On the Wise Play of Rhythmomachia
---------------------------------

I have herd muche tell that the land of Carolingia is one replete
withe grete schollars and thinkers, men and women too who are bothe
wise and lerned. Wyth this wysdome, it semeth that a game most apt
for the peple would be the play of Rhythmomachia, call'd by some
The Philosophers Game.

Rythmomachia is not a game for the simple folk, for to plaie at it,
one must have numbers, for it is a game all of Arithmeticke, and
hee who knows not the numbers wyll never win a game, save he plaie
gainst a simpleton. But for them that have the wysdome, it is a moste
excellent plaie, of grete subtlety even beyond the Chesse.

There are manie forms of Rythmomachy, plaid by schools around the
civilized worlde. I lernt the game from Fulke, who dyd write of it
in a boke manie years past. Yet this boke was of suche lengthe and
thoroughnes as to affright the casual plaier. So I shal attempt to
write at less length about the game, that the gamesters of Carolingia
maie have chance to plaie it without grete study. Yet shal I spend
several letters on it, for I shal teche eche game separately.

I begin with the forme of the game. It is plaied on a board, of the
size and forme of two Chessbordes laid beside eche other, making a
single bord .16. squares in length and .8. across. And thys bord is
not checkered; yet if it is (as yf you have two Chessbordes to play
upon), it matters not. The gamesters eche have a Home, whych is the
bord of .8. by .8. that he sits behind.

Eche plaier has .24. men; and these men are different, one from the
other. Fyrst there is the color -- one gamester plaies White, the other
Blacke. And eche man has the color of his plaier upon his face, as
White's men are all White. Yet eche man has the color of hys Adversary
upon his back, for men may be taken, and then turn coat to the Enemy.
So should all of White's men have Blacke upon their backs, that Blacke
maie play them yf taken.

Eche man has a shape, whych mattereth gretely to the plaie. Of the
.24. men, .8. are Roundes as smooth as a circle, .8. are Triangles,
and .8. are Squares. And these men differ in their Movement; but I
shal speke more of that next time.

Eche man has a number upon his face, and the like number upon hys
backe. And these numbers are all different; White has different
numbers from Blacke. And yt matters not whych color has whych
numbers; yet I have alwaies played that White plaieth with the
Even, and Blacke wyth the Odde. And now I shal tell you of the
Numbers, that you maie alwaies know how to arraie the men, even
yf you have not a guide.

There are twoo Ranks of Roundes. The fyrst is the Plaine Numbers of
the syde, from whych all others flow. And the Plaine Numbers of the
Whyte are .2.4.6.8; and those of Blacke are .3.5.7.9.
And note that these are all the numbers of Even and Odde below .10.

And the seconde Rank of Roundes are made by multiplyinge eche of
the fyrst Ranke by ytself. Thus, the Seconde Ranke of Whyte are
.4. (for this is .2. tymes .2.), .16.36.64. And those
of Blacke are .9. (for thys is .3. tymes .3.), .25.49.81.

There are two Rankes of Triangles. And the fyrst is made by addyng
together the twoo Rankes of Roundes. Thus, the fyrst Rank of Triangles
for Whyte is .6. (for thys is .2. plus .4.), .20. (whych is .4. plus
.16.), .42.72. And that of Blacke is .12.30.56.90.

The Second Rank of Triangles is made by addyng one to the fyrst
Rounde, and then multiplying that by hytself. Thus, the Seconde
Rank of Whyte Triangles is .9. (whych is .2. plus .1. tymes itself),
.25. (whych is .4. plus .1. tymes itself), .49.81. And that
of Blacke is .16.36.64.100.

The fyrst ranke of Squares is muche as that of Triangles, for yt is
made by addying the two Triangles together. And that of Whyte is
.15. (for thys is .6. plus .9.), .45.91.153. And that of
Blacke is .28.66.120.190.

And the Seconde Ranke of Squares is made by doubling the fyrst
Rounde and addyng one, and then multiplying it by itself. Thus, the
Seconde Squares of Whyte are .25. (whych is .2. tymes .2. plus .1.
tymes itself), .81. (whych is .2. tymes .4. plus .1. tymes itself),
.169.289. And that of Blacke is .49. (for thys is .2. tymes .3. plus
.1. tymes itself), .121.225.361.

And know that, though it be not a need, yt is good to place on eche
man a dot below the number, that the .6. may be told from the .9. and
no other numbers confused.

And that you nede not remember the foregoing at all tymes, I shal
give you a mappe of the borde, wyth the men upon yt, so that
you may see them all in place. My owne borde I have made wyth the
numbers of the men upon it, that I might quickly see the arraie of
the men. Yet it is good to know the reason for this array, that
you may play upon a plaine borde with ease.

The last I shal speke of today is of the Kynges. Eche plaier has
a Kyng, whych has the form of a Pyramid. Thys Kyng is made up of
smaller pieces, each of whych is a number whych is a smaller
number tymes itself. The Kynge of Whyte is the .91., whyche is
made up of the Roundes .1. and .4., the Triangles .9. and .16.,
and the Squares .25. and .36. And that of the Blacke is the .190.,
whych is made up of the Round .16., the Triangles .25. and .36.,
and the Squares .49. and .64. And the movement and powers of the
King are different from one forme of the game to another; yet yt
is always a powerful piece, and takyng the Enemies King is alwaies
part of your purpose.

That shal be enow for one missive: I have told you of the formes
of the game, that your craftsmen maie fashion the borde and men.
In my next letter, I shall write of the first forme of the plaie.
Yrs in service, Justin duC, this cool 20 of Oct in our Yeer of
the Lord 1597.

Endnotes
--------

With this section, I begin the first of (probably) four chapters on
Rhythomachy, one of the most ancient and honorable of the games of
period, and one of the few that has absolutely and completely died out
since. Don't be daunted by the game -- while it *is* a little more
complex than most games, it's enormously rewarding, one of those rare
games of fairly straightforward rules and remarkably interesting
strategy. Take heart from the fact that *no* one is good at it any
more, so you're in good company as a beginner. (And note that the
rules, although deeper than, say, Backgammon, really aren't much more
complex than Chess.) The game was popular among the philosophers
throughout most of our period, from early in the millenium right up
through the 17th century.

This discussion is primarily drawn from William Fulke's book, _The
Philosopher's Game_, published in 1563, which is a very loose
translation of a French book by Boissiere in 1554. It is available on
microfilm as entry 15542a in the Short Title Catalog (on Reel 806 of
the STC microfilms, available at Brandeis). An incomplete transcription
(all the words, few of the pictures) can be found at:

	http://www.inmet.com/~justin/fulke.html

In point of fact, I'm not going to be all *that* much shorter than
Fulke himself; I'll be omitting some of the unnecessary mathematical
digressions that aren't very interesting or necessary for the educated
modern reader, and I'll be trying to be slightly less redundant. But
Fulke is actually a pretty good book, clearer than many of the modern
sources on the game, and may be worth a read. (And it's very interesting
for the mathematically inclined.)

Note that, as with all games that had a history of several hundred
years, Rythmomachy evolved a bit over time. You will find, if you look
around, that there are many reconstructions that differ in the
details. I am specifically using only Fulke's version, preferring to
give an accurate snapshot of one version of the game, rather than
mushing several together. This version is of the right general
timeframe for Le Poulet Gauche (albeit rather highbrow for that
venue).

The choices of color I use are a simplification; in fact, Fulke is
explicit that you can use any contrasting colors you like. The
colors chosen happen to match my personal set (as well as some of
Fulke's illustrations). As it says, pieces should be white on one
side, black on the other. I find it useful to paint the edges of
the piece with the color of the initial owner -- it speeds up
sorting out whose piece is whose at setup time.

The formulas for the various ranks, in modern terms, follow. [Typesetting
note: "x ** 2" means "x squared"; it would be ideal if you can take out
the asterisks and instead superscript the 2.]

First rank of Rounds (which are used as "x" in the formulas below):
White -- 2, 4, 6, 8.
Black -- 3, 5, 7, 9.

Second rank of Rounds: x ** 2.

First rank of Triangles: x + (x ** 2).

Second rank of Triangles: (x + 1) ** 2.

First rank of Squares: x + (x ** 2) + ((x + 1) ** 2).

Second rank of Squares: (2x + 1) ** 2.