Rithomachia was played by the intellectuals of Europe, often Church employees and most likely male. Although it is obscure today, rithomachia played an important part in the lives of this elite group. The rules were complex and convoluted and required considerable computational skill. Its players also claimed that its benefits extended well beyond numerical training - it supposedly improved the moral character of players and even gave deep insights into religious truth.
There are many different versions of the rules of play but there are many common features. There are two players who take turns moving pieces on a board, much like chess or checkers. The board is something like a chess board, though rectangular and not square, with 48 pieces, each inscribed with a number. The size of board varies with rule set but eventually it was standardized on 8 by 16.
But now the complication starts. Rithomachia was inspired by a type of mathematics made popular by Boethius. This work was dominated by various types of numerical progression: in particular arithmetic progressions (where the nth number is a+b*n for constant a and b), geometric progressions (a*b^n) and harmonic progressions (1/(a+b/n)). There were also other more complex types of progression such as multiplex progression or the progression of superparticulars. At the start of the game the initial layout of the numbered pieces, in three ranks for each side, was such that the numbers formed various types of progression. One side had odd numbers in the first rank, the other side even. So even before the game had started quite a bit of mathematical knowledge was required. And don't forget that in earlier times arabic numerals were unknown so any calculations were carried out using Roman numerals. (By the way, these proportions played an important part in medieval music theory. Consider the harmonic progression in particular.)
Pieces from the first rank could move one square, those from the second rank two and those from the third three. As a mnemonic the pieces in these ranks were sometimes circles, triangles and squares respectively (a slightly illogical mnemonic!). Like in chess, pieces could be captured, however there were many ways to capture. One way was to move a piece with a number onto another with the same number. Two pieces could take another if it was the case that if they were simultaneously to make legal moves that would land them both on the captured piece and the sum of the numbers on the two pieces sums to that of the captured one. Another way to capture was to occupy all the spaces that another piece could move to making it unable to move - this was called besieging it. Here is another example of a capture rule taken from Lever and Fulke's "The Most Noble and Auncient, and Learned Playe" published in 1563:
Of taking by cossical signs
Obviously there has been a little semantic drift over the centuries. There were also some much more complex capture rules requiring the pieces to be arranged in a progression.
By cossical signs: any man that hath these signs, 3, &, 33, 3&, meeting with his root in his ordinary draught that hath this sign z taketh him up, or else is taken of his, without removing into his place; except he may not take him before he remove.
Now we've warmed up we can get onto the conditions to achieve victory in the game. Typically a player had to line up a series of pieces in certain arrangements, often as a Boethian progression of a certain length. Whoever did this would win. There were many different choices of victory conditions based on such progressions and players would negotiate before to game to decide which were in play for a particular game. An example simple victory condition follows:
Victory of goods
There was even a victory condition based on simulating the armies of the Christians and Turks at war.
Victory of goods is to take a certain number without respect of the men. As if it be covenanted that he which first taketh men amounting to the number of 100 or 200 shall have the victory.
As you can see, this was no game for weenies and yet it was played all over Europe from Medieval times, through the Renaissance into Elizabethan times. Many rithomachia manuals still exist today and doubtless many more were originally published. It seems to me there is only one explanation: Rithomachia is a geek game and the players were predecessors of today's game playing geeks! They spent their time shut up in dark rooms hunched over books and games believing themselves to be superior to the masses because they were experts in difficult and arcane, but largely useless knowledge.
Today Boethian mathematics is almost unheard of. This is the key to the decline of rithmomachia. Boethian mathematics was highly technical but today it's abstruseness seems completely arbitrary and useless. As it was replaced by more modern approaches to mathematics the rules to rithmomachia came to seem more and more arbitrary until interest in the game completely waned. However it is worth noting that this game did have a lifetime of 500 years and was played by such illustrious luminaries as John Dee and praised by Roger Bacon.
Incidentally there was a recent resurgence in the play of rithmomachia when it was released as a shareware computer game called Ambush.
Unlike chess it was probably not played in Iraq and the pieces were typically not made of sapphire.
- The Philosopher's Game, Ann E Moyer, University of Michigan Press (also contains the entire text of the rithmomachia Lever and Fulke's 1563 manual)
- Das mittelalterliche Zahlenkampfspiel, Arno Borst, Carl Winter Universitätsverlag