   

4   

16   

36   

64   

72   

81   

    2    4 

6   

8   

   

 
               
six more rows
               
     9  7   

5   

3   

   

 
100   

90   

81   

49   

25   

9   

12   

16   

190   

120   

64   

56   

30   

36   

66   

28   

361   

225   

        121   

49   

WHITE SIDE

  

Capture methods

This game allows different possible captures and winning configurations to the two players.

There were four capture methods:
Meeting – If a piece could capture another piece with the same value by landing on it, the piece stays in its location and the opponent’s piece is taken from the board;
Assault – If a piece with a small value, multiplied by the number of vacant spaces between it and another larger piece is equal to the larger piece, the larger piece is captured;
Ambuscade – If two pieces’ sum is equal to an enemy piece that is placed between the two, the enemy piece is captured and removed from the board;
Siege – If a piece is surrounded on all four sides, it is removed.

It is notable that pieces do not land on another piece to capture it, but instead remain in their square and remove the other.

  

Types of victory

There were 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 — suggest a connection with the mathematical work of Boëthius.

Common Victories:
De Corpore (Latin: “by body”) – If a player captures a certain number of pieces set by both players, he wins the game.
De Bonis (“by goods”) – If a player captures enough pieces to add up to or exceed a certain value that is set by both players, he wins the game.
De Lite (“by lawsuit”) – If a player captures enough pieces to add up to or exceed a certain value that is set by both players, and the number of digits in his captured pieces’ values are less than a number set by both players, he wins the game.
De Honore (“by honour”) – If a player captures enough pieces to add up to or exceed a certain value that is set by both players, and the number of pieces he captured are less than a certain number set by both players, he wins the game.
De Honore Liteque (“by honour and lawsuit”) – If a player captures enough pieces to add up to or exceed a certain value that is set by both players, the number of digits in his captured pieces’ values are less than a number set by both players, and the number of pieces he captured are less than a certain number set by both players, he wins the game.

Proper Victories:
Victoria Magna (“great victory”) – This occurs when three pieces that are arranged are in an arithmetic progression.
Victoria Major (“greater victory”) – This occurs when four pieces that are arranged have three pieces that are in a certain progression, and another three pieces that are in another type of progression.
Victoria Excellentissima (“most excellent victory”) – This occurs when four pieces that are arranged have all three types of mathematical progressions in three different groups.

Further reading

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