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?

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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

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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

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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)

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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

1

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

1

Open Knight’s Tour on 8x8 Board

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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

26 29 2 21 8 23 6 17 14 11

1 20 27 24 3 18 9 12 5 16

28 25 30 19 22 7 4 15 10 13

Closed Knight’s Tour

on 5x6 Board

Closed Knight’s Tour

on 3x10 Board1

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

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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

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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).

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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.

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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 theorem17

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

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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

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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

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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

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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

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