The Rules for Rithmomachy

These are the rules at Boardspace - see rules variations for a discussion on the alternatives.  Generally, these rules treat all the pieces and movements uniformly, and aim for a fast resolution.

The game is for two players, black and white.  

Equipment::  and 8x16 board, 28 black pieces, 29 white pieces with numbers as shown, arranged as shown.

The pieces are either rounds, triangles, or squares, except that each player has one stack of pieces which is called a pyramid.
White rounds are 2,4,4,6,8,16,36, 64
White triangles are 6,9,20,25,42,49,72,81
White squares are 15,25,45,81,169,153,289
The White pyramid contains 1 and 4 round, 9 and 16 triangle, 25 and 36 square.
Black rounds are 3,5,7,9,9,25,29,81
Black triangles are 12,16,30,36,56,64,90,100
Black squares are 28,49,66,120,121,225,361
The Black pyramid contains 1 round, 25 and 36 triangle, 49 and 64 square.
starting position

Object of the game is to win by any of


Players alternate turns, starting with Black
Movement is only possible to an empty square. 
Horizontal and vertical moves are blocked by intervening pieces.
Moves with a diagonal component can't be blocked - they're like chess knights moves.

round movemen

Rounds move one step diagonally
triangle movement
Triangles move 2 steps horizontally, or a knight's move
square movement
Squares move 3 steps horizontally or vertically, or a longer knights move
pyramid movement
Pyramids move in any of the ways permitted by any of the pieces they still contain.


  • Capturing depends on the geometric or numeric relationship between the captured and capturing pieces.
  • Capturing never involves moving into the square occupied by the captured piece.  Capturing pieces stay where they are.
  • If by luck or design multiple captures are possible after a single move, they all occur.
  • Pyramids can capture or be captured using either their total value, or the value of any component
Capture by Siege
A piece is captured by siege if it is surrounded by enemies or the edge of the board on all 4 orthogonal or all four diagonal directions.
capture by siege
Four whites surround a black piece.
capture by siege
Two whites capture the entire black
pyramid using the edge of the board
Capture by Equality
A piece or stack is captured by equality if the attacking piece (or stack) has the same value, and if the attacking piece could move to the position of the attacked piece if it were vacant.  Note that very few of the black and white pieces have the same value.
capture by
Black 16 captures white 16 by equality (it can move there by one of its triangles). 
capure by
White 64 captures black 64 by equality. 
The reverse is not the case;
black 64 can't move to white 64's location.
Capture by Eruption
A piece or stack is captured by eruption if the value of the smaller, multiplied by the distance between the pieces, is equal to the value of the greater.  Captures by eruption do no depend on the natural movement of the pieces.  The starting and ending locations are both counted, so the minimum distance between pieces is 2.
capture by
6 or 12 would capture by eruption,
depending on whose move it is.
capture by
9 or 81 would capture by eruption, depending on whose turn it is.
Capture by Ambush
A piece is captured by ambush if two pieces, which could move to the enemy to be captured, have a sum, difference, product, or quotient equal to the captured piece value.
capture ambush
White 45 and 4 capture black 49
 (49 = 45+4)

capture ambush
White 4 and 36 capture black 9
( 9 = 36/4)

Glorious Victories

Glorious victories are an arrangement of 3 pieces which meet several conditions
  • all three pieces (A,B,C) are on the opponent's side of the board
  • they form an unobstructed evenly spaced horizontal or vertical line or a triangle.
  • the values of the pieces form an arithmetic, geometric, or harmonic progression
    • arithmetic progression, (B-A) = (C-B)   examples:  2,4,6     9,81,153
    • geometric progression, (A*C) = (B*B)  examples:  2,4,8    20,30,45
    • harmonic progression, ((A+C)*B) = (A*C*2)   examples: 2,3,6   30,45,90

This page has a table of all solutions.

win by arithmetic
White wins by forming an arithmetic progression on Black's
side of the board.  (2-1) = (3-2)
win by geometric
Black wins by forming a geometric progression on White's side
of the board.  (1*36) = (6*6)
win by harmonic
White wins by forming a harmonic progression on Black's
side of the board.  (2+6)*3 = ((2*6)*2)

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