chess
TRANSCRIPT
Basic Rules of ChessChess and Mathematics
Chess and Mathematics
Kim Yong Woo
KAIST
December 23, 2011
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard
with a whole lot of other pieces.Pawn Knight Bishop Rook Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.
Pawn Knight Bishop Rook Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.
Pawn
Knight Bishop Rook Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.Pawn
Knight
Bishop Rook Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.Pawn Knight
Bishop
Rook Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.Pawn Knight Bishop
Rook
Queen King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.Pawn Knight Bishop Rook
Queen
King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Board and pieces
a
1
b c d e f g h
2
3
4
5
6
7
8
Chess is played on an 8x8 chessboard with a whole lot of other pieces.Pawn Knight Bishop Rook Queen
King
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Pawn
a
1
b c d e f g h
2
3
4
5
6
7
8
The pawn can move only one square forward
except at the beginning where itcan move two squares.When capturing, pawns move diagonally
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Pawn
a
1
b c d e f g h
2
3
4
5
6
7
8
The pawn can move only one square forward
except at the beginning where itcan move two squares.When capturing, pawns move diagonally
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Pawn
a
1
b c d e f g h
2
3
4
5
6
7
8
The pawn can move only one square forward except at the beginning where itcan move two squares.
When capturing, pawns move diagonally
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Pawn
a
1
b c d e f g h
2
3
4
5
6
7
8
The pawn can move only one square forward except at the beginning where itcan move two squares.
When capturing, pawns move diagonally
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Pawn
a
1
b c d e f g h
2
3
4
5
6
7
8
The pawn can move only one square forward except at the beginning where itcan move two squares.
When capturing, pawns move diagonally
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight
a
1
b c d e f g h
2
3
4
5
6
7
8
Knights move in an interesting way.
Knights can jump over other pieces.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight
a
1
b c d e f g h
2
3
4
5
6
7
8
Knights move in an interesting way.
Knights can jump over other pieces.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight
a
1
b c d e f g h
2
3
4
5
6
7
8
Knights move in an interesting way.
Knights can jump over other pieces.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight
a
1
b c d e f g h
2
3
4
5
6
7
8
Knights move in an interesting way.
Knights can jump over other pieces.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight
a
1
b c d e f g h
2
3
4
5
6
7
8
Knights move in an interesting way.
Knights can jump over other pieces.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishop
a
1
b c d e f g h
2
3
4
5
6
7
8
Bishops move diagonally.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishop
a
1
b c d e f g h
2
3
4
5
6
7
8
Bishops move diagonally.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishop
a
1
b c d e f g h
2
3
4
5
6
7
8
Bishops move diagonally.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Rook
a
1
b c d e f g h
2
3
4
5
6
7
8
Rooks move in a straight line.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Rook
a
1
b c d e f g h
2
3
4
5
6
7
8
Rooks move in a straight line.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Rook
a
1
b c d e f g h
2
3
4
5
6
7
8
Rooks move in a straight line.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Queen
a
1
b c d e f g h
2
3
4
5
6
7
8
Queens can move both like rooks and bishops.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Queen
a
1
b c d e f g h
2
3
4
5
6
7
8
Queens can move both like rooks and bishops.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Queen
a
1
b c d e f g h
2
3
4
5
6
7
8
Queens can move both like rooks and bishops.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Queen
a
1
b c d e f g h
2
3
4
5
6
7
8
Queens can move both like rooks and bishops.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The King
a
1
b c d e f g h
2
3
4
5
6
7
8
Kings can move to any square surrounding it.
The game ends when the king cannot escape. (Check mate)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The King
a
1
b c d e f g h
2
3
4
5
6
7
8
Kings can move to any square surrounding it.
The game ends when the king cannot escape. (Check mate)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The King
a
1
b c d e f g h
2
3
4
5
6
7
8
Kings can move to any square surrounding it.
The game ends when the king cannot escape. (Check mate)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The King
a
1
b c d e f g h
2
3
4
5
6
7
8
Kings can move to any square surrounding it.
The game ends when the king cannot escape. (Check mate)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
So how can we relate chess to mathematics?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
In recreational mathematics, there are two problems asked.
1 How many pieces of a given type can be placed on a chessboard withoutattacking each other?
2 What is the smallest number of pieces needed to attack every square?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
In recreational mathematics, there are two problems asked.
1 How many pieces of a given type can be placed on a chessboard withoutattacking each other?
2 What is the smallest number of pieces needed to attack every square?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
In recreational mathematics, there are two problems asked.
1 How many pieces of a given type can be placed on a chessboard withoutattacking each other?
2 What is the smallest number of pieces needed to attack every square?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishops Problem
How many pieces of a given type can be placed on a chessboard withoutattacking each other?
a1
b c d e f g h
2
3
4
5
6
7
8
For an n × n chessboard, the answer is 2n − 1.
The number of rotationallyand reflectively distinct solutions is given by,
B(n) =
2n−42
[2
n−22 + 1
]for n even
2n−32
[2
n−32 + 1
]for n odd
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishops Problem
How many pieces of a given type can be placed on a chessboard withoutattacking each other?
a1
b c d e f g h
2
3
4
5
6
7
8
For an n × n chessboard, the answer is 2n − 1. The number of rotationallyand reflectively distinct solutions is given by,
B(n) =
2n−42
[2
n−22 + 1
]for n even
2n−32
[2
n−32 + 1
]for n odd
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Bishops Problem
What is the smallest number of pieces needed to attack every square?
a
1
b c d e f g h
2
3
4
5
6
7
8
The answer is n = 8.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knights Problem
How many pieces of a given type can be placed on a chessboard withoutattacking each other?
a1
b c d e f g h
2
3
4
5
6
7
8
For an 8 × 8 chessboard, the answer is 32.
In general,
K(n) =
{12n2 for n > 2 even
12
(n2 + 1
)for n > 1 odd
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knights Problem
How many pieces of a given type can be placed on a chessboard withoutattacking each other?
a1
b c d e f g h
2
3
4
5
6
7
8
For an 8 × 8 chessboard, the answer is 32. In general,
K(n) =
{12n2 for n > 2 even
12
(n2 + 1
)for n > 1 odd
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knights Problem
What is the smallest number of pieces needed to attack every square?
a1
b c d e f g h
2
3
4
5
6
7
8
The answer is 12.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard.
An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
The knight’s tour is a mathematical problem involving a knight on achessboard. An instance of the more general Hamiltonian path problem, orHamiltonian cycle problem in graph theory.
a1
b c d e f g h
2
3
4
5
6
7
8
On an 8 × 8 chessboard, there are exactly 26,534,728,821,064 directed closedtours.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
Schwenk’s Theorem
1 m and n are both odd and are both not equal to 1.
2 m = 1, 2, 4 and m and n are both not equal to 1.
3 m = 3 and n = 4, 6, 8
Proofs?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
Schwenk’s Theorem
1 m and n are both odd and are both not equal to 1.
2 m = 1, 2, 4 and m and n are both not equal to 1.
3 m = 3 and n = 4, 6, 8
Proofs?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
Schwenk’s Theorem
1 m and n are both odd and are both not equal to 1.
2 m = 1, 2, 4 and m and n are both not equal to 1.
3 m = 3 and n = 4, 6, 8
Proofs?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
Schwenk’s Theorem
1 m and n are both odd and are both not equal to 1.
2 m = 1, 2, 4 and m and n are both not equal to 1.
3 m = 3 and n = 4, 6, 8
Proofs?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
The Knight’s Tour
Schwenk’s Theorem
1 m and n are both odd and are both not equal to 1.
2 m = 1, 2, 4 and m and n are both not equal to 1.
3 m = 3 and n = 4, 6, 8
Proofs?
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 1: m and n are both oddFor the standard black and white chessboard, the knight must move eitherfrom a black square to a white square or from a white square to a black square.
So in a closed tour, the knight must visit the same number of black and whitesquares. (i.e. the total number of squares visited must be even)However, when m and n are both odd, the total number of squares is odd andtherefore, a closed tour does not exist.(except for the trivial case of m = 1 = n)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 1: m and n are both oddFor the standard black and white chessboard, the knight must move eitherfrom a black square to a white square or from a white square to a black square.So in a closed tour, the knight must visit the same number of black and whitesquares. (i.e. the total number of squares visited must be even)
However, when m and n are both odd, the total number of squares is odd andtherefore, a closed tour does not exist.(except for the trivial case of m = 1 = n)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 1: m and n are both oddFor the standard black and white chessboard, the knight must move eitherfrom a black square to a white square or from a white square to a black square.So in a closed tour, the knight must visit the same number of black and whitesquares. (i.e. the total number of squares visited must be even)However, when m and n are both odd, the total number of squares is odd andtherefore, a closed tour does not exist.(except for the trivial case of m = 1 = n)
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 2: The shorter side m is of length 1, 2, 4.
When m = 1, the knight cannot move anywhere since it must change lineswhen moving.When m = 2, the knight can move but it can only visit some squares which aredetermined by the knight’s starting pointThe case when m = 4 requires some more thinking.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 2: The shorter side m is of length 1, 2, 4.When m = 1, the knight cannot move anywhere since it must change lineswhen moving.
When m = 2, the knight can move but it can only visit some squares which aredetermined by the knight’s starting pointThe case when m = 4 requires some more thinking.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 2: The shorter side m is of length 1, 2, 4.When m = 1, the knight cannot move anywhere since it must change lineswhen moving.When m = 2, the knight can move but it can only visit some squares which aredetermined by the knight’s starting point
The case when m = 4 requires some more thinking.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 2: The shorter side m is of length 1, 2, 4.When m = 1, the knight cannot move anywhere since it must change lineswhen moving.When m = 2, the knight can move but it can only visit some squares which aredetermined by the knight’s starting pointThe case when m = 4 requires some more thinking.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Imagine a 4 × n board, which is coloured like the following figure, and let’sassume that a closed knight’s tour exist.
Now let’s define the following:A1 - The set of all black squares.A2 - The set of all white squares.B1 - The set of all green squares.B2 - The set of all red squares.Note that there are an equal number of green squares and red squares.From a square in B1, the knight doesn’t have any choice apart from moving toa square in B2.Since the knight has to visit every square, and there are an equal number ofgreen squares and red squares, from a square in B2, the knight must move to asquare in B1 or else the knight would have to move from B1 to B1 which isimpossible.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Imagine a 4 × n board, which is coloured like the following figure, and let’sassume that a closed knight’s tour exist.
Now let’s define the following:A1 - The set of all black squares.A2 - The set of all white squares.B1 - The set of all green squares.B2 - The set of all red squares.
Note that there are an equal number of green squares and red squares.From a square in B1, the knight doesn’t have any choice apart from moving toa square in B2.Since the knight has to visit every square, and there are an equal number ofgreen squares and red squares, from a square in B2, the knight must move to asquare in B1 or else the knight would have to move from B1 to B1 which isimpossible.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Imagine a 4 × n board, which is coloured like the following figure, and let’sassume that a closed knight’s tour exist.
Now let’s define the following:A1 - The set of all black squares.A2 - The set of all white squares.B1 - The set of all green squares.B2 - The set of all red squares.Note that there are an equal number of green squares and red squares.
From a square in B1, the knight doesn’t have any choice apart from moving toa square in B2.Since the knight has to visit every square, and there are an equal number ofgreen squares and red squares, from a square in B2, the knight must move to asquare in B1 or else the knight would have to move from B1 to B1 which isimpossible.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Imagine a 4 × n board, which is coloured like the following figure, and let’sassume that a closed knight’s tour exist.
Now let’s define the following:A1 - The set of all black squares.A2 - The set of all white squares.B1 - The set of all green squares.B2 - The set of all red squares.Note that there are an equal number of green squares and red squares.From a square in B1, the knight doesn’t have any choice apart from moving toa square in B2.
Since the knight has to visit every square, and there are an equal number ofgreen squares and red squares, from a square in B2, the knight must move to asquare in B1 or else the knight would have to move from B1 to B1 which isimpossible.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Imagine a 4 × n board, which is coloured like the following figure, and let’sassume that a closed knight’s tour exist.
Now let’s define the following:A1 - The set of all black squares.A2 - The set of all white squares.B1 - The set of all green squares.B2 - The set of all red squares.Note that there are an equal number of green squares and red squares.From a square in B1, the knight doesn’t have any choice apart from moving toa square in B2.Since the knight has to visit every square, and there are an equal number ofgreen squares and red squares, from a square in B2, the knight must move to asquare in B1 or else the knight would have to move from B1 to B1 which isimpossible.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.
This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.
Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Now we can draw a closed knight’s tour. Looking at the process step by step:
1 Let’s start at a square of A1 and B1.
2 The second square must be of A2 and B2.
3 The third square must be of A1 and B1.
4 The fourth square must be of A2 and B2.
5 And so on.
This shows that set A1 has the same elements as set B1 and set A2 has thesame elements as set B2.This is obviously not true since the red and green patterns do not have thecheckerboard pattern of a chessboard.Therefore, our assumption was false and no closed knight’s tour can be drawnfor a 4 × n board.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Condition 3: m = 3 and n = 4, 6, 8This condition can be proved by actually attempting to draw a closed knight’stour which will lead to failure. For n even and greater than 8, there is arepeating pattern and therefore could be shown to have closed knight’s toursby induction.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Warnsdorff’s rule
Warnsdorff’s rule is a method for finding a knight’s tour; always proceed to thesquare from which the knight will have the fewest onward moves.
a1
b c d e f g h
2
3
4
5
6
7
8
2
3
5
7
7
7
When calculating the number of onward moves, we do not include squares thathave been visited already. This rule in general can be applied to any graph;each move is made to the adjacent vertex with the least degree.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Warnsdorff’s rule
Warnsdorff’s rule is a method for finding a knight’s tour; always proceed to thesquare from which the knight will have the fewest onward moves.
a1
b c d e f g h
2
3
4
5
6
7
8
2
3
5
7
7
7
When calculating the number of onward moves, we do not include squares thathave been visited already. This rule in general can be applied to any graph;each move is made to the adjacent vertex with the least degree.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Warnsdorff’s rule
Warnsdorff’s rule is a method for finding a knight’s tour; always proceed to thesquare from which the knight will have the fewest onward moves.
a1
b c d e f g h
2
3
4
5
6
7
8
2
3
5
7
7
7
When calculating the number of onward moves, we do not include squares thathave been visited already. This rule in general can be applied to any graph;each move is made to the adjacent vertex with the least degree.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Warnsdorff’s rule
Warnsdorff’s rule is a method for finding a knight’s tour; always proceed to thesquare from which the knight will have the fewest onward moves.
a1
b c d e f g h
2
3
4
5
6
7
8
2
3
5
7
7
7
When calculating the number of onward moves, we do not include squares thathave been visited already.
This rule in general can be applied to any graph;each move is made to the adjacent vertex with the least degree.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Warnsdorff’s rule
Warnsdorff’s rule is a method for finding a knight’s tour; always proceed to thesquare from which the knight will have the fewest onward moves.
a1
b c d e f g h
2
3
4
5
6
7
8
2
3
5
7
7
7
When calculating the number of onward moves, we do not include squares thathave been visited already. This rule in general can be applied to any graph;each move is made to the adjacent vertex with the least degree.
Kim Yong Woo Chess and Mathematics
Basic Rules of ChessChess and Mathematics
Thank you for listening!
Kim Yong Woo Chess and Mathematics