The Battle of Numbers

Our topic is the game called rithmomachia or rithmomachy—literally, the battle of numbers...

Ursula Whitcher
AMS | Mathematical Reviews, Ann Arbor, Michigan

This month, we're going to explore a very old—indeed, medieval—educational game and correct a mathematical error in a sixteenth-century game manual. But before we delve into the past, let me remind you that the Feature Column is seeking new columnists. If you're interested in sharing your writing about intriguing mathematical ideas, please get in touch!

Pleasant Utility and Useful Pleasantness

Our topic is the game called rithmomachia or rithmomachy—literally, the battle of numbers. The game is played with pieces shaped like geometric figures and labeled with different numbers, on a board like a double chessboard.

The twelfth-century scholar Fortolfus described the experience of rithmomachia as the pinnacle of educated leisure:

Indeed, in this art, which you will admire in two ways, is pleasant utility and useful pleasantness. Not only does it not cause tedium, but rather it removes it; it usefully occupies one uselessly idle, and usefully un-occupies the person uselessly busy.

The game's rules are elaborate. Their importance, and their draw for medieval intellectuals, lies in their connection to the quadrivium. Arithmetic, geometry, astronomy, and music were the four advanced arts in the medieval liberal arts curriculum. All four required an understanding of ratios and sequences. Playing rithmomachia allowed medieval people to practice their math skills and show off their erudition.

Some rithmomachia proponents even claimed the game made you a better person. They often quoted the late Roman philosopher Boethius' "demonstration of how every inequality proceeds from equality," which makes grand claims:

Now it remains for us to treat of a certain very profound discipline which pertains with sublime logic to every force of nature and to the very integrity of things. There is a great fruitfulness in this knowledge, if one does not neglect it, because it is goodness itself defined which thus comes to knowable form, imitable by the mind.

Boethius describes a specific procedure for creating different types of sequences and ratios, beginning with the same number:

Let there be put down for us three equal terms, that is three unities, or three twos, or however many you want to put down. Whatever happens in one, happens in the others.

Now these are the three rules: that you make the first number equal to the first, then put down a number equal to the first and the second, finally one equal to the first, twice the second, and the third.

For example, if we begin with $1, 1, 1$ we obtain $1, 2, 4$.

This is the beginning of a geometric sequence where the numbers double at each step: in Boethius's language, it is a duplex. Applying the same rule to the new list of numbers will create a list with more complicated relationships.

Rithmomachia Pieces

Every rithmomachia piece has its own number (or, in some cases, a stack of numbers):

The choice of numbers is not arbitrary; they are generated by rules similar to Boethius' rules for creating inequality from equality. Traditionally, the white gamepieces are considered the "evens" team and the black pieces are considered the "odds" team, though as we will see, this split between even and odd only applies to the circles.

Circles

Each side has eight circle pieces, given by the first four even or odd numbers and their perfect squares. (The odd numbers skip 1, which is a more mystical "unity" in the Boethian scheme.)

Evens

2	4	6	8
4	16	36	64

Odds

3	5	7	9
9	25	49	81

Triangles

The triangles in a rithmomachia set appear in pairs that demonstrate superparticular proportions. These are ratios of the form $n+1:n$, such as $3:2$ or $4:3$. Practically speaking, one can lay out the triangle and circle pieces in a table. The numbers for the first row of triangles are obtained by adding the two circle numbers above that number, in the same column. One may find the numbers for the second row of triangles using ratios. In each column, the ratio of the number in the first triangles row to the number in the last circles row and the ratio of the number in the second triangles row to the number in the first triangles row are the same.

I'll start with partially completed tables, in case you want to try finding the values yourself:

Evens

Circles I	2	4	6	8
Circles II	4	16	36	64
Triangles I	6	20
Triangles II	9
Ratio	3:2

Odds

Circles I	3	5	7	9
Circles II	9	25	49	81
Triangles I
Triangles II
Ratio

In medieval and Renaissance music, different ratios were used to create different musical scales and analyze the differences between musical notes within those scales. For example, the Pythagorean temperament is based on the ratio $3:2$, which appears when finding the first Team Evens triangle values.

Here are all the triangle values:

Evens

Circles I	2	4	6	8
Circles II	4	16	36	64
Triangles I	6	20	42	72
Triangles II	9	25	49	81
Ratio	3:2	5:4	7:6	9:8

Odds

Circles I	3	5	7	9
Circles II	9	25	49	81
Triangles I	12	30	56	90
Triangles II	16	36	64	100
Ratio	4:3	6:5	8:7	10:9

Squares

The triangular rithmomachia gamepieces used ratios of the form $n+1:n$. The squares use ratios of the form $n+(n-1):n$, which we may simplify to the less evocative form $2n-1:n$. This is a special case of the more general superpartient proportions. A superpartient proportion is any ratio of the form $n+a:n$ where $a$ is an integer greater than 1 and $a$ and $n$ are relatively prime (that is, their greatest common divisor is 1).

The numbers for the square pieces may be found by repeating the method for finding the numbers for triangular pieces, but now shifted two rows down. The numbers for the first row of squares are obtained by adding the two triangle numbers above above that number, in the same column. One may find the numbers for the second row of squares using ratios. In each column, the ratio of the number in the first squares row to the number in the last triangles row and the ratio of the number in the second squares row to the number in the first squares row are the same.

Evens

Circles I	2	4	6	8
Circles II	4	16	36	64
Triangles I	6	20	42	72
Triangles II	9	25	49	81
Squares I	15	45	91 (pyramid)	153
Squares II	25	81	169	289
Ratio	5:3	9:5	13:7	17:9

Odds

Circles I	3	5	7	9
Circles II	9	25	49	81
Triangles I	12	30	56	90
Triangles II	16	36	64	100
Squares I	28	66	120	190 (pyramid)
Squares II	49	121	225	361
Ratio	7:4	11:6	15:8	19:10

Pyramids

The pyramids or kings are sums of perfect squares. Ideally, they should be built out of spare pieces of the appropriate color with these values. The
Even team's pyramid has the value $1 + 4 + 9 + 16 + 25 + 36 = 91$. The Odd team's pyramid has the value $16 + 25 + 36 + 49 + 64 = 190$.

Moving Pieces

We have already seen the starting board layout, in the illustration from Claude de Boissière's manual. Black (Team Odds) always moves first. Each shape of piece follows a different movement rule. The following guidelines are based on the 1563 English rithmomachia manual by Lever and Fulke, which was in turn based on de Boissière's book in French.

• The circles move one space diagonally.
• The triangles move two spaces horizontally or vertically. If not taking a piece, they may also make a chess knight's move ("flying").
• The squares move three spaces horizontally or vertically. If not taking a piece, they may also make a "flying" knight-like move that crosses four squares total. This may be either three vertical squares followed by one horizontal square, or three vertical squares followed by one vertical square.
• The pyramids may move in the same way as any of the circles, triangles, or squares.

Lever and Fulke give the following diagram illustrating potential moves:

They illustrate the square's knight-like move by pointing out a square may move from P to Y or T in their diagram.

Capturing Pieces

When a player takes a piece, they change its color to their team's color (ideally, rithmomachia pieces are two-sided!) The transformed piece moves to the row of the board closest to the player, and may now be used like other pieces. There are many ways to take pieces, using different mathematical properties. Lever and Fulke mention Equality, Obsidion (in some editions, Oblivion), Addition, Subtraction, Multiplication, and Division, as well as an optional Proportion rule.

The simplest capture method is Equality. If a piece could move to another piece with the same number, it takes that piece. The Obsidion capture is a trap: if four pieces prevent another piece from moving horizontally or vertically, it is taken.

If two pieces from one team can each move to the same piece of the other team, and those two pieces can add, subtract, multiply, or divide to make the number on the opposing piece, they capture that piece. Whether one of the two attacking pieces has to move into the space they are attacking depends on when the possible capture appears. If a player moves a piece on their turn, bringing it into position for an addition, subtraction, multiplication, or division capture, then they immediately take the other player's piece without having to move their piece again. On the other hand, if a player notices a possible capture at the start of their turn, before they have moved a piece, they must place one of their attacking pieces in the captured piece's space in order to take a piece by addition, subtraction, multiplication, or division.

Pyramids may not be taken by equality. They may be taken by obsidion, by addition, subtraction, multiplication, or division, by the optional proportion capture if this is in play, or by taking the pieces with square numbers that make up the pyramid one by one.

Capture by Proportion

What is the optional rule for taking pieces by proportion? Lever and Fulke refer to arithmetic, geometric, and musical or harmonic proportion, so this optional rule has three sub-rules.

Capture by arithmetic proportion is similar to capture by addition: if two pieces may move into the space of a third and the numbers on all three pieces fit into a partial arithmetic sequence of the form $n, n+a, n+2a$, then the third piece is captured. Three pieces may also capture a fourth by arithmetic proportion. Capture by geometric proportion uses the same idea, but using partial geometric sequences of the form $n, an, a^2n$ or $n, an, a^2n, a^3n$.

Musical proportion only applies to three-term sequences. Lever and Fulke give a "definition" of musical proportion:

Musicall proportion is when the differences of the first and last from the middes, are the same, that is betwene the first and the last, as .3.4.6., betwene .3. and .4. is .1. betwene .4. and .6. is .2. the whole difference is .3. which is the difference betwene .6. and .3. the first and the last.

Unfortunately, this "definition" of musical proportion would apply to any three numbers $a, b, c$. We are comparing $(b-a) + (c-b)$ with $c-a$, but these values are the same! The correct definition of musical proportion (perhaps better known as harmonic proportion) uses ratios. Three numbers $a,b,c$ with $a < b < c$ are in harmonic proportion if $c:a = (c-b):(b-a)$. For example, $4,6,12$ is a musical proportion, because $12:4 = 3:1$ and $(12-6):(6-4) = 6:2 = 3:1$. We can now say that capture by musical proportion happens when two pieces may move into the space of a third and all three pieces fit into a harmonic proportion.

Victory Conditions

Even figuring out how to take pieces in rithmomachia is complex! Thus, players may agree on any of several victory conditions. These are divided into "common" victories, which are based on capturing enough pieces by some measure, and "proper" victories (also known as triumphs) which involve capturing the enemy's pyramid and then arranging three or four of one's one pieces to create an arithmetic, geometric, or harmonic proportion.

Here are the common victories.

• Victory of bodies: The first player to take a certain number of pieces wins.
• Victory of goods: The first player to take pieces adding to at least a certain number wins.
• Victory of quarrel: The first player to take pieces adding to at least a certain number and using a certain total number of digits wins. (This prevents a player from winning by taking a single very high value piece, as might be possible in the victory of goods.)
• Victory of honor: The first player to take a specified number of pieces adding to at least a certain number wins.

Let us quote Lever and Fulke on how to complete a proper victory or triumph:

When the king is taken, the triumph must be prepared to be set in the adversaries campe. The adversaries campe is called al the space, that is betweene the first front of his men, as they were first placed, unto the neither ende of the table, conteyning .40. spaces or as some wil .48. When you entend to make a triumph you must proclaime it, admonishing your adversarie, that he medle not with anye man to take hym, whiche you have placed for youre triumphe. Furthermore, you must bryng all your men that serve for the triumph in their direct motions, and not in theyr flying draughtes.

To triumphe therefore, is to place three or foure men within the adversaries campe, in proportion Arithmeticall, Geometricall, or Musicall, as wel of your owne men, as of your enemyes men that be taken, standing in a right lyne, direct or crosse, as in .D.A.B. or els .5.1.3. if it consist of three numbers, but if it stande of foure numbers, they maye be set lyke a square two agaynst two.

Anyone who attained a proper victory would indeed feel triumphant!

Further Reading

• Rafe Lever and William Fulke, The Philosophers Game, posted by Justin du Coeur.
• Michael Masi, Boethian Number Theory: A Translation of the De Institutione Arithmetica (Rodopi, 1996)
• Ann E. Moyer, The Philosophers’ Game: Rithmomachia in Medieval and Renaissance Europe (Ann Arbor: University of Michigan Press, 2001).