# hist-games: Rithmomachia

Jon at Gothic Green Oak jon at gothicgreenoak.co.uk
Tue May 8 06:42:20 PDT 2007

```OK tell about Rithmomachia  please...

Arianwen ferch Arthur

Hi These are the rules I have written, hope they are of help
jon

Rithmomachia  -
The Philosophers' Game

This is not a game for the faint-hearted. Probably invented in one of the
monastery schools in southern Germany in the 11th C it spread to Britain by
the 12th C reaching a peak in the 14th C after which it gave way to chess
and was forgotten by the 18thC. Rithmomachia means battle of numbers and is
played between two players on a 8 x 16 chequered or undifferentiated board
with square, triangular and circular pieces each with a numerical value.
There is more than one documentary source detailing the rules, each of which
vary in minor details. Playing requires an ability to apply simple maths,
and winning requires an understanding of arithmetic, geometric and harmonic
number progressions, or at least an ability to remember them. The rules
outlined below will allow players to complete a game. However, this is a
complex game based on a more than common grasp of number theory and there is
insufficient room for Rithmomachia here. Parlett (1999) has written well on
this game and what follows is largely drawn from his chapter. Those
interested in the game should turn first to his book for an excellent read
before turning to original sources. However, do not be put off, it is a
wonderful game and a good one for getting inside the mind of the medieval
mathematician.

The Pieces
Each side occupies 24 squares on the board at the start of the game
positioned as shown. The board is theoretically divided down the middle
between the two sides so that each side has a home. On each side 23 of the
occupied squares have a single piece but one has a stack of pieces called a
pyramid. The pieces are square, triangular and circular, known as rounds,
black on one side and white on the other with the numerical value on each
side.

Rounds may move one position orthogonally
Triangles may move two positions diagonally
Squares may move three squares diagonally or orthogonally
Pyramids may move in the direction of any counter it contains

There is no jumping over other pieces and only one piece may occupy a square
at any time.

The position the pieces take on the board at the beginning of play is based
on number series built on even numbers 2, 4, 6, 8 for white and uneven, 3,
5, 7, 9 for black. The number series follow arithmetic, geometric and
harmonic

the progression is arithmetic where the integers a (x) b (y) c,  where
a<b<c, x=b-a and y=c-b,  if x=y

the progression is geometric where the integers a  b  c,  where  a<b<c, if
b/a=c/b

the progression is harmonic where the integers a (x) b (y) c,  where  a<b<c
and x=b-a and y=c-b, if y/x=c/a

White has the numbers 2, 4, 6, 8, 9, 15, 16, 20, 25, 36, 42, 49, 45, 64, 72,
81, 91(pyramid) 153, 169, 289.

Black has the numbers 3, 5, 7, 9, 12, 16, 25, 28, 30, 36, 49, 56, 64,  66,
81, 90, 100, 120, 121, 190 (pyramid), 225, 361.

The pyramids are made up of the following from the top to the base:

White pyramid 91  Black pyramid 190
round 1 round 16
round 4 triangle 25
triangle 9 triangle 36
triangle 16 square 49
square 25 square 64
square 36

Players alternate in moving one of their pieces each turn with the aim of
capturing their opponents pieces and winning by aligning three of their own
pieces in their opponents half of the board in an arithmetic, geometric or
harmonic progression, know as a triumph.

Capturing

There are several ways in which a player may capture an opponent's piece.
Sources vary on the fate of captured pieces. They can be taken off the board
and do not come back into play; they can be removed from their square,
turned over to show the capturing party's colour and put on the back row of
the board to join the capturing side's army; or they can be turned over and
left on the square where they were captured. This final way of dealing with
captured pieces can have an important role in forming triumphs and winning a
game.

A piece is captured by blockade if it is in such a position on the board
that it cannot make a legal move. The edge of the board can be used in
blockading but a piece cannot be blockaded by any piece if its own colour,
i.e. all blockading pieces must be of the opposing side. Also the blockaded
piece must not be in a position where it is able to capture one of those

Capture by Arithmetic Relationship
It is important to note here that pieces do not land on an opponents square
in order to capture. Instead one or more pieces have to move into a position
where they could move to an opponents square and in doing so demonstrate an
arithmetic relationship with the piece on that square. A piece moving into
such a position is said to control an enemy piece.

Equality If a piece from one side moves into a position where it can control
an enemy piece and both pieces have the same numerical value the enemy piece
is captured.

Addition If two pieces from one side control an enemy piece and the sum of
their numerical values is equal to that of the enemy piece then the enemy
piece is captured.

Subtraction If two pieces from one side control an enemy piece and the lower
numerical values when subtracted from the greater equals the numerical value
of the enemy piece then the enemy piece is captured.
Multiplication If two pieces from one side control an enemy piece and the
multiplication of their numerical values is equal to that of the enemy piece
then the enemy piece is captured

Division If two pieces from one side control an enemy piece and the division
of the numerical values of one by the other is equal to that of the enemy
piece then the enemy piece is captured

Progression If two pieces from one side control an enemy piece and the
numerical values of the three pieces form an arithmetic, geometric or
harmonic progression then the enemy piece of the three is captured. The
enemy piece can be anywhere in the progression. If following the rules that
allow a captured piece to remain in situ but turned over, then capture by
progression is followed by winning (see below).

Distance multiplication/division If a piece from one side is so positioned
that an uninterrupted line lies between it and an enemy piece and that
distance in squares (including those both pieces are on) multiplied or
divided by the numerical value of the capturing piece is equal to the
numerical value of the enemy piece then the enemy piece is captured.

Winning
Winning Rithmomachia is achieved by a player forming an arithmetic,
geometric or harmonic progression of three pieces in the enemy half of the
board. These are called Triumphs. The three pieces may be arranged in a
number of different ways, orthogonal or diagonal rows or three points
forming a right angle. The three pieces do not have to be adjacent but it is
important that they are equally spaced such that the middle piece is
separated from the two on either side by an equal number of squares.

In one version of Rithmomachia greater Triumphs can be made after this
beginning with an arrangement of four pieces so that their numerical values
form two separate progressions. The third and greatest triumph is achieved
through the arrangement of four pieces such that their numerical values form
one each of the three progressions, arithmetic, geometric and harmonic. The
progressions for each of these victories are included in the tables within
which those progressions that can be made using the pieces from one side
alone are indicated by shading for black and outlining for white. The
individual numbers of the progressions within an arrangement of four pieces
do not have to run consecutively. It will be noted that white has the
advantage over black in having a much greater opportunity in being able to
make triumphs with four pieces demonstrating two progressions without having
to capture any black pieces.

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