Sources: Fulke's The Philosopher's Game. This is the first of the three variants he describes.
This particular version of Rhythmomachy is my personal favorite. It is in many respects the strangest variant: the rules are considerably different from the "main" versions of Rhythmomachy that were common for much of the Middle Ages and Renaissance. It really cannot be justified outside the 16th century, although it is possible that it was known earlier. However, it is significantly more consistent, and arguably easier to learn and play, than the main version.
The reconstruction here is rather long; however, there is a concise summary at the end. The game actually summarizes very briefly -- much of the length here is in describing all of the little caveats, and a few of the odder concepts. (Along with lots of examples.)
In all of the following descriptions, the man begins on A; I describe where he can move to. You can not jump over another man during a movement, either one of your own, or an opponent's.
For the triangles and squares, there are two distinct forms of movement, normal and "flying". I confess, it isn't absolutely clear to me what flying refers to. It may mean when you are about to be attacked, and are running away; or, it may mean any time except when you are attacking. I currently believe the latter interpretation is more likely. Thus, you may use the flying movement unless the move would result in taking an opposing man.
The Rounds: The round pieces move one step diagonally. Thus, a round on A can move to B, C, D, or E.
The Triangles: The triangles move orthogonally (that is, horizontally or vertically), two spaces. Thus, a triangle on A can move to F, G, H, or I. When flying, a triangle may make the knight's move from Chess -- that is, two spaces either horizontally or vertically, and then one more to the side. Thus, a triangle flying from A could move into any of spaces 1 - 8. Note that you still cannot jump over a man when flying, so you could not move from A to 1 if there was a man standing in F. (Fulke is not completely clear on this point; I infer it from the way Capture by Oblivion works.)
The Squares: The squares move orthogonally, three spaces. Thus, a square on A can move to L, M, N, or O. When flying, a square can make a sort of elongated knight's move -- that is, three spaces horizontally or vertically, then one to the side. Thus, a square flying from A can more to any of 11 - 18. (The same caveats apply as with the triangles, with respect to jumping over other men.)
The Kings: The Kings can move in any of the ways that their constituent pieces can. Since a King starts with Rounds, Triangles, and Squares, it can move with all of the movements of all of those pieces. However, a King can lose some of those constituents during play; that is, the enemy can capture pieces out of the King's pyramid. If the King loses all of the pieces of a given shape, it loses the ability to move like that shape. So if both triangles in a King are captured, it can no longer move like a triangle.
Fulke talks, somewhat cryptically, about the Kings being able to move in all the ways that a Chess Queen can move, within a three-space span. This would seem to obviate the fact that movements go away as the King loses components, though, and seems generally a little inconsistent with how the rest of things run. So I tend to ignore this, since I don't entirely understand it.
The most important, and sometimes confusing, aspect of capture in Rhythmomachy is that you don't actually have to jump onto an enemy man to capture him. In Chess, you capture the opposing piece by moving one of your pieces into his square. In Rhythmomachy, you only have to set up the position -- as soon as you could capture him, you do so. Alternatively, you can capture by jumping onto the enemy man, but you don't have to; you can essentially capture a move earlier. The result is that you must pay much more attention in Rhythmomachy than in Chess: by the time a man is in check, he is already captured. This will be made a little clearer by looking at the simplest example: Capture by Equality.
Capture by Equality: The simplest capture (although the rarest) is Capture by Equality. This just means that a man can capture another man with the same number: a 9 can capture the other 9, an 81 the other 81, a 4 the other 4, and like that.
As alluded to above, you can capture this way by simply leaping onto the other man, if that is a legal move for your piece. If it is your turn, and you have the round 81 on A, and the enemy square 81 is on B, your round can jump onto his square and capture it, because B is a legal move for a round on A. However, you will usually capture just by setting it up. Say instead that it is your turn, your round 81 is on C, and the enemy square 81 is on B. You move your round to A. Since the enemy is now within reach of yours, it is immediately captured, without you moving your piece into the enemy space. (However, note that the enemy man must be in a space that is a legal move for yours.)
Capture is therefore much richer in Rhythmomachy than Chess. Indeed, although the Fulke doesn't say it, you should be able to make multiple captures with a single move -- if you move your man so that it attacks multiple enemies, they should all go away. (This is incredibly difficult to set up, but quite stylish.)
Capture by Addition: The other captures are mostly pretty much alike; I will start by illustrating the simplest case, addition. If you place two of your men, such that both of them could legally jump onto an enemy man, and they add up to its value, the enemy is immediately taken.
So, for example, say that the round black 16 is on A, and you have a round white 9 on B and a round white 7 on F. You move the 7 to C. Since the 9 and the 7 could both legally move to A, and they add up to 16, the 16 on A is immediately captured.
Alternately, say that you have your round 9 and 7 on B and C. On your opponent's turn, he foolishly moves the 16 to A. You may now capture the 16, but you have to actually jump onto it -- either the 9 or the 7 (it doesn't matter which), must jump onto the 16, and thereby take it.
So like with Capture by Equality, there are two ways to do Capture by Addition: moving your pieces so that they could jump onto the enemy, or actually jumping onto it on your turn.
Capture by Subtraction, Multiplication, Division: These work essentially the same as Capture by Addition. In each case, you arrange two of your pieces, whose numbers relate to the enemy piece, so that both of them could legally move onto it. Examples:
If the black 4 is sitting on A, and the white 9 is on B, and you move the white 5 onto C (all of them being round), you capture the 4 by Subtraction.
If the black square 15 is on A, and the white round 3 is on B, and you move the white round 5 onto C, you capture the 15 by Multiplication.
If the black triangle 25 is on A, and the white round 9 is on B, and you move the white square 225 onto C, you capture the 25 by Division.
(Optional) Capture by Arithmetic, Geometric, or Harmonic Proportion: Fulke mentions that you can also capture by one of the proportions (which are described in the Basic Rhythmomachy paper). He does not give much detail, but it seems pretty obvious based on the other captures: you capture by proportion if your two men and the enemy man together form one of the proportions, and you move your men so that they could move onto the enemy. I consider this Capture an advanced variant, and don't recommend it for beginners. Fulke makes it clear that this is optional, and should be agreed on in advance.
Capture by Oblivion: The exception to the above patterns is the simple Capture by Oblivion. This basically means surrounding the enemy piece so that it can not move. Thus, to capture a round piece that is on A by Oblivion, you must place men on B, C, D, and E. To capture a triangle in A by Oblivion, you must place men in F or S, G or V, H or Y, and I or T. (That is, you need to block the triangle from making any legal moves.) To capture a square in A, you must place men at, or closer than, L, M, N, or O. Since the flying moves are the normal moves, plus one step to the side, blocking the normal moves suffices to block the flying ones as well. In Capture by Oblivion, you must actually surround the enemy man -- it is not sufficient to be able to surround him.
Fulke is not clear about some of the finer details here. It isn't clear whether your own men count towards a Capture by Oblivion; I doubt it, but it is possible. (I doubt it because many men begin the game so blocked.) It also isn't clear whether the edges of the board count; again, I doubt it. I recommend playing that a Capture by Oblivion requires actually blocking all four sides with enemy men, but this is open to interpretation.
Turning the Enemy: In all cases, capturing an enemy man does more than take him off the board -- he actually begins to work for you. When you capture an enemy man, flip him over, so that your color is showing. (This is why Rhythmomachy pieces should be double-sided.) Place him on your end of the board, in any of the squares in your first rank. From then on, the piece belongs to you, and can act like one of your men (unless, of course, your opponent recaptures it).
This is actually quite important: due to the way the higher victories are constructed, you can only win some of them with the aid of some of your opponent's pieces. It also tends to shorten the game somewhat; since captured pieces are not only lost by one player, but gained by the other, the game slowly unbalances once someone gets the upper hand.
They go like this:
The Great Triumph: Begin by announcing that you are going to make a triumph, and which men you are going to use to do it; these three men must make up an arithmetic, geometric, or harmonic proportion. From then on, your opponent can not capture those men; however, those men can no longer use their flying movements. Move those men into enemy territory, so that they form a connected line, horizontal, vertical, or diagonal. As soon as they are in place, you win.
The Greater Triumph: This is the same as the Great Triumph, except that you must use four men, which make any two of the proportions together. They do not have to form a straight line; they may instead make a square. (So you could have men in B, C, D, and E.)
The Greatest Triumph: This is the same as the Greater Triumph, except that you must make all three of the proportions using the four men.
Fulke has detailed tables giving all of the possible combinations of men that can be used in the Major Victories; I commend them, especially if you want to try either the Greater or Greatest Triumphs.
The Common Victories
Fulke also lists a number of much simpler victories; he lists them with a different version of the game, but I commend them to beginning and intermediate players, as a way to finish the game more rapidly and without needing to get into the Proportions. You should discuss this with your opponent before beginning the game, and agree on which Victory you are going to play for. They all involve the number and kind of men that you capture; they do not require capture of the King, or the creation of proportions in enemy territory. They are listed below, roughly in order of difficulty.
Victory of Bodies: To achieve victory of bodies, you just need to be the first player to capture an agreed-upon number of men. I recommend 6 men as a reasonable goal for a game of modest length, but you can set it as high or low as you prefer. Note that the strategies get odd here -- you wind up shooting to capture as many easy pieces as possible.
Victory of Goods: To achieve victory of goods, you need to capture a certain number of points first, not a specific number of men; that is, you add up the numbers on all of the men you have captured. A good number seems somewhere between 100 and 200. However, note that this also skews the strategies, because capturing a single really big piece can win the game.
Victory of Quarrel: This one is a bit curious -- you agree to capture a certain number of points and a number of characters. That is, if you agree to capture 100 in 8 characters, you must have at least 100 points, and at least 8 characters showing on those men. So, for instance, if you capture the 2, 4, 6, 8, 25, and 64, it would satisfy this victory: you have 109 points, and there are 8 characters showing on the men.
Victory of Honor: This is the combination of Victory of Bodies and Victory of Goods -- you must make a specified number of points, and a specified number of men. Thus, you might have to capture at least 100 points, and at least 8 men. This helps to adjust the unbalancing effects that Victory of Bodies and Victory Goods have on game strategies by themselves. I particularly recommend this as a good victory for intermediate players, who aren't quite ready for the Major Victories, but want something that will take a little more thought.
Victory of Honor and Quarrel: This combines all of the above: you must capture at least a specified number of men, a number of points, and a number of characters. So for instance, you might have to capture 100 points, 8 men, and 10 characters.
This version is my personal favorite, even though it is the least common from period: it is relatively balanced, and very consistent.
It does require some learning, and I won't kid you: it takes years to get good at it. On the other hand, you can comfort yourself with the fact that no one around you is good at it either, so you don't have to deal with people coming in and just crushing you through experience.