Rithmomachia
Will Overington
Some years ago, I bought a copy of a small book called Discovering
Old Board Games. There is in this country a publisher, Shire Books,
that produces a wide range of more or less uniform books in their
Discovering .... series with all manner of topics, and it is one of these.
Near the back of the book is a delightful medieval board game called
Rithmomachia, which the author of the book doubts will ever be revived.
Well, shall we revive it?
It is played on an 8 x 16 board with 24 pieces for each of the two
sides, namely 8 discs, 8 triangles, 7 squares and a pyramid. The pieces
have numbers on them, different for the two sides.
I have contacted the British Museum, but they are unable to trace the
16th Century book refered to, though there are also some manuscripts cited.
I expect that these will probably be in latin.
However, with the prospect of discussion over the net between interested
persons, (maybe even rec.games.board.rithmomachia in due course) and the
possibility of PC based software to play the game, as well as sets being
made in hardware form, there is every reason to be hopeful of gradual
progress as a long term hobby.
I feel that this will, as well as being a fascinating board game project
in its own right, be an interesting historical research study as well.
To start things off, does anyone know of any other modern interest in
rithmomachia? Is someone out there already selling sets?
Will Overington
Will Overington
As a result of my posting about rithmomachia, I have received 3
letters by e-mail already, one from England, one from Germany
and one, I think, from Denmark.
The gentleman in Germany has seen a book in German in the University
Library and is going to try to find it, and also find out if there
is anybody in Germany making the pieces.
What I know thus far:
The game is played on an 8 x 16 board with 24 pieces for each of the two
sides, namely 8 discs, 8 triangles, 7 squares and a pyramid. The pieces
have numbers on them, different for the two sides.
I have the short item in Discovering Old Board Games, Shire Books,
(at home, not in front of me here) that shows the layout of the pieces
at the start of the game and the seven different ways of deciding who
has won. Before play commences, the players agree what will be the
chosen way to win! Winning involves things like, (and here I am only
remembering what I read some time ago but did not seek to learn, but
I expect that interested readers will like to know anyway) getting
four pieces into a line where 3 of them are your own and one is the
opponents and the product of the numbers on two of them is the number on
one of the others!
The calculations, which could prove irksome to those who do not like
arithmetic, are essentially trivial to anyone who can add up and multiply
integers. Things like, and I may have not quite remembered it correctly,
but like rules that a piece can be taken if the number on it is the
product of the number on the taking piece multiplied by how many squares
away from each other that they are. For example, a 4 could take a 28 that
is 7 squares away.
I cannot remember off hand whether there is a 28, but there is certainly
a 4.
Some numbers appear several times, even on pieces of the same side, in
one case even on pieces of the same shape on the same side.
Some of the numbers are 3 digits, up in the 300s. I remember that the
white pyramid is 91. It is something like 1^2 + 2^2 + 3^2 + 4^2 +5^2 + 6^2,
yes, that's it. As it is a pyramid of squares.
I have made a set of pieces myself in ceramic (that is, fired clay) as
a hobby project. I have used a size of about 30mm as the diameter of the
circlular pieces. I am quite pleased with them, and, now that I am getting
more proficient at the use of the clay, I hope to make a finer set.
I am not certain that the book I have will allow unambiguous interpretation
of the rules. Clearly, a mutual research project will need to look
carefully at historical sources to find out the rules, of which there
may well be variations from place to place and from time to time.
Perhaps the sets of pieces will vary too. We shall see.
In view of the interest in the game, I will endeavour to transcribe a
list of the references given.
In view of the growing interest, could anyone who is interested please
mail me and I will try and build up a mailing list.
The pieces are essentially simple in shape, and so, for example, to ask
a local craft potter such as those who advertise to make cups with your
name upon them and the like, to make a set could perhaps not be too
expensive an idea. Apart from the two pyramids, the pieces are all flat.
Will Overington
Nick Roworth
>Tuesday 25th June 1991
>
>I have the short item in Discovering Old Board Games, Shire Books,
>(at home, not in front of me here) that shows the layout of the pieces
>at the start of the game and the seven different ways of deciding who
>has won. Before play commences, the players agree what will be the
>chosen way to win! Winning involves things like, (and here I am only
>remembering what I read some time ago but did not seek to learn, but
>I expect that interested readers will like to know anyway) getting
>four pieces into a line where 3 of them are your own and one is the
>opponents and the product of the numbers on two of them is the number on
>one of the others!
I have "The Boardgame Book" also written by R.C. Bell.
ISBN 0 85685 447 6 published 1979
This contains the boards and pieces for many old games including Rithmomachia.
The pieces ar as follows
Black
Round - 3,5,7,9,9,25,49,81
Triangle - 16,36,64,100,12,30,56,90
Square - 28,66,120,49,121,225,361
Pyramid - 190 built of Squares 64,49, Triangles 36,25 round 16
White
Round - 2,4,6,8,4,16,36,64
Triangle - 9,25,49,81,6,20,42,72
Square - 15,45,153,25,81,169,289
Pyramid - 91 built of squares 36,25 triangles 16,9 Rounds 4,1
Black has first move, play alternately.
Rounds move onto any adjacent space.
Triangles three spaces in any direction.
Squares four spaces in any direction.
The pyramids in any way of any layer they are composed of.
No pieces can leap.
Winning is by capturing pieces or for advanced players achieving various
formations in the opponents territory.
Methods of Capture
Meeting - if a piece can move onto another piece (of the same number i think)
it is taken instead.
Assault - if a smaller number multiplied by the vacant spaces = larger then
the smaller number can take the larger.
Ambuscade - if two pieces whose sum equals the number on an opponents piece
can move onto the spaces on either side it is captured.
siege - if a piece is surrounde on all sides by enemy pieces it is captured.
Pyramids are considered vunerable if one of there layers can be captured. A
ransom is permitted of an equal piece or any other piece if not available.
Capture is cpermitted if a successful attack is made on its base square 36
white or 64 black. or by siege, of course!
Common Victories
De corpere - an agreed number of pieces.
De Bonis - an agreed value of captured pieces e.g. 180.
De Lite - Value of pieces and number of digits e.g. 160 with eight digits.
De Honere - The number of pieces and thier values e.g. 160 with 5 pieces.
De Honere Liteque - value, digits and pieces e.g. 160, 5 pieces and 9 digits.
Proper Victories
Victoria Magna - 3 counters in one of 3 progressions, arithmetic, geometric
and harmonic.
Victoria Major - a combination of any two progressions in the enemies
territory. Four pieces in a line in the enemy territory e.g. 2,3,4,8
2,3,4, arithmetic 2,4,8, geometric whites piecs and the 3 blacks.
Victoria Excellentissima - 4 numbers in a row embodying all 3 progressions.
there are only six possible 2,3,4,6; 4,6,8,12; 7,8,9,12; 4,6,9,12; 3,5,15,20;
12,15,16,20
If anyone wants any elaboration I shall do my best to interpret the rules as
written.
A simple game it would seem, anyone for a play by email game? :-)
--
Nick Roworth, ORACLE Europe "Bugs Mr. Rico! Zillions of 'em!"
Rijnzathe 6, NL-3454 PV De Meern Starship Troopers - Robert A. Heinlein
Phone: +31-3406-94211
nrow...@nl.oracle.com
Kevin Gallagher
>
>Some years ago, I bought a copy of a small book called Discovering
>Old Board Games. There is in this country a publisher, Shire Books,
>that produces a wide range of more or less uniform books in their
>Discovering .... series with all manner of topics, and it is one of these.
>
>Near the back of the book is a delightful medieval board game called
>Rithmomachia, which the author of the book doubts will ever be revived.
Unfortunately, the author of the book, R.C. Bell, lifted his
discussion, without giving proper credit, from an article written by
David Eugene Smith and Clara C. Eaton in the April, 1911, issue of The
American Mathematical Monthly, Vol. XVIII, No. 4, pp. 73-80. For
example, here's how Bell begins his discussion:
At least three manuscripts of the eleventh century, two of the
twelth and one each of the thirteenth and fourteenth centuries
survive, describing this medieval game which may have
originated at Byzantium or Alexandria, and was based on the
Pythagorean philosophy of numbers. The first mention of the
game was by Hermannus Contractus (A.D. 1013-1054).
The Monthly article reads as follows:
For although we have manuscripts of three writers of the
eleventh century, two of the twelfth, one of the thirteenth,
and Bradwardin's work of the fourteenth, and although we have
several printed treatises on the subject, we know practically
nothing of the origin of the game. We only know that the
medieval writers attributed it to Pythagoras, that no trace of
it has been discovered in Greek literature, and that no
mention of it has been found before the time of Hermannus
Contractus (1013-1054).
Bell also writes:
Outstanding intellectuals of the Middle Ages became addicted
to Rithmomachia and regarded the game superior to chess.
Today it is completely forgotten and is unlikely to enjoy a
revival for there is now no interest in the number theory upon
which it was based.
And in the Monthly article we find:
Such is a brief description of the game of which Boissiere
speaks as "Noblissimus et antiquissimus ludus Pythagoreus qui
Rythmomachia nominatur," a game of which we are told many of
the devotees were men of no mean reputation. Such leaders of
thought as Gerbert, whom his contemporaries called a wizard
but made a Pope; Hermannus, whose infirmity gave him the name
Contractus, by which he is commonly known,...--these are the
names of some of those who played the game, and several of
them composed tractates setting forth its merits. It cannot
be revived, since the interest in the number theory for which
it stood has passed away,...
There are other examples in Bell's books where he has done the same
lifting of other's works without citing his sources.
Another article on the subject is entitled: "Boissiere's Pythagorean
Game", Translated with Notes on the Text by John F. C. Richards. You
MUST obtain a copy of this article. It is also a Monthly article, but
I cannot tell you exactly what issue, except it runs from page 177 to
page 217.
>Well, shall we revive it?
Read Richards' article before you decide. You should be able to find
it in any decent university library mathematics collection.
>To start things off, does anyone know of any other modern interest in
>rithmomachia? Is someone out there already selling sets?
Unlikely. We will probably have to sign up for a woodworking class
and build our own sets.
Florian Hars
>I cannot remember off hand whether there is a 28, but there is certainly
>a 4.
There is a 28, and it is on the other side of the 4!
>white pyramid is 91. It is something like 1^2 + 2^2 + 3^2 + 4^2 +5^2 + 6^2,
>yes, that's it. As it is a pyramid of squares.
It depends... There are rules which have a pyramid which consists of
segments, so the white pyramid has two square (36, 25), two triangular
(16, 9) and two circular (4, 1) segments. These segments might be taken
separately, and they determine the way the pyramid moves (i.e. if it has
no triangular segment, it can move like a triangle).
>I am not certain that the book I have will allow unambiguous interpretation
>of the rules. Clearly, a mutual research project will need to look
>carefully at historical sources to find out the rules, of which there
>may well be variations from place to place and from time to time.
The rules differ on almost everything:
- the size of the board: from 9*8 to 16*8
- the starting position: even 2 opposite to uneven 3 or uneven 9.
- diagonal moves may be allowed or not
- the pyramid may be one piece or divided into segments
- in taking a piece throug "eruptio" division may be allowed (so the
uneven 28 in the above example might take the even 4)
I wrote a thesis on this game for a seminary at the University on this game
some yaers ago where I listed the rules whith most variants and a short
history of it. This text is 28 pages long and in german, so it probably
won't help you very much (and I have it only on paper, so I would have
to send it via snailmail).
We also played the game on that occasion, it is very slow, there is
almost no dynamics in it.
>Will Overington
Tschuess, Florian.
--
Florian Hars Der Mensch ist ein nuetzliches Lebewesen, weil er dazu
fl...@mcshh.hanse.de dient, durch den Soldatentod Petroleumaktien in die
Hoehe zu treiben... (Kurt Tucholsky, 1931)
Brad THOMPSON
> Old Board Games. There is in this country a publisher, Shire Books,
> that produces a wide range of more or less uniform books in their
> Discovering .... series with all manner of topics, and it is one of these.
Could you post the address of Shire books?
> (Rithmomachia) ...is played on an 8 x 16 board with 24 pieces for each of
> the two sides, namely 8 discs, 8 triangles, 7 squares and a pyramid. The
> pieces have numbers on them, different for the two sides.
What other rules are there to the game? One problem I have found with
some medieval games is that the game designers of the time had no concept
of Odds. This made games that looked intriguing become boring very
quickly, once you figured out how to win.
It sounds intriguing. I would like to see more discussion of this,
if we can get some rules for the game.
On other thing. It's quite possible that this is a variation of an
already existing game. For instance Chess has gone by many names and
different rules over the millenia.
Brad 'Cory' Thompson
ADAR...@esoc.bitnet
Date: Friday, 5 Jul 1991 16:37:23 CET
From: <ADAR...@ESOC.BITNET>
Message-ID: <91186.1637...@ESOC.BITNET>
Newsgroups: rec.games.board
Subject: Re: Rithmomachia
References: <+{2{MG{@cck.cov.ac.uk> <73...@microsoft.UUCP>
In article <73...@microsoft.UUCP>, br...@microsoft.UUCP (Brad THOMPSON) says:
>
>In article <+{2{MG{@cck.cov.ac.uk> esz...@cck.cov.ac.uk (Will Overington)
>writes:
>>
>> Some years ago, I bought a copy of a small book called Discovering
>> Old Board Games. There is in this country a publisher, Shire Books,
>> that produces a wide range of more or less uniform books in their
>> Discovering .... series with all manner of topics, and it is one of these.
>
> Could you post the address of Shire books?
>
>> (Rithmomachia) ...is played on an 8 x 16 board with 24 pieces for each of
>> the two sides, namely 8 discs, 8 triangles, 7 squares and a pyramid. The
>> pieces have numbers on them, different for the two sides.
>
> What other rules are there to the game? One problem I have found with
>some medieval games is that the game designers of the time had no concept
>of Odds. This made games that looked intriguing become boring very
>quickly, once you figured out how to win.
>
> It sounds intriguing. I would like to see more discussion of this,
>if we can get some rules for the game.
This was also called Ludus Philosophorum. The only time I tried it I
thought it was essentialy unplayable. It was a long time ago but this
is what I can remember of the rules.
One side is based on the odd integers and the other side is based on
the even integers.
The disks of each side are labeled with the base sequence 1,3,5...
or 2,4,6,8... respectivly.
The squares for each side are labeled with the squares of the sides
base sequence, ie 1,9,25,... or 4,16,36...
The triangles on each side are labelled with the triangular numbers
N*(N-1)/2
The pyramids are made up of a pile of squares, triangles and discs.
The board is placed with the short sides nearest the players, I can
not remember what the start position is.
Play proceeds as in most games by the players each moving a single
piece in turn. Pieces moved like a chess queen (I think).
Pieces could take by combining the number of themselves, the piece they
were taking and the number of intervening squares into an mathemetical
expression using the +,-,* or / operations.
Ie piece M could take piece N if if was distance D away if one of the
following relations held true
M = N + D, M = N - D, M = D - N
M = M * D, M = N / D, M = D / N
I think the inclusion of multiply and divide was optional.
The pyramid was taken a single layer at a time.
The object was to capture the opponents pyramid.
I doubt if the game was ever played just for fun.
Bill Masek
game. You capture or move if you are a factor of the number of the
piece you are playing, or something like that.
(I went to Granada Hills High School in Los Angeles, California. Can someone
close call the school to see if anyone remembers it?)
--
bill masek charles river data systems, inc
(617) 491-5320 (h) (508) 626-1122 (w)