This document summarizes research on knight's tours on chessboards. It defines closed and open knight's tours, provides examples of tours on boards of different sizes, and discusses some related topics like magic squares, Latin squares, and the mathematical proofs investigating the existence of tours on certain board sizes. It also cites several sources for further information on knight's tour puzzles and combinatorics.

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
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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?
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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?
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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
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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
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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)
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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

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
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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
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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
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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
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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
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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).
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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.
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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
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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
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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
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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

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
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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

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
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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
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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
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