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<DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>OK tell about
Rithmomachia please...<BR><BR>Arianwen ferch Arthur<BR><BR><BR>Hi These
are the rules I have written, hope they are of
help<BR>jon<BR><BR><BR><BR>Rithmomachia -<BR>The Philosophers'
Game<BR><BR>This is not a game for the faint-hearted. Probably invented in one
of the<BR>monastery schools in southern Germany in the 11th C it spread to
Britain by<BR>the 12th C reaching a peak in the 14th C after which it gave way
to chess<BR>and was forgotten by the 18thC. Rithmomachia means battle of numbers
and is<BR>played between two players on a 8 x 16 chequered or undifferentiated
board<BR>with square, triangular and circular pieces each with a numerical
value.<BR>There is more than one documentary source detailing the rules, each of
which<BR>vary in minor details. Playing requires an ability to apply simple
maths,<BR>and winning requires an understanding of arithmetic, geometric and
harmonic<BR>number progressions, or at least an ability to remember them. The
rules<BR>outlined below will allow players to complete a game. However, this is
a<BR>complex game based on a more than common grasp of number theory and there
is<BR>insufficient room for Rithmomachia here. Parlett (1999) has written well
on<BR>this game and what follows is largely drawn from his chapter.
Those<BR>interested in the game should turn first to his book for an excellent
read<BR>before turning to original sources. However, do not be put off, it is
a<BR>wonderful game and a good one for getting inside the mind of the
medieval<BR>mathematician.<BR><BR>The Pieces<BR>Each side occupies 24 squares on
the board at the start of the game<BR>positioned as shown. The board is
theoretically divided down the middle<BR>between the two sides so that each side
has a home. On each side 23 of the<BR>occupied squares have a single piece but
one has a stack of pieces called a<BR>pyramid. The pieces are square, triangular
and circular, known as rounds,<BR>black on one side and white on the other with
the numerical value on each<BR>side.<BR><BR>Rounds may move one position
orthogonally<BR>Triangles may move two positions diagonally<BR>Squares may move
three squares diagonally or orthogonally<BR>Pyramids may move in the direction
of any counter it contains<BR><BR>There is no jumping over other pieces and only
one piece may occupy a square<BR>at any time.<BR><BR>The position the pieces
take on the board at the beginning of play is based<BR>on number series built on
even numbers 2, 4, 6, 8 for white and uneven, 3,<BR>5, 7, 9 for black. The
number series follow arithmetic, geometric and<BR>harmonic<BR><BR>the
progression is arithmetic where the integers a (x) b (y) c,
where<BR>a<b<c, x=b-a and y=c-b, if x=y<BR><BR>the progression is
geometric where the integers a b c, where a<b<c,
if<BR>b/a=c/b<BR><BR>the progression is harmonic where the integers a (x) b (y)
c, where a<b<c<BR>and x=b-a and y=c-b, if y/x=c/a<BR><BR>White
has the numbers 2, 4, 6, 8, 9, 15, 16, 20, 25, 36, 42, 49, 45, 64, 72,<BR>81,
91(pyramid) 153, 169, 289.<BR><BR>Black has the numbers 3, 5, 7, 9, 12, 16, 25,
28, 30, 36, 49, 56, 64, 66,<BR>81, 90, 100, 120, 121, 190 (pyramid), 225,
361.<BR><BR>The pyramids are made up of the following from the top to the
base:<BR><BR>White pyramid 91 Black pyramid 190<BR>round 1 round
16<BR>round 4 triangle 25<BR>triangle 9 triangle 36<BR>triangle 16 square
49<BR>square 25 square 64<BR>square 36<BR><BR>Players alternate in moving one of
their pieces each turn with the aim of<BR>capturing their opponents pieces and
winning by aligning three of their own<BR>pieces in their opponents half of the
board in an arithmetic, geometric or<BR>harmonic progression, know as a
triumph.<BR><BR>Capturing<BR><BR>There are several ways in which a player may
capture an opponent's piece.<BR>Sources vary on the fate of captured pieces.
They can be taken off the board<BR>and do not come back into play; they can be
removed from their square,<BR>turned over to show the capturing party's colour
and put on the back row of<BR>the board to join the capturing side's army; or
they can be turned over and<BR>left on the square where they were captured. This
final way of dealing with<BR>captured pieces can have an important role in
forming triumphs and winning a<BR>game.<BR><BR>Capture by Blockade<BR>A piece is
captured by blockade if it is in such a position on the board<BR>that it cannot
make a legal move. The edge of the board can be used in<BR>blockading but a
piece cannot be blockaded by any piece if its own colour,<BR>i.e. all blockading
pieces must be of the opposing side. Also the blockaded<BR>piece must not be in
a position where it is able to capture one of
those<BR>blockading.<BR><BR>Capture by Arithmetic Relationship<BR>It is
important to note here that pieces do not land on an opponents square<BR>in
order to capture. Instead one or more pieces have to move into a
position<BR>where they could move to an opponents square and in doing so
demonstrate an<BR>arithmetic relationship with the piece on that square. A piece
moving into<BR>such a position is said to control an enemy
piece.<BR><BR>Equality If a piece from one side moves into a position where it
can control<BR>an enemy piece and both pieces have the same numerical value the
enemy piece<BR>is captured.<BR><BR>Addition If two pieces from one side control
an enemy piece and the sum of<BR>their numerical values is equal to that of the
enemy piece then the enemy<BR>piece is captured.<BR><BR>Subtraction If two
pieces from one side control an enemy piece and the lower<BR>numerical values
when subtracted from the greater equals the numerical value<BR>of the enemy
piece then the enemy piece is captured.<BR>Multiplication If two pieces from one
side control an enemy piece and the<BR>multiplication of their numerical values
is equal to that of the enemy piece<BR>then the enemy piece is
captured<BR><BR>Division If two pieces from one side control an enemy piece and
the division<BR>of the numerical values of one by the other is equal to that of
the enemy<BR>piece then the enemy piece is captured<BR><BR>Progression If two
pieces from one side control an enemy piece and the<BR>numerical values of the
three pieces form an arithmetic, geometric or<BR>harmonic progression then the
enemy piece of the three is captured. The<BR>enemy piece can be anywhere in the
progression. If following the rules that<BR>allow a captured piece to remain in
situ but turned over, then capture by<BR>progression is followed by winning (see
below).<BR><BR>Distance multiplication/division If a piece from one side is so
positioned<BR>that an uninterrupted line lies between it and an enemy piece and
that<BR>distance in squares (including those both pieces are on) multiplied
or<BR>divided by the numerical value of the capturing piece is equal to
the<BR>numerical value of the enemy piece then the enemy piece is
captured.<BR><BR>Winning<BR>Winning Rithmomachia is achieved by a player forming
an arithmetic,<BR>geometric or harmonic progression of three pieces in the enemy
half of the<BR>board. These are called Triumphs. The three pieces may be
arranged in a<BR>number of different ways, orthogonal or diagonal rows or three
points<BR>forming a right angle. The three pieces do not have to be adjacent but
it is<BR>important that they are equally spaced such that the middle piece
is<BR>separated from the two on either side by an equal number of
squares.<BR><BR>In one version of Rithmomachia greater Triumphs can be made
after this<BR>beginning with an arrangement of four pieces so that their
numerical values<BR>form two separate progressions. The third and greatest
triumph is achieved<BR>through the arrangement of four pieces such that their
numerical values form<BR>one each of the three progressions, arithmetic,
geometric and harmonic. The<BR>progressions for each of these victories are
included in the tables within<BR>which those progressions that can be made using
the pieces from one side<BR>alone are indicated by shading for black and
outlining for white. The<BR>individual numbers of the progressions within an
arrangement of four pieces<BR>do not have to run consecutively. It will be noted
that white has the<BR>advantage over black in having a much greater opportunity
in being able to<BR>make triumphs with four pieces demonstrating two
progressions without having<BR>to capture any black
pieces.</FONT><BR><BR><BR><BR></FONT></DIV></BODY></HTML>