Rithmomachy
Rithmomachy Rulebook
Table of Contents
Table of Contents
Overview of a Turn
Rithmomachia is a board game played between two people, called Black (or Odd) and White (or Even). These different names (White and Even, for example) will be used interchangeably. The two players alternate turns, each turn consisting of a move of a friendly piece and possibly captures of enemy pieces. According to de Boissière, White/Even has the first turn. Other versions of rithmomachia state that Black/Odd moves first.
A move and a capture are distinguished: a move is made by a friendly piece moving to an unoccupied position, while a capture is the removal of an enemy piece from the board, where no friendly piece moves. Movement is described in the Movement section below and captures in the Captures section.
What is a player's turn?
A turn consists of three stages.
- Initial captures, if any. For instance, such captures could be the result of the opponent's previous move. In any case, more than one capture is allowed.
- A legal move. Exactly one move is made per turn, onto an unoccupied position on the board. This space could be the one occupied by an enemy piece just captured.
- Final captures, if any. This could be a capture that was made possible by the move just made.
If it is a player's turn and no legal move can be made, then that player loses the game. This is in contrast to chess, where a stalemate results in a draw. There are no draws in this version of rithmomachia.
Board and Pieces
The board is an 8 by 16 rectangular grid of 128 squares. Rows are lettered a to p and columns are numbered 1 to 8, so each position is named by a letter and a number (for example, a1 or p8). Each piece is named by its shape, its value and its position: the shape is given by its first letter, capital letters for White/Even pieces (C, P, S, T) and lower case letters for Black/Odd pieces (c, p, s, t).
Each piece has a value. The shapes are:
- Circle (C / c): a circle piece.
- Triangle (T / t): a triangle piece.
- Square (S / s): a square piece.
- Pyramid (P / p): a multi-valued square piece, described in the Pyramid section below.
Both sides have piece values 9, 16, 25, 36, 49, 64 and 81 in common. These values, all square numbers, are visible in the initial setup described below.
Each player controls 24 pieces: 8 circles, 8 triangles and 8 squares. One of the squares is the pyramid. The total value of the White pieces is 1312, and the total value of the Black pieces is 1752.
White (Even) pieces and their values
| Shape | Values |
|---|---|
| Circles | 64, 36, 16, 8, 6, 4, 4, 2 |
| Triangles | 81, 72, 49, 42, 25, 20, 9, 6 |
| Squares | 289, 169, 153, 81, 45, 25, 15 |
| Pyramid (square) | 91 |
The White pyramid is the square 91. Its value is made up of six sub-pieces, which are described in the Pyramid section below.
Black (Odd) pieces and their values
| Shape | Values |
|---|---|
| Circles | 81, 49, 25, 9, 9, 7, 5, 3 |
| Triangles | 100, 90, 64, 56, 36, 30, 16, 12 |
| Squares | 361, 225, 121, 66, 120, 49, 28 |
| Pyramid (square) | 190 |
The Black pyramid is the square 190. Its value is made up of five sub-pieces, which are described in the Pyramid section below.
Duplicate values
Some piece values appear more than once in a colour, and these duplicates are distinguished by a subscript when needed.
- White 4: circles on c6 and d5, and the square sub-piece 4 of the White pyramid.
- White 6: circle on d4 and triangle on c7.
- White 9: triangle on c8 and the square sub-piece 9 of the White pyramid.
- White 16: circle on c5 and the square sub-piece 16 of the White pyramid.
- White 25: square on a8, triangle on b6, and the square sub-piece 25 of the White pyramid. (The source's own tally counts two White 25s; the pyramid sub-piece is listed separately from the square and triangle here, as the source's piece list does.)
- White 36: circle on c4 and the square sub-piece 36 of the White pyramid.
- White 81: square on a7 and triangle on c1.
- Black 9: circles on m6 and n3.
- Black 16: triangle on n1 and the square sub-piece 16 of the Black pyramid.
- Black 25: circle on n4 and the square sub-piece 25 of the Black pyramid.
- Black 36: triangle on o3 and the square sub-piece 36 of the Black pyramid.
- Black 49: square on p1, circle on n5, and the square sub-piece 49 of the Black pyramid.
- Black 64: triangle on o6 and the square sub-piece 64 of the Black pyramid. Other piece values are unique to their colour.
Since the duplicate values are distinguished by position, the setup below lists every piece by its starting square.
Setting Up the Pieces
The initial position is shown below. White (Even) occupies rows a to d, at the bottom of the board, and Black (Odd) occupies rows m to p, at the top. Rows e to l are empty. Columns are numbered 1 to 8 from left to right.
The pyramids are at b2 for White and at o8 for Black.
Initial position, by row
Each piece is written as shape and value followed by its starting square. A subscript distinguishes duplicate values of the same colour.
Row p (Black): square 49 at p1 (subscript 1), square 121 at p2, square 225 at p7, square 361 at p8.
Row o (Black): square 28 at o1, square 66 at o2, triangle 36 at o3, triangle 30 at o4, triangle 56 at o5, triangle 64 at o6, square 120 at o7, pyramid 190 at o8.
Row n (Black): triangle 16 at n1, triangle 12 at n2, circle 9 at n3 (subscript 2), circle 25 at n4, circle 49 at n5, circle 81 at n6, triangle 90 at n7, triangle 100 at n8.
Row m (Black): circle 3 at m3, circle 5 at m4, circle 7 at m5, circle 9 at m6 (subscript 1).
Rows l to e: empty.
Row d (White): circle 8 at d3, circle 6 at d4, circle 4 at d5 (subscript 1), circle 2 at d6.
Row c (White): triangle 81 at c1, triangle 72 at c2, circle 64 at c3, circle 36 at c4, circle 16 at c5, circle 4 at c6 (subscript 2), triangle 6 at c7, triangle 9 at c8.
Row b (White): square 153 at b1, pyramid 91 at b2, triangle 49 at b3, triangle 42 at b4, triangle 20 at b5, triangle 25 at b6, square 45 at b7, square 15 at b8.
Row a (White): square 289 at a1, square 169 at a2, square 81 at a7, square 25 at a8 (subscript 1).
Note that the White square 25 at a8 and the White triangle 25 at b6 are both White pieces of value 25, and the two White pieces of value 81 are the square at a7 and the triangle at c1.
Notation of the setup
Pieces are referred to by shape, value and square. In the notation used throughout this rulebook, a White circle of value 2 on d6 is written C2 d6, a Black circle of value 3 on m3 is written c3 m3, and a White square of value 289 on a1 is written S289 a1. A move is written as the piece, the starting square and the ending square, for example C2 d6-d7 for the White circle of value 2 moving from d6 to d7.
Setup patterns
The values of the pieces in the setup follow arithmetic patterns. The top row of pieces in the initial position, the row nearest the centre of the board, is 8, 6, 4, 2 for White (row d) and 9, 7, 5, 3 for Black (row m). Each other piece value in the initial setup can be computed from the values of these top-row pieces.
Pyramid placement
The White pyramid stands on b2 and is worth 91. The Black pyramid stands on o8 and is worth 190. The pyramid is described in full in the Pyramid section below.
Movement
Move notation
A move is written as shape and value, then the starting square, then the ending square. In coordinate notation, C2:(5, 3)-(6, 3) describes moving the White circle of value 2 from column 5, row 3 to column 6, row 3. In the algebraic notation used in this rulebook the same move is written C2 d6-d7, or C2 d6d7 without the dash. For example, the move S81 a7-a4 (written 81 a7a4 in the compact form) is the White square of value 81 moving three squares along its row, from a7 to a4.
Movement rules
- The players move alternately, and each move must land on an empty space on the game board.
- No piece is allowed to jump over another. This is a move possible in some other versions of rithmomachia. De Boissière did not approve of it, calling the chess knight, which can jump over other pieces, "the mad warrior".
- Each shape has a different movement. The movement of a piece does not depend on its value.
- A circle moves exactly one space vertically or horizontally (but not diagonally) in any direction.
- A triangle moves exactly two spaces in any horizontal or vertical direction.
- A square moves exactly three spaces, but only vertically or horizontally.
- The pyramid moves like a square (see the Pyramid section below).
- No piece can move, or capture, diagonally.
- When a piece cannot move because of the positions of other pieces, it is blocked.
The source's note on repetition: other versions of rithmomachia include a "no repetition rule" ("no move can return the game to a previous state"). De Boissière's version does not use this rule, so it is not part of these rules.
Parity consequence of the movement rules. A triangle moves two spaces at a time, so a triangle that starts in column 1 can never move to an even-numbered column, and a triangle that starts in column 2 can never move to an odd-numbered column. Triangles on a given colour of square therefore never reach squares of the other colour.
Pyramid movement
The pyramid is a multi-valued piece with sub-pieces. In de Boissière's version, the pyramid piece moves just like a square piece. The pyramid can move to an unoccupied position if and only if one of its component sub-pieces can legally move to that position. Other versions of rithmomachia allow circle and triangle sub-pieces, which give the pyramid more flexibility; de Boissière's version does not.
The pyramid's sub-pieces are not separate pieces and do not move on their own. They are involved in captures, as described in the Captures section.
White pyramid. The White pyramid is initially the square 91 (S91). It has six white sub-pieces: squares of value 36, 25, 16, 9, 4 and 1. Their values sum to 91. Each sub-piece is a square and is not counted as a separate piece on the board. Algebraically, White's pyramid can be written P = S36 + S25 + S16 + S9 + S4 + S1, with the list of its values [1, 4, 9, 16, 25, 36] and its total 91.
Black pyramid. The Black pyramid is initially the square 190 (s190). It has five black sub-pieces: squares of value 16, 25, 36, 49 and 64. Their values sum to 190. Algebraically, Black's pyramid can be written p = s64 + s49 + s36 + s25 + s16, with the list of its values [16, 25, 36, 49, 64] and its total 190.
A sub-piece can only be used, and can only be captured by the enemy, as part of its pyramid.
Captures
How pieces capture
A player's turn consists of captures and one piece move. Captures are not mandatory. Capturing an enemy piece does not require a friendly piece to move. Capture requires the permanent removal of the captured piece or sub-piece: a captured piece can never be placed back on the board, even as the opposite colour. If a pyramid sub-piece is captured, that sub-piece is removed and the value of the pyramid is deducted accordingly.
Repeated captures. Repeated captures are allowed. If removing one enemy piece exposes a second enemy piece for capture (by the same friendly piece or by another), the second piece may also be captured. There is no rule that stops this. When at least two pieces are captured in this way in one sequence, it is called a chain capture.
Captures, apart from pyramid captures, are performed in one of six ways:
- numbering (or encounter)
- addition (or ambush)
- subtraction (also called an ambush)
- multiplication (or assault)
- division (also called an assault)
- siege (or surrounding)
Numbering, addition and subtraction are local captures: the capturing piece or pieces can, in one move, land on the captured piece. Multiplication and division are non-local: the capture can be made at a distance that is not the move distance of the capturing piece.
Capture types:
| Capture type | Description |
|---|---|
| numbering | 1 friendly takes 1 enemy, local |
| addition | 2 friendlies take 1 enemy, local |
| subtraction | 2 friendlies take 1 enemy, local |
| multiplication | 1 friendly takes 1 enemy, non-local |
| division | 1 friendly takes 1 enemy, non-local |
| siege | many friendlies take 1 enemy, local |
Captures are carried out during the turn as described in the turn structure. For example, if during White's turn one or more Black pieces have been captured, White removes the captured pieces and must then make a move of some piece to an unoccupied board position. This is in contrast to chess, where a capturing piece must replace the captured piece.
Numbering
If a piece can move to a position where it lands on an enemy piece with the same value, the enemy piece may be captured. For a numbering capture to occur the values of the two opposing pieces must be equal. The only values for which there is a piece of each colour with that value are 9, 16, 25, 36, 49, 64 and 81.
A numbering capture is written (attacking piece+value) x (captured piece+value). Where the context is clear it may be written without the coordinates.
Addition
Two pieces on the same side can capture together when these conditions are met:
- there is an enemy piece that either capturing piece can legally land on, and
- the value of the enemy piece is the sum of the values of the two capturing pieces.
The enemy piece is then captured. For example, if White moves a piece of value x to a square orthogonally next to a Black circle of value y and another Black circle of value z, with x = y + z, Black can capture that White piece on the next move.
Notation: [(attacking piece 1 + value 1) + (attacking piece 2 + value 2)] x (captured piece 3 + value 3). For example, in the position of Figure 13 of the source, Black captures a White circle by addition with [c3 d7 + c5 e6] x C8 d6, where the captured White circle is of value 3 + 5 = 8.
Subtraction
Two pieces on the same side can capture together when these conditions are met:
- there is an enemy piece that either capturing piece can legally land on, and
- the value of the enemy piece is the difference of the values of the two capturing pieces.
The enemy piece is then captured. Notation: [(attacking piece 1 + value 1) - (attacking piece 2 + value 2)] x (captured piece 3 + value 3). For example, Black captures a White circle of value 2 by subtraction with [c9 e3 - c7 e5] x C2 d4, that is, (9 - 7) x 2.
Multiplication
A friendly piece of value v can capture an enemy piece of value m times v if the number of vacant squares along an unoccupied straight line (vertical or horizontal) drawn between the two pieces is exactly m, where m > 0.
- If m = 1, capture by multiplication coincides with capture by numbering, and the capturing piece must be able to land on the captured piece.
- For other values of m, the capturing piece does not need to be able to land on the captured piece on its next move.
Notation: [(attacking piece 1 + value 1) * distance] x (captured piece 2 + value 2). For example, in Figure 14 of the source, Black's circle c5 (value 5) captures a White square of value 15 (S15 at i4) by multiplication, written [c5 m4 * 3] x S15 i4: there are exactly three vacant squares between them, and 5 times 3 is 15.
Division
A piece of value d times v can capture an enemy piece of value v if the number of vacant squares along an unoccupied straight line (vertical or horizontal) drawn between the two pieces is exactly d, where d > 0. Like capture by multiplication, capture by division reduces to capture by numbering when d = 1.
Notation: [(attacking piece 1 + value 1) ÷ distance] x (captured piece 2 + value 2). For example, the White square S15 at i4 captures Black's circle c5 by division, written [S15 i4 ÷ 3] x c5 m4, since three vacant squares lie between them.
Division and multiplication are symmetric: a White piece can capture a Black piece by division if and only if that Black piece could capture that White piece by multiplication.
Siege
If an enemy piece is surrounded by opposing pieces in such a way that it has no legal moves, the besieged enemy piece may be captured. The enclosure can be made by the edge of the board, with the enemy piece partly surrounded by friendly pieces, or away from the edge, with the enemy piece entirely surrounded by friendly pieces.
Notation: if attacking piece 1 is the last piece to move to help surround and block captured piece 2, the capture is written (attacking piece 1 + value 1) ∩ (captured piece 2 + value 2). For example, if White has just moved the circle C8 to d3, the capture of the Black circle c3 by siege is written C8 d3 ∩ c3 c3 (siege), as in the source's notation. (Figure 16 of the source shows a Black circle at c3 with no legal moves, surrounded by White.)
Captures of and with the pyramid
A pyramid can be captured using its total value. In addition, any one of its component sub-pieces can be captured.
- If only a sub-piece of the pyramid is captured, only that sub-piece is removed from the board, never to return, and its value is deducted from the value of the pyramid. The pyramid stays on the board.
- If the pyramid is captured using its full value, the entire piece and all its component sub-pieces are removed from the board.
A pyramid can capture an enemy piece using its total value. Alternatively, it can capture an enemy piece using the value of any one of its component sub-pieces. For example, in the source's Figure 17, the Black pyramid's square sub-piece of value 36 (s36) can capture the White circle of value 36 (C36). In a second example, the White pyramid (S91) stands against the edge of the board and is surrounded on the other three sides by Black pieces of values 3, 7 and 81. Black can capture the White pyramid as a whole by siege, but cannot capture it by addition.
How a Game Is Won
As set out in the turn structure, if it is a player's turn and that player has no legal move, that player loses the game.
The source's "How to Win a Game" heading lists two further ways a game can be won, which it calls the common and proper victories. (The source's sentence introducing them reads "A player loses when it is their turn and no legal move can be made, or when…"; the two types listed after it are the winning conditions, and are transcribed here as the conditions under which a player wins.) Both kinds of victory are settled in advance: where a victory uses a number that White and Black must agree on, the players set it before the game.
Common victories
Common victories are also called minor victories, because they are somewhat easier to manage. White and Black agree beforehand on one of the following conditions.
- Body (de Bonis victory). Assume White and Black agree on a number of captured pieces, N0. The first player to capture N0 of the enemy's pieces, irrespective of their values, is the winner. De Boissière gives N0 = 4. Another source suggests a larger proportion of the pieces, say 50% (so N0 = 12).
- Goods (de Lite victory). Assume White and Black agree on a total value of captured pieces, N1. De Boissière gives N1 = 100. Another source suggests a larger proportion of the total value, say 50%. The total value of the pieces depends on the colour (see the piece lists above), but in either case lies in the range 1300 to 1800. The first player to capture enemy pieces whose total equals or exceeds N1 is the winner. An alternative is for the players to agree that this victory occurs only if N1 is attained exactly by a combination of the captured values.
- Law-suit. Assume White and Black agree on two numbers, N2 and N3, for the number and total value of captured pieces. De Boissière gives N2 = 4 and N3 = 100. The first player to capture at least N2 of the enemy's pieces whose total equals or exceeds N3 is the winner. The winner may capture more than N2 pieces, but of them there must be N2 captured enemy pieces whose total is at least N3.
- Honor. This is similar to the law-suit, but the total value must be attained exactly. The first player to capture N2 enemy pieces whose total equals N3 is the winner. The winner may capture more than N2 pieces, but of them there must be N2 captured enemy pieces whose total is exactly N3.
- Honor and law-suit together. The first player to capture N2 enemy pieces (no more, no less) whose total equals N3 is the winner. To win by this method the player must capture exactly N2 enemy pieces, and their total must be exactly N3, no more and no less.
Proper victories
A proper victory is also called a major victory. De Boissière's text on the proper victory reads: "…those who make war win a victory after routing the enemy's battle-line and arranging everything in accordance with their own will, so too the harmony and arrangement of numbers…" (Claude de Boissière, pages 206-207 of the Richards translation cited by the source).
A proper victory is won by placing a fixed number of friendly pieces in the enemy territory, such that their values form a coordinating progression: an arithmetic, geometric or harmonic progression. The three kinds of progression are:
- Arithmetic: a sequence a, b, c with b − a = c − b. Example: 2, 4, 6.
- Geometric: a sequence a, b, c with b/a = c/b. Example: 2, 4, 8.
- Harmonic: a sequence a, b, c with 1/a − 1/b = 1/b − 1/c, so that the reciprocals form an arithmetic sequence. Examples: 9, 15, 45 and 9, 16, 72, both of which White has.
There are three types of proper victory:
- Small victory: place three friendly pieces in enemy territory whose values a, b, c form an arithmetic, geometric or harmonic progression.
- Big victory: place four pieces in enemy territory; one subset of three of them forms a coordinating progression, and another subset of three forms another coordinating progression.
- Great victory: among the four pieces there are subsets of three that form an arithmetic progression, a geometric progression and a harmonic progression. A player wins by a great victory when they have a big victory and, moreover, among the four pieces in enemy territory there is a subset of three forming an arithmetic progression, a subset of three forming a geometric progression, and a subset of three forming a harmonic progression.
Remark: in some versions of rithmomachia the three pieces of a coordinating progression must be arranged in a straight line (vertical, horizontal or diagonal), and other versions allow them to form a right-angled triangle. No such linear or geometric condition applies in de Boissière's version.
Example: in the source's Figure 21, White wins by geometric progression, with three pieces in enemy territory at i1, i4 and i7 whose values are 2, 4 and 8 in geometric progression.
Summary of Rules and Setup
This summary restates the rules above, from section 6.2 of the source.
- Board: The game is played on an 8 by 16 rectangular board.
- Players and pieces: Each player controls 24 pieces (White/Even and Black/Odd), comprising 8 circles, 8 triangles and 8 squares. One of the squares is a special multi-valued piece called a pyramid.
- Movement:
- Circles move orthogonally one space.
- Triangles move exactly two spaces orthogonally.
- Squares and pyramids move exactly three spaces orthogonally.
- No piece may jump over another; movement must be unobstructed.
- Capturing:
- Capture by numbering: the attacking piece is one move away.
- Capture by addition or subtraction: one move away for each of the attacking pieces.
- Capture by multiplication or division: the attacking piece is at a distance d. The source's wording is "its value equals d (divided by/times) the enemy's value"; read against the Multiplication and Division sections above, the attacker's value multiplied by d gives the enemy's value for multiplication, and the attacker's value divided by d gives it for division.
- Siege capture: an enemy piece is flanked on all sides by friendly pieces.
- Victory conditions:
- Proper victory: three of your pieces in arithmetic, geometric or harmonic progression on the opponent's side of the board.
- Common victory: some pre-agreed proportion of the opponent's pieces, by count or by total value.