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How To Play Rithmomachia

Updated on October 23, 2014
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Jeff Johnston is a medieval reenactor and avid history fan. He is also the publisher at Living History Publications.

The Philosopher's Game

Rithmomachia, or the Philosophers Game is a complex mathematical game not for the faint of heart. It has often been reputed that the game was invented by the great mathematician and philosopher Pythagoras, however there is no real evidence of this, although certainly without the works of Pythagoras and others of his stature the game would not be possible.

It is certainly evident that its popularity in the 11th and 12th centuries in monastic schools throughout Germany and France was epic. As early as 1030 there were competitions between schools in Rithmomachia, notably one between Würzburg and Worms, two schools well known for their mathematical prowess. Some think a monk named Asilo created the game specifically for this competition based on the work of Boethius in order to teach his mathematical theories. Like so many other games, the real origins of Rithomachia have been lost.

Rithmomachia is a two player game, generally one plays white the other black. The board is a sixteen by eight grid. The pieces are complex, but they break down to four basic types, Rounds, Squares, Triangles and Pyramids. Pyramids are not an individual piece exactly, but rather a stack made up of several other pieces they are stacked with the highest value piece on the bottom and the lowest on the top. The game is unbalanced, but fans of the game are divided as to which side has an actual advantage.

The game teaches you to watch for mathematical functions within the game. It requires strategy, mathematical prowess, and quick thinking. It's a great way to teach kids to look for math everywhere.

How To Win

Before beginning the players must agree on victory conditions. There are several methods of victory that fit into two basic catagories, Common Victories and Proper Victories.

Common Victories:

  • The Victory of Body: Taking a set number of the opponents pieces.

  • The Victory of Assets: To take a set value of the opponents pieces.

  • The Victory of Procedes: To take a set number of digits for example you could agree that whoever takes the 9 25 and 81 pieces wins.

Proper Victories:

In proper victories the object is to take the opponents pyramid then form an arithmetic, geometric or harmonic progression in the opponents side of the board.

  • The Mediocre Victory: Achieving a single progression using at least three pieces.

  • The Great Victory: Achieving two progressions at once using at least four pieces. In this case two pieces could fit into two separate progressions for example 5 10 15 25 would work as 5 10 15 is an arithmatic progression of 5s and 5 15 25 is an arithmatic progression of 10s. Of course both progressions aren’t required to be the same type you could have an arithmatic and a geometric.

  • The Excellent Victory: Achieving three progressions using at least four pieces.

The Pieces

White pieces:

  • Rounds 3, 5, 7, 9, 9, 25, and 81

  • Triangles 12, 30, 56, 90, 16, 36, 64, and 100

  • Squares 28, 66, 120, 49, 121, 225, and 361

  • The white pyramid is made up of a Square 64, Square 49, Triangle 36, Triangle 25, and a Round 16 for a value of 190

Black pieces:

  • Rounds 2, 4, 6, 8, 16, 36, and 64

  • Triangles 6, 20, 42, 72, 9, 25, and 81

  • Squares 15, 45, 153, 25, 81, 169, and 289

  • The black pyramid is made up of a Square 36, Square 25, Triangle 16, Triangle 9, Round 4, and a Round 1 for a value of 91

Board Set Up

A standard Rythmomachia board is an 8X16 grid, often checkered. It can be played by setting two chessboards side by side with one board as the white side and the other as the black side. The board above shows the starting position of the pieces, you may note the black and white are reversed on this picture.

Valid Moves

There are two types of moves in the game, regular and irregular, regular moves requires the path be unobstructed, which means no men in their path, irregular can move regardless of anything it its path. Circles only move regularly, there are no valid irregular moves for circles, they can move one space diagonally. Triangles move two spaces horizontally or vertically for their regular moves, and like a rook in chess (two spaces horizontally or vertically then turn 90º in any direction and one move one space) for irregular moves. Square pieces move three spaces horizontally or vertically for regular moves and for irregular three horizontal or vertical moves and then a 90º turn in any direction and move one space.

Capturing An Opponent's Piece

There are several ways to capture an opposing players pieces. A capture can take place either before or after a regular move:

  • The Encounter: During a turn if an opponent’s piece is within one regular move of your piece and it is of equivalent value you can capture that piece and take its place.

  • The Imprisonment: If a piece has all possible regular moves blocked by an opponent’s pieces that piece may be taken by the opponent.

  • The Ambush: The ambush requires two capturing pieces to be within one regular move of the victim, if the sum, difference, product, or ratio of the two capturing pawns then one of the two pieces can capture the opponent pawn and take its place. Many variants only allow for the capture to take place if the two capturing pieces sums add up to the captured piece and ignores the ratio, difference, or product.

  • The Progression: When a piece can be made up of part of an arithmetic, harmonic, or geometric progression with at least two opposing pieces that piece can be taken, assuming both opposing pieces are within one regular move of the victim. For example if a 20 and 15 were both within one move of 25 the 25 can be taken as they are part of an arithmetic progression by fives.

  • The Assault: If a piece encounters an opponent in the same row, column or diagonal that has the number of unoccupied spaces between time equal to the product or ratio of the two numbers the opponents piece can be captured.

  • The Power: When a number is equal to the power or roots of an opposing pawn it may be taken if it is within one regular move of said opposing pawn. For example if 3 is within one regular move of 81 the three can take the 81 because 3 is the fourth root of 81.

    Pyramids can capture or be captured using either any one of the individual pieces, or the combined value of all remaining pieces.

    It is not mandatory to capture a piece if it is possible, there may be strategic advantage to not capture a piece and as such a player may choose to leave it alone.

The Avacal Games Guild is a group of SCA players in the Principality of Avacal who are interested in Medieval Games.

The Badge of the Avacal Games Guild
The Badge of the Avacal Games Guild | Source

© 2014 Jeff Johnston

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    • esmonaco profile image

      Eugene Samuel Monaco 3 years ago from Lakewood New York

      I've never heard of this game, but do love a good challenge, this sounds like a fun and very interesting game. Thanks :)

    • CuAllaidh profile image

      Jeff Johnston 3 years ago from Alberta Canada

      @Merrci: Yes the rules are complex and it requires that you are always paying attention and thinking.

    • CuAllaidh profile image

      Jeff Johnston 3 years ago from Alberta Canada

      @PAINTDRIPS: The SCA is great, its my activities with the SCA that got me hooked on medieval games, and mead making actually. This is not a real popular game in the SCA as far as I know, a few know of it, and fewer play it, but it's not overly popular :D

    • PAINTDRIPS profile image

      Denise McGill 3 years ago from Fresno CA

      What a fascinating game. I never heard of it either but played a few medieval games in my time with SCA. Talk about luminosity... this would really give your brain a work out. Wish I knew about it when I was homeschooling my kids... they would have loved it.

    • Merrci profile image

      Merry Citarella 3 years ago from Oregon's Southern Coast

      It looks like it might take some time to learn, but how rewarding it must be when it clicks! I'm impressed!

    • CuAllaidh profile image

      Jeff Johnston 3 years ago from Alberta Canada

      @RinchenChodron: It is an interesting challenging game. I research medieval games for fun, came across this one, took me ages to fully redact the rules, my brain hurt ;)

    • profile image

      RinchenChodron 3 years ago

      I have never heard of this game, but it does look challenging and fun. Thanks for the introduction.