How to Play Rithmomachy

By Jaques of London · Est. 1795 · Updated 2026-08-17

Rithmomachy, also called the Philosophers' Game, is a two-player abstract strategy game played on a rectangular board with numbered pieces of three shapes: rounds, triangles, and squares. Every capture and every winning condition depends on arithmetic, making it one of the most intellectually demanding traditional board games ever devised.

Players who enjoy chess for its depth but want something rooted in pure mathematics find Rithmomachy uniquely rewarding. It is played at home, in schools with a strong mathematics programme, and at historical games conventions. A single game can last anywhere from thirty minutes to several hours depending on the victory condition agreed before play.

A little history

Rithmomachy is documented in European manuscripts from at least the eleventh century, with the earliest known detailed description attributed to the monk Asilo of Wuerzburg around 1030. It was widely played in medieval universities as a teaching aid for Boethian arithmetic and the study of number theory derived from Nicomachus. Its precise origins before that manuscript tradition are unclear. The game remained popular in educated European circles through the sixteenth century, fell into obscurity thereafter, and has been periodically revived by historians of mathematics and games.

Setting up

1. Place the board lengthwise between the two players so each has a short end in front of them.
2. White pieces occupy the two rows nearest the white player; black pieces occupy the two rows nearest the black player. The exact starting positions follow the traditional layout by piece shape and number.
3. White rounds (numbered 2, 4, 6, 8, 10, 12, 14, 16) sit in the first row, alternating with empty squares.
4. White triangles (numbered 3, 9, 25, 45, 49, 81, 121, 169 in the standard scheme) fill the second row in their assigned positions.
5. White squares occupy the outer positions of the second row according to the traditional arrangement. Black pieces mirror this layout from the opposite end.
6. Agree before play which victory condition will be used: Lesser Victory, Greater Victory, or Excellence (see Winning).
7. White moves first.

How to play

1. On your turn you must move exactly one of your pieces. You may not pass.
2. Rounds move one square diagonally in any direction.
3. Triangles move two squares orthogonally (horizontally or vertically) in a straight line.
4. Squares move three squares orthogonally in a straight line.
5. No piece may jump over another piece during its normal move, except when making a specific capture.
6. There are four types of capture. You may use any legal capture available on your turn instead of a normal move.
7. Encounter capture: move your piece onto a square occupied by an opponent piece whose number equals your piece's number. The opponent piece is removed.
8. Assault capture: position your piece so that your number, multiplied or divided by the number of squares between your piece and the target, equals the target's number. The target is removed without your piece moving.
9. Ambuscade (siege) capture: two of your pieces flank an opponent piece on the same rank or file, and the sum or product of your two numbers equals the opponent's number. The flanked piece is removed.
10. Eruption (deceit) capture: your piece can reach the opponent's square using the movement rules of a different shape than your own, and the numbers match as in an encounter capture. This is the rarest capture type and must be agreed as legal before the game begins.
11. A captured piece is removed from the board permanently.
12. You may not move into a square occupied by your own piece.
13. Play continues, alternating turns, until one player achieves the agreed victory condition or both players agree the position is drawn.

Winning and scoring

There are three standard victory conditions, agreed before play. The Lesser Victory requires you to place a set number of your pieces (typically three) in a straight line on the opponent's half of the board. The Greater Victory requires those same pieces to form an arithmetic, geometric, or harmonic progression with their numbers. Excellence, the most demanding condition, requires three or more pieces in the opponent's half forming all three types of progression simultaneously among different subsets of the group. Capturing all opponent pieces of one shape is sometimes used as a simpler alternative victory condition for beginners.

The fine points

The most common dispute concerns assault captures: the count of squares between attacker and target is measured as the number of empty squares separating them, not including either occupied square. So a piece on column 1 assaulting a piece on column 4 has two squares between them. Always count the gap, not the distance. A second frequent argument involves the eruption capture: because historical sources describe it inconsistently, it is strongly recommended to decide explicitly before the game whether eruption captures are permitted, and if so, whether any piece may use any other shape's movement or only adjacent shape categories.

In the harmonic progression required for Excellence, three numbers a, b, c satisfy the condition when (a minus b) divided by (b minus c) equals a divided by c. This calculation trips up experienced players regularly. Write it down and check it before claiming victory. A false victory claim that the opponent successfully challenges should result in the claimant forfeiting their turn, under the most commonly adopted house rule.

How to win more often

• Prioritise moving your triangles into the midfield early. Their two-square orthogonal movement gives them reach without the vulnerability of squares.
• Set up assault captures from a distance rather than advancing pieces into contact. Assault captures let you take a piece without exposing your own.
• Keep your numbers in mind at all times. The numbers on your pieces are fixed assets: losing a piece that features heavily in your planned winning progression can make victory impossible.
• Build towards your winning formation from the first move. Unlike chess, positional play and endgame construction happen simultaneously in Rithmomachy.
• Watch for ambuscade opportunities. Two modestly placed pieces flanking a high-value opponent piece can remove a critical number from their winning set.

Variations

Beginner Capture Only

New players use only encounter captures, where your piece moves directly onto a square with an equal-numbered opponent piece. All other capture types are disabled. This removes the arithmetic pressure of assault and ambuscade and lets players learn movement and board geography first.

Simplified Victory

Rather than requiring number progressions, the winner is simply the first player to place any three pieces in a straight horizontal line anywhere on the opponent's half of the board. This dramatically shortens the game and suits younger or less mathematically confident players.

Piece Capture Victory

The game ends when one player has captured all eight of the opponent's pieces of any single shape. This variant keeps the full range of captures active but removes the need to calculate progressions for the end condition, giving the game a cleaner tactical feel.

Full Classical Rules

All four capture types including eruption are legal, and the victory condition is Excellence only. This is the closest reconstruction of how the game appears in sixteenth-century printed sources. It is long, demanding, and recommended only for players already comfortable with the standard rules.

Rithmomachy: frequently asked questions