The story of Rithmomachy is extraordinary; once considered the most intellectually compelling game of all, even more so than Chess, played by men of learning across central Europe, and rivalling Chess for popularity, it gradually lost appeal, and by the eighteenth century, had entirely disappeared, except as described in moldering tomes.
The origins of the game are a matter of conjecture; some have claimed origin in Ancient Greece, but this seems unlikely (and the notion is probably predicated on the mathematical nature of the game). Parlett believes it originates sometime in the 11th or 12th century, but an earlier origin is conceivable, since the rules of the game are closely related to the Pythagorean numerology of Boethius, a Christian philosopher of the 6th century, best known for his work De Consolatione Philosophiae (On the Consolation of Philosophy).
Indeed, for its players in Medieval times, the numerological aspect of the game must have been one of its main appeals; keep in mind that, in the period, mathematics was prized less for practical reasons (e.g., applicability to science and engineering), than for its beautiful regularity and depth and therefore its ability to cast light on the nature of the mind of God. Thus, a student of Chess was merely someone striving to master a game, but a student of Rithmomachy would be engaged in pursuit of the divine mysteries -- as well as striving to master a game, of course.
Unfortunately, while there are a number of descriptions of the game, none agree on the precise rules, and it is likely that there were many variations in different places and at different times. But we do know some things:
It was played on a double-sized checkerboard, that is, one 8x16 squares in extent. One player (white) had pieces with numerical values deriving from even numbers, and the other (black) from odd numbers. Each had eight round pieces, represented as circles, four containing the even or odd numbers between 1 and 9, the other containing their squares (thus, white had 2, 4, 6, 8, 4, 16, 36, and 64; black had 3, 5, 7, 9, 9, 25, 49, 81). Each had eight triangular pieces, four of them based on the the "original" numbers times the next number numerically, and the other four the square of "the next number numerically." Thus, white has 6 (2x3), 20 (4x5), 42 (6x7), and 72 (8x9), and also 9 (3x3), 25 (5x5), 49 (7x7) and 91 (9x9). Similarly for black, but starting with the odd numbers as base. Each also had eight square pieces, derived from more complicated equations (2 m squared, minus m; and n plus m squared together -- where n represents the base numbers of each side, odds or evens, with m the next number sequentially).
To complicate things, white's 91 piece is omitted and instead constructed of a pile of six pieces (rounds 1 and 4, triangles 9 and 16, and squares 25 and 36); black's 190 is similarly constructed of other pieces (round 16, triangles 25 and 36, squares 49 and 64). These composite pieces are called pyramids.
Pieces are set up on either side of the board in a set pattern similar to that from the screenshot above; in some versions, they do not begin at the outermost rank of squares, but two squares toward the center of the board.
As in Chess, players alternate moves. Rounds move one square; triangles two; and squares three. You may not move fewer squares than indicated by the piece's shape, and intervening spaces must be vacant. The pyramids may move as permitted by any of the shapes that constitute the pyramid -- but since pieces may be removed from the pyramid during play, if all pieces of a shape have been lost from the pyramid, it may no longer move as appropriate for that shape.
Different sources vary on the nature of movement, however; for example, Selenus says that rounds may move forward only, triangles only orthogonally, and squares and pyramids in any direction; while Illmer says that rounds may move backward or forward, but only orthogonally, triangles only diagonally, and squares and pyramids in any direction. (Other versions are known as well.)
And now for the truly arcane piece: There are several methods of capture. The simplest is by blockade -- my pieces are situated in such a way that one of your pieces has no legal moves, ignoring the presence of your own pieces.
More important (and common) is the arcane rule that if one or two of my pieces are capable of moving to the square occupied by one of your pieces, and either one of them alone, or two of them in combination, have some mathematical relationship to your piece, I may capture it. Thus, let us call your piece B, and my pieces A1 and A2; I can capture your piece if any of the following is true:
- A1 or A2 = B
- A1 + A2 = B
- A1-A2 = B
- A1 * A2 = B
- A1 / A2 = B
- If the three pieces form an arithmetical progression, with B in any order in that progression -- that is, the difference between any pair divided by the difference between the other pair is one
- If the three pieces form a geometrical progression, with B in any order in that progression -- that is, if the difference between any pair divided by the difference between the other pair is equal to the ratio between the smallest and middle numbers
- If the three pieces form a harmonic progression, with B in any order in that progression -- that is, if the difference between any pair divided by the difference between the other pair is equal to the ratio between the smallest and largest of the numbers
What's interesting about this is your ability to capture is realistically based on your ability to do math in your head.
In addition, a single piece can capture an enemy piece if it that piece is in the first piece's permitted direction of movement, there are no intervening pieces, and the distance times the first piece's value equals the second piece's value -- or, contrariwise, if the first piece's value divided by the distance equals the second piece's value.
If you can make more than one capture on your turn, you may make all.
Unfortunately, sources also vary on what 'capture' means: in some cases, the piece is removed; in others, the piece is flipped over, where it has the same number in the other player's color, and either left in situ or re-entered by the new owner at his board edge.
Also, it is possible to capture individual component pieces within a pyramid -- or the pyramid as a whole.
There are also a whole slew of different victory conditions in different works; the Wikipedia article linked above lists several.
The reason for Rithmomachy's decline and disappearance can only be conjectured; Parlett suggests that as mathematics lost its mystical connotations and became perceived as a practical tool, it lost its fascination for players -- and that, also, players came to realize that the game was more complex than but without the strategic depth of Chess.
If you want to give the game a try, the shareware Ambush, linked above, is a Rithmomachy variant -- unfortunately, not a particularly faithful one. Game Cabinet also provides a translation of the Boissiére rules, which date from 1556.
Update: Apparently, a British company will sell you a nicely made wooden set of the game for a cool 115 quid. I'm tempted. Okay, not very.