RITHMOMACHIA --A GAME FOR MAGES? by Daniel U. Thibault (Based on an article by Michel Boutin in Jeux & Strategie #26, April-May 1984; the translation is mine) GURPS applicability =================== Recommended for a Fantasy setting, this game would be played by the Mages and other members of the intellectual elite. It is deeply steeped into numerological mysticism, and might even be used for divinatory purposes. It would be perfectly unintelligible to the non-initiated, and thus presents quite a few interesting role-playing possibilities! 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, 10-20 lbs). Historical Background ===================== This complex chess-like game appeared in the western world around the year 1000. The game knew a great burst of popularity in the XVth 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 King-like motion), Rithmomachia started fading rapidly, at the close of the XVIth century. The rules given here are those established in 1556 by Claude de Boissiere, a Frenchman. Introduction ============ 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 than you'll find opting for the black pieces!). Components ========== - A 16 by 8 chessboard - 57 pawns, 29 black ones and 28 white ones, broken down as follows: White pieces - 8 round pieces, of values 3 / 5 / 7 / 9 / 9 / 25 / 49 / 81 (odd numbers 3 to 9 and their squares) - 8 triangular pieces, of values 12 / 30 / 56 / 90 / 16 / 36 / 64 / 100 (3*4 / 5*6 / 7*8 / 9*10 and the squares of 4 / 6 / 8 / 10) - 7 square pieces, of values 28 / 66 / 120 / [190] / 49 / 121 / 225 / 361 (4*7 / 6*11 / 8*15 / [10*19] and the squares of 7 / 11 / 15 / 19) - the preceding 190 piece is broken down into a pyramid of round 16 (4*4) triangular 25 (5*5) triangular 36 (6*6) square 49 (7*7) square 64 (8*8) A pyramid is made up of stacked pieces. The pawns destined to be stacked are of progressivly smaller sizes, the larger value at the bottom, the smaller on top. Black pieces - 8 round pieces, of values 2 / 4 / 6 / 8 / 4 / 16 / 36 / 64 (even numbers 2 to 8 and their squares) - 8 triangular pieces, of values 6 / 20 / 42 / 72 / 9 / 25 / 49 / 81 (2*3 / 4*5 / 6*7 / 8*9 and the squares of 3 / 5 / 7 / 9) - 7 square pieces, of values 15 / 45 / 153 / 25 / 81 / 169 / 289 (3*5 / 5*9 / [7*13] / 9*17 and the squares of 5 / 9 / 13 / 17) (note that 3+2=5 / 5+4=9 / 7+6=13 / 9+8=17) - the preceding 91 piece is broken down into a pyramid of round 1 (1*1) round 4 (2*2) triangular 9 (3*3) triangular 16 (4*4) square 25 (5*5) square 36 (6*6) 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. Set-up ====== 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 Victory conditions ================== 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 Victory of body To take a set number of pawns The Victory of assets To take a set total value in pawns The Victory of proceeds To take a set number of digits The Victory of body and assets The Victory of body and proceeds The Victory of assets and proceeds The Victory of body, assets, and proceeds The Proper Victories are broken down into: Mediocre Victories Great Victories Excellent Victory 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 = (c-b)/(b-a). This number is the progression's ratio. For example: 4 - 6 - 12 is an Harmonic Progression of ratio 3, as 12/4 = 3 and (12-6)/(6-4) = 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|___|___| This is a Mediocre Victory by |___|___|___|___| Arithmetic Progression (of ratio 20) |___|___|___|_56| _______________ |___|___|___|___| |__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|___| This is an Excellent Victory: |___|___|___|___| Arithmetic (6 - 9 - 12, ratio 3) |__9|___|__6|___| 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. Movement ======== 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 ___________________________ |___|___|___|___|___|___|___| |___|___|_+_|_*_|_+_|___|___| |___|_+_|___|___|___|_+_|___| |___|_*_|___|_X_|___|_*_|___| |___|_+_|___|___|___|_+_|___| |___|___|_+_|_*_|_+_|___|___| |___|___|___|___|___|___|___| Square piece movement template ___________________________ |___|___|_+_|_*_|_+_|___|___| |___|___|___|___|___|___|___| |_+_|___|___|___|___|___|_+_| |_*_|___|___|_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 ======== 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 re-introduced 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|___|___| B1: Round white D3: Square black 2 |___|___|___|___| 3 |___|___|___|_25| If white moves his piece to C2, he could 4 |___|___|___|___| 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|___|___| B1, A4: Round black C2: Triangular white 2 |___|___|_12|___| 3 |___|___|___|___| If black moves his 4 to B3, he can take the 4 |__4|___|___|___| 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| D1: Tri. white B3: Round black A4: Tri. black 2 |___|___|___|___| 3 |___|__2|___|___| By moving his 2 out of the way (to A2 or 4 |__6|___|___|___| 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|___|___| B1: Round white B3: Square black 2 |___|___|___|___| 3 |___|_81|___|___| If the 3 is moved to A2 or C2, it can take 4 |___|___|___|___| 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| A1: Round white D1: Square black C3: Tri. black 2 |___|___|___|___| 3 |___|___|_20|___| If black moves his 20 to A3, he can take the 4 |___|___|___|___| 25 as 15-20-25 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|___|___| B2: Tri. white D4: Tri. black 2 |_15|_30|___|_36| 3 |__4|___|___|___| If black moves his 72 to B4, he imprisons 4 |___|___|___|_72| the white 30 as all of his possible Regular moves will then be blocked by black or white pawns. Pyramids ======== 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