Rithmomachia
The Philosophers’ Game
Omar Ortiz
Department of Mathematics and Statistics
The University of Melbourne
[email protected]
23 May 2013, Melbourne
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History
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History
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History
• Practically nothing is known about the origins of Rithmomachia.
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History
• Practically nothing is known about the origins of Rithmomachia.
• Medieval writers attributed the game to Pythagoras, however no trace
of it has been found in Greek literature.
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History
• Practically nothing is known about the origins of Rithmomachia.
• Medieval writers attributed the game to Pythagoras, however no trace
of it has been found in Greek literature.
• The first record of Rithmomachia is found in works of Hermannus
Contractus (1013-1054).
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History
• Practically nothing is known about the origins of Rithmomachia.
• Medieval writers attributed the game to Pythagoras, however no trace
of it has been found in Greek literature.
• The first record of Rithmomachia is found in works of Hermannus
Contractus (1013-1054).
• In the 12th and 13th centuries, Rythmomachia spread quickly through
schools and monasteries in Germany, France and England, used
mainly as a teaching aid.
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History
• Practically nothing is known about the origins of Rithmomachia.
• Medieval writers attributed the game to Pythagoras, however no trace
of it has been found in Greek literature.
• The first record of Rithmomachia is found in works of Hermannus
Contractus (1013-1054).
• In the 12th and 13th centuries, Rythmomachia spread quickly through
schools and monasteries in Germany, France and England, used
mainly as a teaching aid.
• The game began to lose popularity in the following centuries. Leibniz
(1646-1716) knew only the name of the game but nothing of its rules.
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The game
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The game
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The game
• Two player board game.
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The game
• Two player board game.
• The board is rectangular with 8 × 16
boxes.
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The game
• Two player board game.
• The board is rectangular with 8 × 16
boxes.
• Unlike chess and checkers, the boxes
in the board are usually all of the
same colour.
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The game
• Two player board game.
• The board is rectangular with 8 × 16
boxes.
• Unlike chess and checkers, the boxes
in the board are usually all of the
same colour.
• One set of pieces is colour white, and
the other is colour black.
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The game
• Two player board game.
• The board is rectangular with 8 × 16
boxes.
• Unlike chess and checkers, the boxes
in the board are usually all of the
same colour.
• One set of pieces is colour white, and
the other is colour black.
• Each player begins with 24 pieces: 8
disks, 8 triangles, 7 squares and 1
pyramid, set on the board as shown in
the picture.
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The game
• Two player board game.
• The board is rectangular with 8 × 16
boxes.
• Unlike chess and checkers, the boxes
in the board are usually all of the
same colour.
• One set of pieces is colour white, and
the other is colour black.
• Each player begins with 24 pieces: 8
disks, 8 triangles, 7 squares and 1
pyramid, set on the board as shown in
the picture.
• A number is inscribed on each piece.
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The pieces
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The pieces
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The pieces
• The first row of white disks (lower half
of the board in the picture) are
labelled with the first four even
numbers: 2, 4, 6, 8.
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The pieces
• The first row of white disks (lower half
of the board in the picture) are
labelled with the first four even
numbers: 2, 4, 6, 8.
• The first row of black disks (upper
half of the board in the picture) are
labelled with the first four odd
numbers: 3, 5, 7, 9.
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The pieces
• The first row of white disks (lower half
of the board in the picture) are
labelled with the first four even
numbers: 2, 4, 6, 8.
• The first row of black disks (upper
half of the board in the picture) are
labelled with the first four odd
numbers: 3, 5, 7, 9.
• The unity 1 is regarded not as a
number but as “the origin of all
numbers”, reason why it is not
inscribed on a piece.
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The pieces
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The pieces
• The disks in the second row are the
squares of the disks in the first row.
For the whites: 4, 16, 36, 64.
For the blacks: 9, 25, 49, 81.
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The pieces
• The disks in the second row are the
squares of the disks in the first row.
For the whites: 4, 16, 36, 64.
For the blacks: 9, 25, 49, 81.
• Each triangle in the third row is the
sum of the two disks in front of it.
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The pieces
• The disks in the second row are the
squares of the disks in the first row.
For the whites: 4, 16, 36, 64.
For the blacks: 9, 25, 49, 81.
• Each triangle in the third row is the
sum of the two disks in front of it.
• The other four white triangles are
squares of odd numbers: 32 = 9,
52 = 25, 72 = 49, 92 = 81.
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The pieces
• The disks in the second row are the
squares of the disks in the first row.
For the whites: 4, 16, 36, 64.
For the blacks: 9, 25, 49, 81.
• Each triangle in the third row is the
sum of the two disks in front of it.
• The other four white triangles are
squares of odd numbers: 32 = 9,
52 = 25, 72 = 49, 92 = 81.
• The other four black triangles are the
square of even numbers: 42 = 16,
62 = 36, 82 = 64, 102 = 100.
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The pieces
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The pieces
• The square pieces in the third row are
formed by adding the respective
triangles.
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The pieces
• The square pieces in the third row are
formed by adding the respective
triangles.
For the whites: 9 + 6 = 15,
25 + 20 = 45, 49 + 42 = 91,
81 + 72 = 153.
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The pieces
• The square pieces in the third row are
formed by adding the respective
triangles.
For the whites: 9 + 6 = 15,
25 + 20 = 45, 49 + 42 = 91,
81 + 72 = 153.
For the blacks: 16 + 12 = 28,
36 + 30 = 66, 64 + 56 = 120,
100 + 90 = 190.
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The pieces
• The square pieces in the third row are
formed by adding the respective
triangles.
For the whites: 9 + 6 = 15,
25 + 20 = 45, 49 + 42 = 91,
81 + 72 = 153.
For the blacks: 16 + 12 = 28,
36 + 30 = 66, 64 + 56 = 120,
100 + 90 = 190.
• One square on each side (91 for the
whites and 190 for the blacks) is
replaced by a pyramid.
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The pieces
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The pieces
• The white square pieces in the last
row are the square of odd numbers:
52 = 25, 92 = 81m 132 = 169,
172 = 289.
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The pieces
• The white square pieces in the last
row are the square of odd numbers:
52 = 25, 92 = 81m 132 = 169,
172 = 289.
• The black square pieces in the last
row are the square of the odd
numbers: 72 = 49, 112 = 121,
152 = 225, 192 = 361.
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The pyramids
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The pyramids
• The white pyramid is formed by
stacking square pieces 36 and 25,
triangles 16 and 9, disk 4, and a conic
tip without label corresponding to the
number 1.
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The pyramids
• The white pyramid is formed by
stacking square pieces 36 and 25,
triangles 16 and 9, disk 4, and a conic
tip without label corresponding to the
number 1.
• This structure of the white pyramid
comes from the partition:
91 = 62 + 52 + 42 + 32 + 22 + 12
(a perfect pyramid).
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The pyramids
• The white pyramid is formed by
stacking square pieces 36 and 25,
triangles 16 and 9, disk 4, and a conic
tip without label corresponding to the
number 1.
• This structure of the white pyramid
comes from the partition:
91 = 62 + 52 + 42 + 32 + 22 + 12
(a perfect pyramid).
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The pyramids
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The pyramids
• The black pyramid is formed by
stacking square pieces 64 and 49,
triangles 36 and 25, and disk 16.
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The pyramids
• The black pyramid is formed by
stacking square pieces 64 and 49,
triangles 36 and 25, and disk 16.
• This structure of the black pyramid
comes from the partition:
190 = 82 + 72 + 62 + 52 + 42
(tricurta or thrice curtailed pyramid).
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The pyramids
• The black pyramid is formed by
stacking square pieces 64 and 49,
triangles 36 and 25, and disk 16.
• This structure of the black pyramid
comes from the partition:
190 = 82 + 72 + 62 + 52 + 42
(tricurta or thrice curtailed pyramid).
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Other relations
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Other relations
• Another form of finding the values of
the triangles in the second row is by
means of a relation known as
superparticularis: adding the
respective triangle in the third row to
an aliquot part of it determined by the
first disk at the top. For example
9 = 6 + 6/2, 25 = 20 + 20/4, etc.
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Other relations
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Other relations
• The square pieces in the last row can
also be found in the following way:
Let n be the respective disk in the
first row and s the respective square
in the third row. Then the square in
the last row is
2n + 1
· s.
n+1
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Other relations
• The square pieces in the last row can
also be found in the following way:
Let n be the respective disk in the
first row and s the respective square
in the third row. Then the square in
the last row is
2n + 1
· s.
n+1
For example
25 =
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2·2+1
· 15.
2+1
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Movement
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Movement
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Movement
• White always moves first.
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Movement
• White always moves first.
• After the initial move, the players alternately move one piece at a
time.
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Movement
• White always moves first.
• After the initial move, the players alternately move one piece at a
time.
• Pieces may not be moved to an occupied box or leap over other
pieces.
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Movement
• White always moves first.
• After the initial move, the players alternately move one piece at a
time.
• Pieces may not be moved to an occupied box or leap over other
pieces.
• Pieces move in accordance with their shape, independently of their
value (inscribed number).
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Movement: Disks
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Movement: Disks
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Movement: Disks
• Disks move one box diagonally in any
direction.
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Movement: Disks
• Disks move one box diagonally in any
direction.
• In the picture, the white disk can
move to any of the boxes labelled ∗.
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Movement: Disks
• Disks move one box diagonally in any
direction.
• In the picture, the white disk can
move to any of the boxes labelled ∗.
• The black disk can only move to the
box labelled +.
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Movement: Triangles
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Movement: Triangles
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Movement: Triangles
• Triangles move two boxes vertically or
horizontally in any direction.
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Movement: Triangles
• Triangles move two boxes vertically or
horizontally in any direction.
• In the picture, the black triangle can
move to any of the boxes labelled +.
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Movement: Triangles
• Triangles move two boxes vertically or
horizontally in any direction.
• In the picture, the black triangle can
move to any of the boxes labelled +.
• The white triangle can only move to
the boxes labelled ∗.
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Movement: Squares
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Movement: Squares
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Movement: Squares
• Squares move three boxes vertically or
horizontally in any direction.
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Movement: Squares
• Squares move three boxes vertically or
horizontally in any direction.
• In the picture, the white square can
move to any of the boxes labelled ∗.
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Movement: Squares
• Squares move three boxes vertically or
horizontally in any direction.
• In the picture, the white square can
move to any of the boxes labelled ∗.
• The black square can only move to
the boxes labelled +.
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Movement: Pyramids
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Movement: Pyramids
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Movement: Pyramids
• Recall that the pyramids are built of two squares, two triangles and
one disk each.
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Movement: Pyramids
• Recall that the pyramids are built of two squares, two triangles and
one disk each.
• Pyramids can move and capture as any of their components.
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Movement: Pyramids
• Recall that the pyramids are built of two squares, two triangles and
one disk each.
• Pyramids can move and capture as any of their components.
• Pyramids can be captured as a whole (rarely happens) or by
components.
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Movement: Pyramids
• Recall that the pyramids are built of two squares, two triangles and
one disk each.
• Pyramids can move and capture as any of their components.
• Pyramids can be captured as a whole (rarely happens) or by
components.
• If all the components of a given shape are captured, then the pyramid
loses the movement of that shape.
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Captures
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Captures
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Captures
• The game consists in capturing the opponent’s pieces.
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Captures
• The game consists in capturing the opponent’s pieces.
• There is a variety of capture methods.
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Captures
• The game consists in capturing the opponent’s pieces.
• There is a variety of capture methods.
• Pieces do not land on another piece to capture it, but instead remain
in their box and remove the other.
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Captures
• The game consists in capturing the opponent’s pieces.
• There is a variety of capture methods.
• Pieces do not land on another piece to capture it, but instead remain
in their box and remove the other.
• Captured pieces can be returned to the board on the side and colour
of the capturing player. For this, all pieces are white on one face and
black on the other, with the same number on both sides.
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Capture methods: Meeting
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Capture methods: Meeting
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Capture methods: Meeting
• The attacking piece most have the
same value as the target piece.
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Capture methods: Meeting
• The attacking piece most have the
same value as the target piece.
• When the attacking piece is placed
one movement apart from the target
piece, the capture is effected.
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Capture methods: Meeting
• The attacking piece most have the
same value as the target piece.
• When the attacking piece is placed
one movement apart from the target
piece, the capture is effected.
• In the picture, white disk 16 captures
black triangle 16.
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Capture methods: Ambuscade
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Capture methods: Ambuscade
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Capture methods: Ambuscade
• In the ambuscade method two pieces
attack and one is captured.
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Capture methods: Ambuscade
• In the ambuscade method two pieces
attack and one is captured.
• A basic arithmetic operation
(addition, subtraction, multiplication
or division) of the values of the
attacking pieces most equal the value
of the target piece.
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Capture methods: Ambuscade
• In the ambuscade method two pieces
attack and one is captured.
• A basic arithmetic operation
(addition, subtraction, multiplication
or division) of the values of the
attacking pieces most equal the value
of the target piece.
• When the attacking pieces are both
placed one movement apart form the
target piece, the capture is effected.
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Capture methods: Ambuscade
• In the ambuscade method two pieces
attack and one is captured.
• A basic arithmetic operation
(addition, subtraction, multiplication
or division) of the values of the
attacking pieces most equal the value
of the target piece.
• When the attacking pieces are both
placed one movement apart form the
target piece, the capture is effected.
• In the picture, white square 45 and
white triangle 9 capture black circle 5
via the equation 45/9 = 5.
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Capture methods: Assault
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Capture methods: Assault
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Capture methods: Assault
• Let A be the value of the attacking
piece,
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Capture methods: Assault
• Let A be the value of the attacking
piece,
• T the value of the target piece, and
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Capture methods: Assault
• Let A be the value of the attacking
piece,
• T the value of the target piece, and
• n ≥ 2 the number of vacant boxes
between the attacking and the target
pieces (horizontally, vertically or
diagonally).
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Capture methods: Assault
• Let A be the value of the attacking
piece,
• T the value of the target piece, and
• n ≥ 2 the number of vacant boxes
between the attacking and the target
pieces (horizontally, vertically or
diagonally).
• Then the capture is effected if
n±1 A = T .
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Capture methods: Assault
• Let A be the value of the attacking
piece,
• T the value of the target piece, and
• n ≥ 2 the number of vacant boxes
between the attacking and the target
pieces (horizontally, vertically or
diagonally).
• Then the capture is effected if
n±1 A = T .
• In the picture, white triangle 20
captures black circle 5 since A = 20,
T = 5, n = 4 and
4−1 20 = 5.
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Capture methods: Siege
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Capture methods: Siege
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Capture methods: Siege
• The attacking pieces are placed in
positions such that the target piece
cannot make any movement.
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Capture methods: Siege
• The attacking pieces are placed in
positions such that the target piece
cannot make any movement.
• The blockage may be assembled with
pieces of different colour.
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Capture methods: Siege
• The attacking pieces are placed in
positions such that the target piece
cannot make any movement.
• The blockage may be assembled with
pieces of different colour.
• In the picture, the white pyramid is
captured by the blacks.
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Capture methods: Siege
• The attacking pieces are placed in
positions such that the target piece
cannot make any movement.
• The blockage may be assembled with
pieces of different colour.
• In the picture, the white pyramid is
captured by the blacks.
• Pyramids can rarely be fully captured
except by siege.
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Capture methods: Siege
• The attacking pieces are placed in
positions such that the target piece
cannot make any movement.
• The blockage may be assembled with
pieces of different colour.
• In the picture, the white pyramid is
captured by the blacks.
• Pyramids can rarely be fully captured
except by siege.
• Siege is the only capture method that
does not take the value of pieces into
account.
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Victories
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Victories
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Victories
• There is a variety of victory conditions for determining when a game
ends and who the winner is.
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Victories
• There is a variety of victory conditions for determining when a game
ends and who the winner is.
• The particular kind of victory for which the game is to be won is
agreed by the players at the beginning.
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Victories
• There is a variety of victory conditions for determining when a game
ends and who the winner is.
• The particular kind of victory for which the game is to be won is
agreed by the players at the beginning.
• There are two classes of victories:
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Victories
• There is a variety of victory conditions for determining when a game
ends and who the winner is.
• The particular kind of victory for which the game is to be won is
agreed by the players at the beginning.
• There are two classes of victories:
• Common victories. Based on the amount or values of the captured
pieces.
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Victories
• There is a variety of victory conditions for determining when a game
•
•
•
•
ends and who the winner is.
The particular kind of victory for which the game is to be won is
agreed by the players at the beginning.
There are two classes of victories:
Common victories. Based on the amount or values of the captured
pieces.
Proper victories. Consist on arranging pieces in a certain way to
form a numerical progression.
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Common victories
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Common victories
• De Corpore (Latin: “by body”): Both players agree on a certain
amount of pieces to capture. The first player to reach this number,
wins.
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Common victories
• De Corpore (Latin: “by body”): Both players agree on a certain
amount of pieces to capture. The first player to reach this number,
wins.
• De Bonis (“by goods”): The players agree on a number for the sum
of the values. The first player to capture enough pieces to add up or
exceed this number, wins.
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Common victories
• De Corpore (Latin: “by body”): Both players agree on a certain
amount of pieces to capture. The first player to reach this number,
wins.
• De Bonis (“by goods”): The players agree on a number for the sum
of the values. The first player to capture enough pieces to add up or
exceed this number, wins.
• De Lite (“by lawsuit”): Players agree on a number for the sum of
values and a number for the total amount of digits in all the captured
pieces.
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Common victories
• De Corpore (Latin: “by body”): Both players agree on a certain
amount of pieces to capture. The first player to reach this number,
wins.
• De Bonis (“by goods”): The players agree on a number for the sum
of the values. The first player to capture enough pieces to add up or
exceed this number, wins.
• De Lite (“by lawsuit”): Players agree on a number for the sum of
values and a number for the total amount of digits in all the captured
pieces.
• De Honore (“by honour”): Players set an amount of pieces to
capture and a number for the sum of their values.
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Common victories
• De Corpore (Latin: “by body”): Both players agree on a certain
amount of pieces to capture. The first player to reach this number,
wins.
• De Bonis (“by goods”): The players agree on a number for the sum
of the values. The first player to capture enough pieces to add up or
exceed this number, wins.
• De Lite (“by lawsuit”): Players agree on a number for the sum of
values and a number for the total amount of digits in all the captured
pieces.
• De Honore (“by honour”): Players set an amount of pieces to
capture and a number for the sum of their values.
• De Honore Liteque (“by honour and lawsuit”): Players fix an
amount of pieces to capture, a number for the sum of their values
and a number for the amount of digits on the pieces.
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Proper victories
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Proper victories
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Proper victories
• Proper victories require placing three or four pieces in specific
arrangements in the opponent’s side of the board, with the values of
the pieces forming a numerical progression.
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Proper victories
• Proper victories require placing three or four pieces in specific
arrangements in the opponent’s side of the board, with the values of
the pieces forming a numerical progression.
• Frequently, returned pieces are needed to achieve a proper victory.
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Proper victories
• Proper victories require placing three or four pieces in specific
arrangements in the opponent’s side of the board, with the values of
the pieces forming a numerical progression.
• Frequently, returned pieces are needed to achieve a proper victory.
• The pieces forming the progression may be arranged in a (horizontal,
vertical or diagonal) line or in the sides of a rectangle.
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Proper victories
• Proper victories require placing three or four pieces in specific
arrangements in the opponent’s side of the board, with the values of
the pieces forming a numerical progression.
• Frequently, returned pieces are needed to achieve a proper victory.
• The pieces forming the progression may be arranged in a (horizontal,
vertical or diagonal) line or in the sides of a rectangle.
• There are three kinds of proper victories: Magna, Major and
Excellentissima.
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Progressions
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
• For example: 2, 5, 8, 11 is an arithmetic progression of ratio 3.
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
• For example: 2, 5, 8, 11 is an arithmetic progression of ratio 3.
• In a geometric progression, the ratios between successive numbers are
given by a single value called the ratio of the progression.
Omar Ortiz (Melbourne University)
Rithmomachia
23 May 2013
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
• For example: 2, 5, 8, 11 is an arithmetic progression of ratio 3.
• In a geometric progression, the ratios between successive numbers are
given by a single value called the ratio of the progression.
• For example: 3, 12, 48 is a geometric progression of ratio 4.
Omar Ortiz (Melbourne University)
Rithmomachia
23 May 2013
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
• For example: 2, 5, 8, 11 is an arithmetic progression of ratio 3.
• In a geometric progression, the ratios between successive numbers are
given by a single value called the ratio of the progression.
• For example: 3, 12, 48 is a geometric progression of ratio 4.
• In a harmonic progression, the ratio of two successive differences is
equal to the ratio of the end numbers. If the progression is a, b, c, we
have c/a = (c − b)/(b − a).
Omar Ortiz (Melbourne University)
Rithmomachia
23 May 2013
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Progressions
• Recall that in an arithmetic progression, the differences between
successive numbers are given by a single value called the ratio of the
progression.
• For example: 2, 5, 8, 11 is an arithmetic progression of ratio 3.
• In a geometric progression, the ratios between successive numbers are
given by a single value called the ratio of the progression.
• For example: 3, 12, 48 is a geometric progression of ratio 4.
• In a harmonic progression, the ratio of two successive differences is
equal to the ratio of the end numbers. If the progression is a, b, c, we
have c/a = (c − b)/(b − a).
• For example: 4, 6, 12 is a harmonic progression of ratio 3, as 12/4 =
3 and (12-6)/(6-4) = 6/2 = 3.
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Victoria Magna
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Victoria Magna
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Victoria Magna
• Three pieces are arranged in
arithmetic, geometric or harmonic
progression.
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Victoria Magna
• Three pieces are arranged in
arithmetic, geometric or harmonic
progression.
• In the picture, the white pieces form
the geometric progression 4, 20, 100
of ratio 5.
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Victoria Magna
• Three pieces are arranged in
arithmetic, geometric or harmonic
progression.
• In the picture, the white pieces form
the geometric progression 4, 20, 100
of ratio 5.
• Here the white triangle 100 is a
returned piece.
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Victoria Major
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Victoria Major
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Victoria Major
• Four pieces are arranged such that
three pieces are in a certain
progression, and another three pieces
are in another type of progression.
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Victoria Major
• Four pieces are arranged such that
three pieces are in a certain
progression, and another three pieces
are in another type of progression.
• In the picture, the white pieces form
two different progressions:
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Victoria Major
• Four pieces are arranged such that
three pieces are in a certain
progression, and another three pieces
are in another type of progression.
• In the picture, the white pieces form
two different progressions:
• The arithmetic progression 12, 16, 20
of ratio 4.
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Victoria Major
• Four pieces are arranged such that
three pieces are in a certain
progression, and another three pieces
are in another type of progression.
• In the picture, the white pieces form
two different progressions:
• The arithmetic progression 12, 16, 20
of ratio 4.
• The geometric progression 9, 12, 16 of
ration 4/3.
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Victoria Major
• Four pieces are arranged such that
three pieces are in a certain
progression, and another three pieces
are in another type of progression.
• In the picture, the white pieces form
two different progressions:
• The arithmetic progression 12, 16, 20
of ratio 4.
• The geometric progression 9, 12, 16 of
ration 4/3.
• Here the white triangle 12 is a
returned piece.
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Victoria Excellentissima
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Victoria Excellentissima
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
• In the picture, the white pieces form
three different progressions:
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
• In the picture, the white pieces form
three different progressions:
• The arithmetic progression 3, 6, 9 of
ratio 3.
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
• In the picture, the white pieces form
three different progressions:
• The arithmetic progression 3, 6, 9 of
ratio 3.
• The geometric progression 4, 6, 9 of
ration 3/2.
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
• In the picture, the white pieces form
three different progressions:
• The arithmetic progression 3, 6, 9 of
ratio 3.
• The geometric progression 4, 6, 9 of
ration 3/2.
• The harmonic progression 3, 4, 6 of
ratio 2.
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Victoria Excellentissima
• Four pieces that are arranged having
all three types of progressions
(arithmetic, geometric and harmonic)
in three different groups
• In the picture, the white pieces form
three different progressions:
• The arithmetic progression 3, 6, 9 of
ratio 3.
• The geometric progression 4, 6, 9 of
ration 3/2.
• The harmonic progression 3, 4, 6 of
ratio 2.
• Here the white circle 3 is a returned
piece.
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Variants
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Variants
• Many variants on the rules of Rithmomachia can be found in the
literature. Some variants include additional capture methods and
victory kinds.
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Variants
• Many variants on the rules of Rithmomachia can be found in the
literature. Some variants include additional capture methods and
victory kinds.
• However, the distribution of the pieces and its values is almost the
same in all different versions.
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Variants
• Many variants on the rules of Rithmomachia can be found in the
literature. Some variants include additional capture methods and
victory kinds.
• However, the distribution of the pieces and its values is almost the
same in all different versions.
• At the present time there is no governing body or Rithmomachia
federation.
Omar Ortiz (Melbourne University)
Rithmomachia
23 May 2013
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Variants
• Many variants on the rules of Rithmomachia can be found in the
literature. Some variants include additional capture methods and
victory kinds.
• However, the distribution of the pieces and its values is almost the
same in all different versions.
• At the present time there is no governing body or Rithmomachia
federation.
• As a consequence, different people play with different set of rules.
Omar Ortiz (Melbourne University)
Rithmomachia
23 May 2013
29 / 30
Variants
• Many variants on the rules of Rithmomachia can be found in the
literature. Some variants include additional capture methods and
victory kinds.
• However, the distribution of the pieces and its values is almost the
same in all different versions.
• At the present time there is no governing body or Rithmomachia
federation.
• As a consequence, different people play with different set of rules.
• The rules described in this presentation follow the references given
next.
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The End
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The End
References:
• David Eugene Smith and Clara C. Eaton. Rithmomachia, the great
medieval number game. American Math. Monthly. Vol XVIII (4)
1911.
• Ann E. Moyer. The Philosophers Game. Rithmomachia in Medieval
and Renaissance Europe. University of Michigan Press. 2001.
• Venezuelan Rithmomachia Club. How to play Ritmomachia? Online
notes. 2013 (spanish).
• Wikipedia’s article: Rithmomachy.
Omar Ortiz (Melbourne University)
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The End
References:
• David Eugene Smith and Clara C. Eaton. Rithmomachia, the great
medieval number game. American Math. Monthly. Vol XVIII (4)
1911.
• Ann E. Moyer. The Philosophers Game. Rithmomachia in Medieval
and Renaissance Europe. University of Michigan Press. 2001.
• Venezuelan Rithmomachia Club. How to play Ritmomachia? Online
notes. 2013 (spanish).
• Wikipedia’s article: Rithmomachy.
Thank you!
Omar Ortiz (Melbourne University)
Rithmomachia
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