Article translated and adapted by Daniel U. Thibault (D.U.Thibault@Bigfoot.com)
(Based on an article by Michel Boutin in Jeux & Stratégie #26, AprilMay 1984)
Recommended for a Fantasy setting, this game would be played by the mages and other members of the intellectual elite. It is deeply steeped in numerological mysticism, and might even be used for divinatory purposes. It would be perfectly unintelligible to the noninitiated, and thus presents quite a few interesting roleplaying possibilities!
The skill description for GURPS is:
Rithmomachia: A Hobby skill, Mental/Hard (or Very Hard), no default, prerequisites Literacy and Mathematics (at IQ). Mathematical Ability helps.
A game set varies from the simplest (pouch of rough wooden pieces and a cloth chessboard; 20 $, 1 lb.) to the exquisite (pieces and board of marble or precious wood inlaid with gold and silver; 1,000 $ and up, 1020 lbs.).
This complex chesslike game appeared in the western world around the year 1000. The game knew a great burst of popularity in the 15th century, because of some rules changes. When chess also saw its rules change (particularly when the Queen started to move in its modern fashion instead of its previous Kinglike motion), Rithmomachia started fading rapidly, at the close of the 16th century. The rules given here are those established in 1556 by Claude de Boissière, a Frenchman.
Rithmomachia is Latin for Battle of Numbers; the game is at once a battle of pawns and a battle of numbers. To play, one must be able to do a lot of quick mental arithmetic, as we shall see. This is why this game was, throughout its period, very elitist, being played mostly by high ranking churchmen and nobles. The game is flawed, in that the two sides are not equal (though you'll probably find as many experts swearing the white pieces are the better ones as you'll find opting for the black pieces!).
A pyramid is made up of stacked pieces. The pawns destined to be stacked are of progressively smaller sizes, the larger value at the bottom, the smaller on top.
As one can see, the game is a numerologist's dream! This went very well with the mystic of numbers that pervaded the Middle Ages. On with the rules!
All the pawns are reversible, being white on one side and black on the other. This is because they can be captured and flipped as the game progresses.
BLACK SIDE
25 
81 
169 
289 

15 
45 
25 
20 
42 
49 
91 
153 
9 
6 
4 
16 
36 
64 
72 
81 
2 
4 
6 
8 

six more rows 

9 
7 
5 
3 

100 
90 
81 
49 
25 
9 
12 
16 
190 
120 
64 
56 
30 
36 
66 
28 
361 
225 
121 
49 
WHITE SIDE
Both players must agree, when starting a game of Rithmomachia, on which victory conditions to use. There are seven common victories and seven proper victories.
The Common Victories are:
The Proper Victories are broken down into:
In all Proper Victories, the object is to take the opposing pyramid, and then position three or four pawns, in line or square formation, in the opponent's half of the board, so as to form an arithmetic, geometric, or harmonic progression.
In an Arithmetic Progression, the differences between successive numbers are given by a single value (called the ratio of the progression). For example: 2  5  8  11 is an Arithmetic Progression of ratio 3.
In a Geometric Progression, the ratios between successive numbers are given by a single value (called the ratio of the progression). For example: 3  12  48 is a Geometric Progression of ratio 4.
In an Harmonic Progression, the ratio of two successive differences is equal to the ratio of the end numbers. If the Progression is a  b  c, we have c/a = (cb)/(ba). This number is the progression's ratio. For example: 4  6  12 is an Harmonic Progression of ratio 3, as 12/4 = 3 and (126)/(64) = 6/2 = 3.
The Mediocre Victories are achieved by obtaining one of the progressions.
The Great Victories are achieved by obtaining two progressions at once.
The Excellent Victory is achieved by obtaining all three progressions at once.
Examples:
16 

36 

56 
This is a Mediocre Victory by Arithmetic Progression (of ratio 20)
4 
6 
12 
36 
This is a Great Victory by Geometric Progression (4  12  36, ratio 3) and Harmonic Progression (4  6  12)
12 
4 

9 
6 

This is an Excellent Victory: Arithmetic (6  9  12, ratio 3) Geometric (4  6  9, ratio 3/2) and Harmonic (4  6  12, ratio 3)
Note that in all these examples, the victories were achieved with captured enemy pawns.
There are two types of moves: Regular and Irregular. In a Regular move, the piece slides from its starting point to its end point; the intervening squares must be unobstructed. In an Irregular move, the piece jumps from its starting point to its end point, regardless of obstacles.
Round piece movement template (* Regular, + Irregular)
* 
* 

X 

* 
* 

Triangular piece movement template (* Regular, + Irregular)
+ 
* 
+ 

+ 
+ 

* 
X 
* 

+ 
+ 

+ 
* 
+ 

Square piece movement template (* Regular, + Irregular)
+ 
* 
+ 

+ 
+ 

* 
X 
* 

+ 
+ 

+ 
* 
+ 
Pyramids can move like round, triangular, or square pieces as long as they still have a representative of that shape in their stack.
Captures must precede or follow a Regular move. When the capture precedes the move, the capturing pawn takes the captured pawn's place. When the capture follows the move, the captured pawn's place is left empty. It is possible to achieve several captures at once, both before and after the move! When capturing several pawns before a move, the capturing pawn chooses which captured pawn's place to take. Captures are NOT mandatory.
Captured pawns are flipped and may be reintroduced on the board, on any free square of the player's board edge. Putting a captured pawn down replaces a move. It is possible to capture in this way.
There are six ways to capture:
The Encounter
The capturing pawn comes within one Regular move of the victim.
A 
B 
C 
D 

1 
25 

2 

3 
25 

4 
B1: Round white D3: Square black
If white moves his piece to C2, he could capture black.
The Ambush
When a number is equal to the sum, difference, product, or ratio of two opposing pawns, it can be taken on condition that both capturing pawns be within a Regular move of the victim.
A 
B 
C 
D 

1 
8 

2 
12 

3 
25 

4 
4 
B1, A4: Round black C2: Triangular white
If black moves his 4 to B3, he can take the 12 as 8+4=12 and both his pieces are within a Regular move of the white piece.
The Assault
A number encounters an opposing pawn in the same row, column, or diagonal so that the number of intervening squares is equal to their product or ratio. The intervening squares must be unoccupied.
A 
B 
C 
D 

1 
12 

2 

3 
2 

4 
6 
D1: Triangular white B3: Round black A4: Triangular black
By moving his 2 out of the way (to A2 or C4), black captures white as 12/6=2, the number of intervening squares.
The Power
When a number is equal to one of the powers or roots of an opposing pawn, the latter can be taken on condition that the capturing pawn be within a Regular move of the victim.
A 
B 
C 
D 

1 
3 

2 

3 
81 

4 
B1: Round white B3: Square black
If the 3 is moved to A2 or C2, it can take 81 as 3 is the fourth root of 81.
The Progression
When a number can be made part of an Arithmetic, Geometric, or Harmonic progression with at least two opposing pawns, it can be taken on condition that both capturing pawns be within a Regular move of the victim.
A 
B 
C 
D 

1 
25 
15 

2 

3 
20 

4 
A1: Round white D1: Square black C3: Triangular black
If black moves his 20 to A3, he can take the 25 as 152025 is an Arithmetic progression (of ratio 5) and both his 15 and 20 will be within a Regular move of the 25.
The Imprisonment
If a pawn is so surrounded that it cannot accomplish a Regular move, it can be captured.
A 
B 
C 
D 

1 
2 

2 
15 
30 
36 

3 
4 

4 
72 
B2: Triangular white D4: Triangular black
If black moves his 72 to B4, he imprisons the white 30 as all of his possible Regular moves will then be blocked by black or white pawns.
Pyramids can be taken apart one component pawn at a time, or all at once. A pyramid's value is given, at all times, by the sum of the values of its component pieces. The pyramid can itself capture, using either its total value, or the value of any one of its component pieces. The only restriction is that it cannot dislocate itself when moving.
Urhixidur