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Boolean Rithmomachia

Boolean Rithmomachia
by L. Lynn Smith

The following game is in homage to the Medieval game of numbers.  It was thought that the ideas of this game should be updated and could even be utilized in the teaching of computer math.


In order to bring this game into the 21st Century, it was thought that it should be played upon a 3D field.  To make this play as simple as possible, a 4x4x4 field was selected.


The playing pieces consist of two of each of the following.  Such pieces, except for the Pyramids, are two sided and have different colors of either side, black or white, but the same value(1 - 14) represented by the binary code.  The Pyramids are whole pieces of solid color, two black and two white, one of each particular value(0 and 15) represented by the binary code.

Value   Shape
0000   Pyramid
0001   Circle
0010   Circle
0011   Triangle
0100   Circle
0101   Triangle
0110   Triangle
0111   Square
1000   Circle
1001   Triangle
1010   Triangle
1011   Square
1100   Triangle
1101   Square
1110   Square
1111   Pyramid


The pieces of each player are arranged in two seperate planes, seperated by two planes. Players can occupy opposite sides of the playing field, either horizontally or vertically, in the following manner:


C=Circle P=Pyramid S=Square T=Triangle

Players have the option of which values may be placed in their respective cells.  This form of set-up could be considered one phase of the game as each player may take turns placing a piece upon the field.  Or they may agree to a standard form of set-up.


Circles move one orthogonal or one diagonal, Triangles move one diagonal or one triagonal, Squares move one orthogonal or one triagonal, Pyramids move one orthogonal or one diagonal or one triagonal.  The Pyramid may by landing on any piece then move that piece(regardless of owner) by any normally legal Pyramid step to an adjacent empty cell, such a piece does not change ownership solely by this action but may if it then becomes part of a capture equation.  All other pieces must move to empty cells.

[The triagonal movement is a move from one cell to the next cell which is adjacent by only one of its eight corners.  Whereas, the orthogonal move is a change along one axis and the diagonal move is an equal change along two axes, the triagonal move is an equal change along all three axes of the 3D field.]


The capture of opponent pieces involves flipping the piece to its opposite color.  Pyramids are removed from the playing field when captured.  The game is over when one player no longer has possession of either Pyramid.

Captures are preformed by legally moving a particular piece then flipping all appropriate enemy pieces.


Captures are preformed by applying Boolean equation to and through adjacent pieces.

Capture by the NOT equation.  When the appropriate enemy piece is located in a cell which, if empty, would be a legal move for the capturing piece and it is its NOT equivalent, that enemy piece is then captured.

NOT equivalents:

0000 - 1111
0001 - 1110
0010 - 1101
0011 - 1100
0100 - 1011
0101 - 1010
0110 - 1001
0111 - 1000
1000 - 0111
1001 - 0110
1010 - 0101
1011 - 0100
1100 - 0011
1101 - 0010
1110 - 0001
1111 - 0000

This is the simplest take as the attacking piece needs no assistance in capturing its opponent.  Examples:  Circle(0001) is able to capture Square(1110) if it becomes orthogonally or diagonally adjacent. Triangle(0110) is able to capture Triangle(1001) if it is diagonally or triagonally adjacent.

The remaining captures involve the use of two pieces by which to take the target.  It is necessary that the second piece used in this form of capture be also owned by the player.  

These captures take two forms, Compiling and Filtering.

Compiling, each of the two pieces used to determine the capture must be adjacent to the target piece by their individual legal move.

Filtering, the played piece must be legally adjacent to one which is legally adjacent to the target.

Several takes can be made during a player's turn, if the pieces and positions allow.

There are five forms of operations which can be preformed in order to make such captures; AND, OR, XOR, NAND and NOR.

The following table can be used as a quick reference during the game:

X | Y |  AND  |  OR  |  XOR  |  NAND  |  NOR  |
0 | 0 |   0   |  0   |   0   |   1    |   1   |
0 | 1 |   0   |  1   |   1   |   1    |   0   |
1 | 0 |   0   |  1   |   1   |   1    |   0   |
1 | 1 |   1   |  1   |   0   |   0    |   0   |

In such equations, the played piece will be referred to as X, the Compiler or Filter piece will be Y and the target piece will be the answer.  If the answer is True then the piece is captured, if False the piece is left as is.

If X = Circle(0100) and the Compiler or Filter = Triangle(1100) then the target must equal the Circle(0100) to be captured using the AND Equation.

Using the OR Equation, the previously stated values could capture Triangle(1100).

Using the XOR Equation, they could capture Circle(1000).

Using the NAND Equation, they could capture Square(1011).

Using the NOR Equation, they could capture Triangle(0011).

The Pyramid is also used to preform such captures.  If its moves involves the displacement of a piece, it still may preform the appropriate captures moves.  The displaced piece can become either a Compiler, Filter or even a Target.

All captures are mandatory, it is the obligation of either player to assist the other in determining any possible captures which may have resulted from a particular move.  Neither player is obliged to suggest any particular move to an opponent.


When an opponent's last Pyramid is captured, or there are no legal moves or potential captures, victory is determined thus:

Pyramid Victory:

The player who has the most number of Pyramids remaining on the playing field.

Piece Victory:

The player who has the largest number of pieces, including Pyramids, on the playing field.

Value Victory:

The player whose value of pieces, including the Pyramids, on the playing field is the highest total.

The Simple Victory consists of one of the above victories.
The Compound Victory consists of two of the above victories.
The Perfect Victory consists of all three victories.

It is possible that not all of the victories will be available.  Example:  Both players may have the same value of pieces on the field but not the same number, so neither would be awarded the Value Victory.

Addendum 01.03.2002

It is not necessary to use all formulae in every game.  The players may choose to play a NOT-NAND game, a game using the NOT and NAND equations.  Or they may play an OR-XOR-NOR game. The desired formulae to be used should be clearly stated before the start of each game.

Addendum 07.11.2002

Allowing the Pyramid to "push" another piece can be optional.  Denial of the "push" can increase the opportunity for the game ending with a player having Pyramids but no legal moves.  There is no direct penalty for a game ending in such a manner.

Addendum 02.16.2003

Here is an interesting way to select the equations to be used during the game:  Each player secretly writes down two choices, then simultaneously reveals them.  This allows for the possibility of either a two-function, three-function or four-function game.

This 'user submitted' page is a collaboration between the posting user and the Chess Variant Pages. Registered contributors to the Chess Variant Pages have the ability to post their own works, subject to review and editing by the Chess Variant Pages Editorial Staff.

By Larry L. Smith.
Web page created: 2008-10-05. Web page last updated: 2008-10-05