SlideShare a Scribd company logo
1 of 23
Download to read offline
Chessboard Puzzles 
Part 3: Knight’s Tour 
Dan Freeman 
April 10, 2014 
Villanova University 
MAT 9000 Graduate Math Seminar
Introduction 
• In the first two presentations, we looked at the 
concepts of chessboard domination and 
independence 
• Tonight we will examine an entirely different 
idea called the knight’s tour 
• I will answer some questions from last time and 
then jump right into the new material 
2
Questions from Last Time 
• Could the solutions to the n-queens problem be 
formed into a group? 
• When would one use independence in a game 
and how would it be beneficial? 
• What influenced you to investigate chessboard 
puzzles? 
3
Questions from Last Time 
• I know the math must be wildly complex, but I 
would love to see independence / dependence 
with the standard 16-piece collection of pieces, 
i.e. is domination / independence possible with 
two of each piece and 8 pawns? 
• Is there a known number of different 
arrangements of knight independence? 
• What about rectangular chessboards? 
4
Knight Movement 
• Recall that knights move two squares in one 
direction (either horizontally or vertically) and 
one square in the other direction 
• Knights’ moves resemble an L shape 
• Knights are the only pieces that are allowed to 
jump over other pieces 
• In the example below, the white and black 
knights can move to squares with circles of the 
corresponding color 
5
Knight’s Tour Defined 
• A knight’s tour is a succession of moves made 
by a knight that traverses every square on a 
chessboard once and only once 
• There are two kinds of knight’s tours, a closed 
knight’s tour and an open knight’s tour: 
– A closed knight’s tour is one in which the knight’s last 
move in the tour places it a single move away from 
where it started 
– An open knight’s tour is one in which the knight’s last 
move in the tour places it on a square that is not a 
single move away from where it started 
6
Euler’s 8x8 Closed Knight’s Tour 
• Below is an example of a closed knight’s tour 
on an 8x8 board that Euler constructed from an 
incomplete open tour (only 60 squares made 
up the tour) 
7 
22 25 50 39 52 35 60 57 
27 40 23 36 49 58 53 34 
24 21 26 51 38 61 56 59 
41 28 37 48 3 54 33 62 
20 47 42 13 32 63 4 55 
29 16 19 46 43 2 7 10 
18 45 14 31 12 9 64 5 
15 30 17 44 1 
6 11 8 
Closed Knight’s Tour 
on 8x8 Board by Euler
Open Knight’s Tour on 5x5 Board 
• Because there are a different number of black 
squares (12) and white squares (13) on a 5x5 
board, no closed knight’s tour exists on this 
size board 
• However, an open knight’s tour does exist (see 
two different examples below) 
8 
1 
14 9 20 3 
24 19 2 15 10 
13 8 23 4 21 
18 25 6 11 16 
7 12 17 22 5
Open Knight’s Tour on 8x8 Board 
9
Smallest Closed Knight’s Tours 
• The smallest boards for which closed knight’s 
tours are possible are the 5x6 and 3x10 boards 
10 
5 26 1 
16 11 20 
2 15 4 19 30 17 
25 6 27 12 21 10 
14 3 8 23 18 29 
7 24 13 28 9 22 
Closed Knight’s Tour 
on 5x6 Board 
26 29 2 21 8 23 6 17 14 11 
20 27 24 3 18 9 12 5 16 
28 25 30 19 22 7 4 15 10 13 
Closed Knight’s Tour 
on 3x10 Board 
1
Smallest Open Knight’s Tour 
• The smallest board for which an open knight’s 
tour is possible is a 3x4 board 
11 
1 4 7 10 
12 9 2 5 
3 6 11 8 
Open Knight’s Tour on 
3x4 Board 
1
de Moivre’s 8x8 
Open Knight’s Tour 
• de Moivre used an effective technique for 
completing knight’s tours that starts on the 
edges of the board and works its way inward 
• This technique is illustrated below 
12 
34 49 22 11 36 39 24 1 
21 10 35 50 23 12 37 40 
48 33 62 57 38 25 2 13 
9 20 51 54 63 60 41 26 
32 47 58 61 56 53 14 3 
19 8 55 52 59 64 27 42 
46 31 6 17 44 29 4 15 
7 18 45 30 5 16 43 28 
Open Knight’s Tour on 
8x8 Board by de Moivre 
1
6x6 Closed Knight’s Tour 
• de Moivre’s technique can also be used to find 
a closed knight’s tour on a 6x6 board 
• This technique is illustrated on a YouTube 
video 
13 
34 7 24 15 32 1 
23 14 33 36 25 16 
6 35 8 17 2 31 
13 22 29 26 9 18 
28 5 20 11 30 3 
21 12 27 4 19 10 
Closed Knight’s Tour on 
6x6 Board 
1
Knight’s Tour Combinatorics 
• The number of unique directed closed knight’s 
tours on an 8x8 board is 26,534,728,821,064 
• There are (only) 19,724 directed closed 
knight’s tours on a 6x6 board 
• The number of directed open tours for an 8x8 
board is unknown 
• However, the number of directed open tours 
are known for 1 ≤ n ≤ 7 
14 
n Permutations 
1 1 
2 0 
3 0 
4 0 
5 1,728 
6 6,637,920 
7 165,575,218,320
Pósa’s Proof that No Closed 
Knight’s Tour Exists on 4xn Board 
• As a teenager, Louis Pósa proved that a 4xn 
chessboard has no closed knight’s tour 
• He used a simple coloring proof, as follows: 
– First, suppose there does exist a closed knight’s tour 
on an arbitrary 4xn board. With the standard black 
and white coloring of the board, we know that a 
knight must alternate between black and white 
squares along the tour. 
– Now color the top and bottom rows of the board red 
and the two middles rows blue (see below). 
15 
Pósa’s Coloring 
on 4x7 Board
Pósa’s Proof Continued 
– Note that a knight on a red square can only move to 
a blue square, not another red square. Thus, since 
there are the same number of red squares and blue 
squares, a knight cannot move from a blue square to 
another blue square, because it would not be able to 
make up for this by visiting two red squares 
consecutively. 
– Therefore, the knight must strictly alternate between 
red and blue squares. But this is impossible 
because, by assumption, the knight alternated 
between black and white squares in the traditional 
coloring pattern to form a tour, which would imply that 
the two coloring patterns are the same. Of course, 
they are not so we have a contradiction. Thus, no 
knight’s tour exists on a 4xn board. 
16
Schwenk’s Theorem 
• An mxn chessboard with m ≤ n has a closed 
knight’s tour unless one or more of the 
following three conditions hold: 
– m and n are both odd; 
– m = 1, 2 or 4; or 
– m = 3 and n = 4, 6 or 8. 
• The complete proof is rather involved and uses 
an induction argument to show that a closed 
knight’s tour exists on 3xn boards for 
n ≥ 10, n even 
• In addition, the proof consists of building larger 
tours from 5x5, 6x6, 5x8, 6x7, 6x8, 7x8 and 
8x8 boards to show that tours exist for all mxn 
boards not excluded by the theorem 
17
Magic Squares 
• A magic square is an array of numbers in 
which the sum of each row, each column and 
the two main diagonals all equal the same 
value 
• A very old and famous 3x3 magic square 
appears below; each row, column and main 
diagonal sums to 15 
18 
4 9 2 
3 5 7 
8 1 6 
3x3 Magic Square
Magic Squares from 
a Knight’s Move 
• Muhammad ibn Muhammad used a knight’s 
move to construct magic squares 
• Starting in the upper-right hand corner, he 
would make knight moves going down and to 
the left, wrapping around the board when 
necessary 
• If he ran into a square that was already visited, 
he would move two squares to the left 
19 
Muhammad ibn 
Muhammad’s 5x5 
Magic Square 
13 25 7 19 1 
17 4 11 23 10 
21 8 20 2 14 
5 12 24 6 18 
9 16 3 15 22
Magic Squares from 
a Knight’s Tour 
• Completely ignorant of Muhammad ibn 
Muhammad’s work, Balof and Watkins 
constructed magic squares using knight’s tours 
• The only difference between the two methods 
was that Balof and Watkins used a knight’s 
move when the knight was blocked 
20 
Balof and Watkin’s 
7x7 Magic Square 
1 24 47 21 37 11 34 
12 35 2 25 48 15 38 
16 39 13 29 3 26 49 
27 43 17 40 14 30 4 
31 5 28 44 18 41 8 
42 9 32 6 22 45 19 
46 20 36 10 33 7 23
Magic Squares from 
a Knight’s Tour 
• Balof and Watkins’ method works in general to 
produce an nxn magic square as long as n is 
not divisible by 2, 3 or 5 
• If n is not divisible by 2 or 3 but is divisible by 5, 
then only the sums for the two main diagonals 
are not the same 
21
Latin Squares and Knight’s Tours 
• An nxn Latin square is a square with n distinct 
labels, which can be numbers, letters, colors, 
etc., that appear inside each cell, with each 
label appearing in each row and each column 
once and only once 
• A very intriguing website showcases an odd 
relationship between Latin squares and 
knight’s tours 
22
Sources Cited 
• J.J. Watkins. Across the Board: The Mathematics of 
Chessboard Problems. Princeton, New Jersey: 
Princeton University Press, 2004. 
• “Knight’s Tour.” Wikipedia, Wikimedia Foundation. 
http://en.wikipedia.org/wiki/Knight%27s_tour 
• "Learn How to Perform the Knight's Tour.“ YouTube. 
https://www.youtube.com/watch?v=Ma1C6wcR0Jg 
• “A001230 – OEIS.” http://oeis.org/A001230 
• “A165134 – OEIS.” http://oeis.org/A165134 
• “The Knight’s Tour.” Borders Chess Club. 
http://www.borderschess.org/KnightTour.htm 
23

More Related Content

What's hot

GamingAnywhere: An Open Cloud Gaming System
GamingAnywhere: An Open Cloud Gaming SystemGamingAnywhere: An Open Cloud Gaming System
GamingAnywhere: An Open Cloud Gaming SystemAcademia Sinica
 
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and Trunking
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and TrunkingPanduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and Trunking
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and TrunkingThorne & Derrick International
 
Embedded system in Smart Cards
Embedded system in Smart CardsEmbedded system in Smart Cards
Embedded system in Smart CardsRebecca D'souza
 
Getting Started with Raspberry Pi
Getting Started with Raspberry PiGetting Started with Raspberry Pi
Getting Started with Raspberry Piyeokm1
 
hawk eye technology
hawk eye technologyhawk eye technology
hawk eye technologymousam meher
 
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)병호 신
 

What's hot (7)

GamingAnywhere: An Open Cloud Gaming System
GamingAnywhere: An Open Cloud Gaming SystemGamingAnywhere: An Open Cloud Gaming System
GamingAnywhere: An Open Cloud Gaming System
 
Chess
ChessChess
Chess
 
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and Trunking
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and TrunkingPanduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and Trunking
Panduit LSF Halogen Free (Zero Halogen) Cable & Wiring Duct and Trunking
 
Embedded system in Smart Cards
Embedded system in Smart CardsEmbedded system in Smart Cards
Embedded system in Smart Cards
 
Getting Started with Raspberry Pi
Getting Started with Raspberry PiGetting Started with Raspberry Pi
Getting Started with Raspberry Pi
 
hawk eye technology
hawk eye technologyhawk eye technology
hawk eye technology
 
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)
Mmorpg의 재미 평가 모델에 관한 연구(2쇄수정)
 

Viewers also liked

Chessboard Puzzles Part 3 - Knight's Tour
Chessboard Puzzles Part 3 - Knight's TourChessboard Puzzles Part 3 - Knight's Tour
Chessboard Puzzles Part 3 - Knight's TourDan Freeman
 
Chessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationChessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationDan Freeman
 
Chessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationChessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationDan Freeman
 
A biased random-key genetic algorithm for the Steiner triple covering problem
A biased random-key genetic algorithm for the Steiner triple covering problemA biased random-key genetic algorithm for the Steiner triple covering problem
A biased random-key genetic algorithm for the Steiner triple covering problemgomgcr
 
Chessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceChessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceDan Freeman
 
Chessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsChessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsDan Freeman
 
Chessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceChessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceDan Freeman
 
Chess camp 3. checkmates with many pieces
Chess camp 3. checkmates with many piecesChess camp 3. checkmates with many pieces
Chess camp 3. checkmates with many piecesNelson ruiz
 
Knights tour
Knights tour Knights tour
Knights tour sasank123
 
Knight’s tour algorithm
Knight’s tour algorithmKnight’s tour algorithm
Knight’s tour algorithmHassan Tariq
 
Management of Tremor
Management of Tremor Management of Tremor
Management of Tremor PS Deb
 
The n Queen Problem
The n Queen ProblemThe n Queen Problem
The n Queen ProblemSukrit Gupta
 
Queue- 8 Queen
Queue- 8 QueenQueue- 8 Queen
Queue- 8 QueenHa Ninh
 
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...Simplify360
 
backtracking algorithms of ada
backtracking algorithms of adabacktracking algorithms of ada
backtracking algorithms of adaSahil Kumar
 
Movement disorders lecture
Movement disorders lectureMovement disorders lecture
Movement disorders lecturetest
 

Viewers also liked (20)

Chessboard Puzzles Part 3 - Knight's Tour
Chessboard Puzzles Part 3 - Knight's TourChessboard Puzzles Part 3 - Knight's Tour
Chessboard Puzzles Part 3 - Knight's Tour
 
Knight's Tour
Knight's TourKnight's Tour
Knight's Tour
 
Chessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationChessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - Domination
 
Chessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - DominationChessboard Puzzles Part 1 - Domination
Chessboard Puzzles Part 1 - Domination
 
A biased random-key genetic algorithm for the Steiner triple covering problem
A biased random-key genetic algorithm for the Steiner triple covering problemA biased random-key genetic algorithm for the Steiner triple covering problem
A biased random-key genetic algorithm for the Steiner triple covering problem
 
Chessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceChessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - Independence
 
Chessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsChessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and Variations
 
Chessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - IndependenceChessboard Puzzles Part 2 - Independence
Chessboard Puzzles Part 2 - Independence
 
Chess camp 3. checkmates with many pieces
Chess camp 3. checkmates with many piecesChess camp 3. checkmates with many pieces
Chess camp 3. checkmates with many pieces
 
Knights tour
Knights tour Knights tour
Knights tour
 
Knight’s tour algorithm
Knight’s tour algorithmKnight’s tour algorithm
Knight’s tour algorithm
 
Management of Tremor
Management of Tremor Management of Tremor
Management of Tremor
 
The n Queen Problem
The n Queen ProblemThe n Queen Problem
The n Queen Problem
 
Queue- 8 Queen
Queue- 8 QueenQueue- 8 Queen
Queue- 8 Queen
 
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...
 
Rehiyon 8 - CARAGA
Rehiyon 8 - CARAGA Rehiyon 8 - CARAGA
Rehiyon 8 - CARAGA
 
Tremors
TremorsTremors
Tremors
 
backtracking algorithms of ada
backtracking algorithms of adabacktracking algorithms of ada
backtracking algorithms of ada
 
Caraga region xiii
Caraga   region xiiiCaraga   region xiii
Caraga region xiii
 
Movement disorders lecture
Movement disorders lectureMovement disorders lecture
Movement disorders lecture
 

Similar to Chessboard Puzzles Part 3 - Knight's Tour

Chessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsChessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsDan Freeman
 
chesslhetsl23rd-200422233339.pptx
chesslhetsl23rd-200422233339.pptxchesslhetsl23rd-200422233339.pptx
chesslhetsl23rd-200422233339.pptxfernandopajar1
 
CHESS 3rd quarter P.E Grade 8
CHESS 3rd quarter P.E Grade 8CHESS 3rd quarter P.E Grade 8
CHESS 3rd quarter P.E Grade 8CNHS-CMSP
 

Similar to Chessboard Puzzles Part 3 - Knight's Tour (7)

Chessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and VariationsChessboard Puzzles Part 4 - Other Surfaces and Variations
Chessboard Puzzles Part 4 - Other Surfaces and Variations
 
Hex Chess
Hex ChessHex Chess
Hex Chess
 
ppt-knight'stour
ppt-knight'stourppt-knight'stour
ppt-knight'stour
 
chesslhetsl23rd-200422233339.pptx
chesslhetsl23rd-200422233339.pptxchesslhetsl23rd-200422233339.pptx
chesslhetsl23rd-200422233339.pptx
 
Chess final 1
Chess final 1Chess final 1
Chess final 1
 
Chess
ChessChess
Chess
 
CHESS 3rd quarter P.E Grade 8
CHESS 3rd quarter P.E Grade 8CHESS 3rd quarter P.E Grade 8
CHESS 3rd quarter P.E Grade 8
 

Chessboard Puzzles Part 3 - Knight's Tour

  • 1. Chessboard Puzzles Part 3: Knight’s Tour Dan Freeman April 10, 2014 Villanova University MAT 9000 Graduate Math Seminar
  • 2. Introduction • In the first two presentations, we looked at the concepts of chessboard domination and independence • Tonight we will examine an entirely different idea called the knight’s tour • I will answer some questions from last time and then jump right into the new material 2
  • 3. Questions from Last Time • Could the solutions to the n-queens problem be formed into a group? • When would one use independence in a game and how would it be beneficial? • What influenced you to investigate chessboard puzzles? 3
  • 4. Questions from Last Time • I know the math must be wildly complex, but I would love to see independence / dependence with the standard 16-piece collection of pieces, i.e. is domination / independence possible with two of each piece and 8 pawns? • Is there a known number of different arrangements of knight independence? • What about rectangular chessboards? 4
  • 5. Knight Movement • Recall that knights move two squares in one direction (either horizontally or vertically) and one square in the other direction • Knights’ moves resemble an L shape • Knights are the only pieces that are allowed to jump over other pieces • In the example below, the white and black knights can move to squares with circles of the corresponding color 5
  • 6. Knight’s Tour Defined • A knight’s tour is a succession of moves made by a knight that traverses every square on a chessboard once and only once • There are two kinds of knight’s tours, a closed knight’s tour and an open knight’s tour: – A closed knight’s tour is one in which the knight’s last move in the tour places it a single move away from where it started – An open knight’s tour is one in which the knight’s last move in the tour places it on a square that is not a single move away from where it started 6
  • 7. Euler’s 8x8 Closed Knight’s Tour • Below is an example of a closed knight’s tour on an 8x8 board that Euler constructed from an incomplete open tour (only 60 squares made up the tour) 7 22 25 50 39 52 35 60 57 27 40 23 36 49 58 53 34 24 21 26 51 38 61 56 59 41 28 37 48 3 54 33 62 20 47 42 13 32 63 4 55 29 16 19 46 43 2 7 10 18 45 14 31 12 9 64 5 15 30 17 44 1 6 11 8 Closed Knight’s Tour on 8x8 Board by Euler
  • 8. Open Knight’s Tour on 5x5 Board • Because there are a different number of black squares (12) and white squares (13) on a 5x5 board, no closed knight’s tour exists on this size board • However, an open knight’s tour does exist (see two different examples below) 8 1 14 9 20 3 24 19 2 15 10 13 8 23 4 21 18 25 6 11 16 7 12 17 22 5
  • 9. Open Knight’s Tour on 8x8 Board 9
  • 10. Smallest Closed Knight’s Tours • The smallest boards for which closed knight’s tours are possible are the 5x6 and 3x10 boards 10 5 26 1 16 11 20 2 15 4 19 30 17 25 6 27 12 21 10 14 3 8 23 18 29 7 24 13 28 9 22 Closed Knight’s Tour on 5x6 Board 26 29 2 21 8 23 6 17 14 11 20 27 24 3 18 9 12 5 16 28 25 30 19 22 7 4 15 10 13 Closed Knight’s Tour on 3x10 Board 1
  • 11. Smallest Open Knight’s Tour • The smallest board for which an open knight’s tour is possible is a 3x4 board 11 1 4 7 10 12 9 2 5 3 6 11 8 Open Knight’s Tour on 3x4 Board 1
  • 12. de Moivre’s 8x8 Open Knight’s Tour • de Moivre used an effective technique for completing knight’s tours that starts on the edges of the board and works its way inward • This technique is illustrated below 12 34 49 22 11 36 39 24 1 21 10 35 50 23 12 37 40 48 33 62 57 38 25 2 13 9 20 51 54 63 60 41 26 32 47 58 61 56 53 14 3 19 8 55 52 59 64 27 42 46 31 6 17 44 29 4 15 7 18 45 30 5 16 43 28 Open Knight’s Tour on 8x8 Board by de Moivre 1
  • 13. 6x6 Closed Knight’s Tour • de Moivre’s technique can also be used to find a closed knight’s tour on a 6x6 board • This technique is illustrated on a YouTube video 13 34 7 24 15 32 1 23 14 33 36 25 16 6 35 8 17 2 31 13 22 29 26 9 18 28 5 20 11 30 3 21 12 27 4 19 10 Closed Knight’s Tour on 6x6 Board 1
  • 14. Knight’s Tour Combinatorics • The number of unique directed closed knight’s tours on an 8x8 board is 26,534,728,821,064 • There are (only) 19,724 directed closed knight’s tours on a 6x6 board • The number of directed open tours for an 8x8 board is unknown • However, the number of directed open tours are known for 1 ≤ n ≤ 7 14 n Permutations 1 1 2 0 3 0 4 0 5 1,728 6 6,637,920 7 165,575,218,320
  • 15. Pósa’s Proof that No Closed Knight’s Tour Exists on 4xn Board • As a teenager, Louis Pósa proved that a 4xn chessboard has no closed knight’s tour • He used a simple coloring proof, as follows: – First, suppose there does exist a closed knight’s tour on an arbitrary 4xn board. With the standard black and white coloring of the board, we know that a knight must alternate between black and white squares along the tour. – Now color the top and bottom rows of the board red and the two middles rows blue (see below). 15 Pósa’s Coloring on 4x7 Board
  • 16. Pósa’s Proof Continued – Note that a knight on a red square can only move to a blue square, not another red square. Thus, since there are the same number of red squares and blue squares, a knight cannot move from a blue square to another blue square, because it would not be able to make up for this by visiting two red squares consecutively. – Therefore, the knight must strictly alternate between red and blue squares. But this is impossible because, by assumption, the knight alternated between black and white squares in the traditional coloring pattern to form a tour, which would imply that the two coloring patterns are the same. Of course, they are not so we have a contradiction. Thus, no knight’s tour exists on a 4xn board. 16
  • 17. Schwenk’s Theorem • An mxn chessboard with m ≤ n has a closed knight’s tour unless one or more of the following three conditions hold: – m and n are both odd; – m = 1, 2 or 4; or – m = 3 and n = 4, 6 or 8. • The complete proof is rather involved and uses an induction argument to show that a closed knight’s tour exists on 3xn boards for n ≥ 10, n even • In addition, the proof consists of building larger tours from 5x5, 6x6, 5x8, 6x7, 6x8, 7x8 and 8x8 boards to show that tours exist for all mxn boards not excluded by the theorem 17
  • 18. Magic Squares • A magic square is an array of numbers in which the sum of each row, each column and the two main diagonals all equal the same value • A very old and famous 3x3 magic square appears below; each row, column and main diagonal sums to 15 18 4 9 2 3 5 7 8 1 6 3x3 Magic Square
  • 19. Magic Squares from a Knight’s Move • Muhammad ibn Muhammad used a knight’s move to construct magic squares • Starting in the upper-right hand corner, he would make knight moves going down and to the left, wrapping around the board when necessary • If he ran into a square that was already visited, he would move two squares to the left 19 Muhammad ibn Muhammad’s 5x5 Magic Square 13 25 7 19 1 17 4 11 23 10 21 8 20 2 14 5 12 24 6 18 9 16 3 15 22
  • 20. Magic Squares from a Knight’s Tour • Completely ignorant of Muhammad ibn Muhammad’s work, Balof and Watkins constructed magic squares using knight’s tours • The only difference between the two methods was that Balof and Watkins used a knight’s move when the knight was blocked 20 Balof and Watkin’s 7x7 Magic Square 1 24 47 21 37 11 34 12 35 2 25 48 15 38 16 39 13 29 3 26 49 27 43 17 40 14 30 4 31 5 28 44 18 41 8 42 9 32 6 22 45 19 46 20 36 10 33 7 23
  • 21. Magic Squares from a Knight’s Tour • Balof and Watkins’ method works in general to produce an nxn magic square as long as n is not divisible by 2, 3 or 5 • If n is not divisible by 2 or 3 but is divisible by 5, then only the sums for the two main diagonals are not the same 21
  • 22. Latin Squares and Knight’s Tours • An nxn Latin square is a square with n distinct labels, which can be numbers, letters, colors, etc., that appear inside each cell, with each label appearing in each row and each column once and only once • A very intriguing website showcases an odd relationship between Latin squares and knight’s tours 22
  • 23. Sources Cited • J.J. Watkins. Across the Board: The Mathematics of Chessboard Problems. Princeton, New Jersey: Princeton University Press, 2004. • “Knight’s Tour.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Knight%27s_tour • "Learn How to Perform the Knight's Tour.“ YouTube. https://www.youtube.com/watch?v=Ma1C6wcR0Jg • “A001230 – OEIS.” http://oeis.org/A001230 • “A165134 – OEIS.” http://oeis.org/A165134 • “The Knight’s Tour.” Borders Chess Club. http://www.borderschess.org/KnightTour.htm 23