chessboard puzzles part 3 - knight's tour

Chessboard Puzzles: Knight’s Tour

Part 3 of a 4-part Series of Papers on the Mathematics of the Chessboard

by Dan Freeman

May 13, 2014

Dan Freeman Chessboard Puzzles: Knight’s TourMAT 9000 Graduate Math Seminar

Table of Contents

Table of Figures...............................................................................................................................3

Introduction......................................................................................................................................4

Definition of Knight’s Tour.............................................................................................................4

Closed Knight’s Tours.....................................................................................................................5

Open Knight’s Tours.......................................................................................................................7

Schwenk’s Theorem........................................................................................................................8

Proof of the Knights Independence Number Formula...................................................................10

Knight’s Tour Combinatorics........................................................................................................12

Magic Square Construction from Knight’s Tours.........................................................................13

Knight’s Tour Latin Squares.........................................................................................................19

Conclusion.....................................................................................................................................24

Sources Cited.................................................................................................................................26

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Table of Figures

Image 1: Knight Movement.............................................................................................................5Image 2: Euler’s Closed Knight’s Tour of 8x8 Board.....................................................................6Image 3: Closed Knight's Tour on 5x6 Board.................................................................................6Image 4: Closed Knight's Tour on 3x10 Board...............................................................................6Image 5: Two Distinct Open Knight's Tours on 5x5 Board............................................................7Image 6: Open Knight’s Tour on 3x4 Board...................................................................................7Image 7: de Moivre's Open Knight's Tour on 8x8 Board................................................................8Image 8: Pósa’s Coloring on 4x7 Board..........................................................................................9Image 9: Block Construction of (4k + 3)x(4k + 3) Chessboard.....................................................12Image 10: Lo-shu Magic Square....................................................................................................13Image 11: Muhammad ibn Muhammad’s Construction of Lo-shu Magic Square........................14Image 12: Muhammad ibn Muhammad’s 5x5 Magic Square Using Diagonal Move...................15Image 13: Muhammad ibn Muhammad’s 5x5 Magic Square Using Knight’s Move...................15Image 14: Balof and Watkins’s 7x7 Magic Square Using Knight’s Tour.....................................16Image 15: Euler’s 8x8 Semi-magic Open Knight’s Tour..............................................................17Image 16: Jaenisch’s 8x8 Semi-magic Closed Knight’s Tour.......................................................18Image 17: Wenzelides’ 8x8 Semi-magic Closed Knight’s Tour...................................................18Image 18: Kuma’s 12x12 Magic Closed Knight’s Tour...............................................................19Image 19: Four Mini Knight’s Tours Used to Construct Knight’s Tour Latin Square.................20Image 20: Thomasson’s Knight’s Tour Latin Square....................................................................21Image 21: Each Horizontal Pair of Numbers in the Knight’s Tour Latin Square Sums to 9........21Image 22: Knight’s Tour Latin Square Divided into 1x4 Blocks..................................................22

Table 1: Number of Permutations of Open Knight’s Tours for 1 ≤ n ≤ 8....................................13Table 2: 1x4 Number Blocks on Left Side of Knight’s Tour Latin Square..................................23Table 3: 1x4 Number Blocks on Right Side of Knight’s Tour Latin Square................................23

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Introduction

This paper analyzes a classic puzzle in recreational mathematics known as the knight’s

tour. This idea is quite different from the concepts of domination and independence that we

analyzed in the first two papers in this series. For one, the knight’s tour problem is more of an

existence problem than an optimization problem in that the main goal of the puzzle is to

determine whether or not a rectangular chessboard of a given size has at least one knight’s tour.

An extension to this problem is the counting of the number of permutations of knight’s tours on a

given size chessboard, a fascinating problem in it and of itself. In addition, there are other

numerical structures in mathematics such as the magic square and Latin square that have

interesting relationships – to say the least – with knight’s tours and these associations are an

active area of research.

This paper starts off by defining two different types of knight’s tours and then offers

several examples of each type of tour. It then proceeds to provide a solution to the knight’s tour

problem and takes a glimpse into the fascinating combinatorics associated with knight’s tours.

The back end of the paper focuses on how knight’s tours can be used to construct magic squares

and then analyzes some unexpected properties that result from using such tours to build a Latin

square.

Definition of Knight’s Tour

Recall that knights move two squares in one direction (either horizontally or vertically)

and one square in the other direction, thus making the move resemble an L shape. Knights are

the only pieces that are allowed to jump over other pieces. In Image 1, the white and black

knights can move to squares with circles of the corresponding color [3].

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Image 1: Knight Movement

A knight’s tour is a succession of moves made by a knight that traverses every square on

a mxn1 chessboard once and only once [1, p. 5]. There are two kinds of knight’s tours, a closed

knight’s tour and an open knight’s tour, defined as follows:

• 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 [1, p. 6].

• 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 [1, p. 6].

The following two sections will examine several examples of closed and open knight’s

tours and a well-known heuristic for constructing knight’s tours.

Closed Knight’s Tours

Image 2 is an example of a closed knight’s tour on an 8x8 board that Euler carefully

constructed from an incomplete open tour (only 60 squares made up the original tour) [1, p. 32].

For this and all subsequent knight’s tours in this paper, the knight begins its tour at the square

labeled with the number 1 (indicated by a knight image), then moves to the square with the

number 2, then the square with the number 3, and so on, until it reaches the mnth square.

1 Throughout this paper, m and n refer to arbitrary positive integers denoting the number of rows and columns of a chessboard, respectively.

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The smallest boards in terms of number of squares for which closed knight’s tours are

possible are 5x6 and 3x10 boards (both have 30 squares) [1, p. 6]. Examples of these tours are

shown in Images 3 and 4.

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Image 2: Euler’s Closed Knight’s

Tour of 8x8 Board

1

1

Image 3: Closed Knight's Tour

on 5x6 Board

Image 4: Closed Knight's Tour on 3x10 Board

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Open Knight’s Tours

As one might expect, for a given size chessboard, an open knight’s tour may exist while a

closed tour may not exist. After all, a closed tour must end where it started while an open tour

can end anywhere on the board. For mxn chessboards in which both m and n are odd, no closed

tour exists while it is often the case that an open tour does exist. Because a knight alternates

between black and white squares in its movement and because an mxn board with both m and n

odd has a different number of black squares and white squares, it follows that no closed knight’s

tour can exist on such a board. For example, no closed knight’s tour exists on a 5x5 board

because there are 12 black squares and 13 white squares, but an open’s tour does exist. Two

examples are shown in Image 7. As you can see, the number of lighter-colored squares

outnumbers the darker-colored squares in each board, making a closed tour impossible [1, p. 8-

9].

The smallest board for which an open knight’s tour is possible is the 3x4 board [1, p. 6].

This board has just 12 squares unlike the smallest boards for which a closed knight’s tour exists,

which have 30 squares. An open tour on a 3x4 board is shown in Image 8.

Image 6: Open Knight’sTour on 3x4 Board

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Image 5: Two Distinct Open Knight'sTours on 5x5 Board


At this point, one might naturally ask how to go about constructing knight’s tours. There

are several ways to do this, but one of the most common techniques is attributed to de Moivre,

who created knight’s tours by starting on the edge of the board and working his way inward,

keeping in the same direction (either clockwise or counterclockwise) throughout the tour. He

would stay as close to the edge of the board as possible and only move inward when all other

squares had already been visited [1, p. 27]. An open knight’s tour on an 8x8 board by de Moivre

is shown in Image 9 [1, p. 28]. I have found the YouTube video at the following link,

https://www.youtube.com/watch?v=Ma1C6wcR0Jg, to be quite helpful in explaining the

mechanics of this heuristic for building knight’s tours [4].

Image 7: de Moivre's Open Knight'sTour on 8x8 Board

Schwenk’s Theorem

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, as illustrated for a 4x7 board in

Image 8 [1, p. 43].

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https://www.youtube.com/watch?v=Ma1C6wcR0Jg


Image 8: Pósa’s Coloring on 4x7 Board

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 closed knight’s tour exists on a 4xn board [1, p. 43].

In 1991, Allen Schwenk published a solution to the closed knight’s tour problem in

Mathematics Magazine. That is, he rigorously proved that a closed tour exists unless a

chessboard meets at least one of three criteria. This is known as Schwenk’s theorem and states

that an mxn chessboard with m ≤ n has a closed knight’s tour unless one or more of the following

three conditions hold:

1) m and n are both odd;

2) m = 1, 2 or 4; or

3) m = 3 and n = 4, 6 or 8 [1, pp. 44-45].

Now we have already taken care of the first scenario in Schwenk’s theorem in which m

and n are both odd in the previous section on open knight’s tours, and we have also already

addressed the case in which m = 4 in the second condition by Pósa’s coloring proof.

Furthermore, if m = 1, a knight cannot move, and if m = 2, a knight can only move horizontally,

making it impossible for it to visit every square on the board [1, p. 39]. Thus, we have now

shown that if the second condition in the theorem holds, then a closed knight’s tour cannot exist.

As one can imagine, the complete proof of Schwenk’s theorem is rather involved, as not

only does one need to exclude chessboards that meet at least one of the three conditions above

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from having a closed knight’s tour, but one must also show that all other chessboards do, in fact,

have a closed tour. Elementary ideas from graph theory can be used to show that no closed tours

exist on the 3x4, 3x6 and 3x8 boards. In order to show that a closed tour does exist on 3xn

boards where n ≥ 10, n even, one starts with closed tours for 3x10 and 3x12 boards and uses an

induction argument to build tours for larger even n. 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 one of the three conditions in the theorem [1, pp. 45-46].

As an additional note, Paul Cull and Jeffery De Curtins, computer science professors at

Oregon State University, showed that every mxn chessboard with min(m, n) ≥ 5 has an open

knight’s tour [2, p. 284]. So, in effect, the open knight’s tour problem has been resolved as well.

Proof of the Knights Independence Number Formula

At this point, we are well-equipped to prove the knights independence number formula

that we first encountered in the second paper in this series. Recall that the formula for the

knights independence number is as follows:

4 if n = 2

β(Nnxn) = ½*n2 if n ≥ 4, n even

½*(n2 + 1) if n odd [1, p. 181]

For the case n = 2, place a knight on each of the 4 squares to produce a maximum

independent set of knights. So β(N2x2) = 4. For the case n = 4, we can split the 4x4 board into

two 2x4 rectangles, each of which can contain at most 4 independent knights. This implies that

β(N4x4) = 2*4 = 8 [1, p. 181].

Ralph Greenberg showed in 1964 that the maximum number of independent knights that

one can place on an 8x8 board is 32. This is simply by virtue of the fact that knights alternate

between black and white squares when they move and the fact that there are 32 black squares

and 32 white squares on an 8x8 board. Not surprisingly, simply placing knights on all of the

black squares or on all of the white squares (that is, exactly half of all the squares on the board)

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works in general for lager even-sized boards to produce an independent set of knights. Martin

Gardner noted that such an independent set of knights is maximum if the board has a closed

knight’s tour. Similarly, in the case where n is odd one can simply place knights on whichever of

the two colors has more squares to produce an independent set of knights. Gardner likewise

pointed out that such a set is maximum if the board has an open knight’s tour. This argument

along with Schwenk’s Theorem implies that β(Nnxn) = ½*n2 for n even, n ≥ 4 [1, pp. 180-181].

We will split the odd n case into two subcases: 1) n of the form 4k + 1 and 2) n of the

form 4k + 3. John Watkins proved in his book Across the Board: The Mathematics of

Chessboard Problems that a (4k + 1)x(4k + 1) chessboard has an open knight’s tour, starting with

a 5x5 board and extending this to boards of size 9x9, 13x13 and so on [1, p. 50]. Since an open

tour exists, it then follows that a maximum number of independent knights is the number of

squares with the more frequently occurring color, that is, precisely half of one greater than the

total number of squares, or ½*(n2 + 1). In other words, β(Nnxn) = ½*(n2 + 1) [1, p. 181].

For the second odd subcase in which n is of the form 4k + 3, we will use a construction

that divides the chessboard into 2x4, 3x3, 3x4 and 4x3 blocks of squares. Each block is

organized so that pairs of squares with the same label (we will use the letters a, b, c, d, e and f)

can contain at most independent knight [1, p. 181]. This construction is illustrated in Image 11

[1, p. 182].

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Image 9: Block Construction of

(4k + 3)x(4k + 3) Chessboard

By the construction above, each 2x4 block can contain at most 4 independent knights, the

3x3 block con have at most 5 independent knights, and the 3x4 and 4x3 blocks can have at most

6 independent knights. Since there are eight 2x4 blocks, one 3x3 block, two 3x4 blocks and two

4x3 blocks, an 11x11 chessboard can have at most 4*8 + 6*2 + 6*2 + 5 = 61 independent

knights. In general, we can have at most 4*(2*k2) + 6*(2*k) + 5 = ½*((4k + 3)2 + 1) = ½*(n2 +

1). This completes the proof of the knights independence number formula [1, p. 181].

Knight’s Tour Combinatorics

Combinatorics associated with knight’s tours is a fascinating subtopic and largely

remains an unsolved problem. For starters, the number of unique directed closed knight’s tours

on an 8x8 board is a whopping 26,534,728,821,064. When the direction of the tour is not

specified, this number cuts in half to 13,267,364,410,532. For the next smaller square

chessboard on which a closed knight’s tour is possible (6x6), the number of directed closed

knight’s tours drops considerably to 19,724. For square chessboards larger than 8x8, the number

of distinct closed tours remains unknown [5].

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The number of directed open knight’s tours have been verified for 1 ≤ n ≤ 8 (see Table 1

below). Interestingly, the number of open tours on an 8x8 board (which is even larger than the

number of closed tours by 3 orders of magnitude) has just been found by Alex Chernov on May

10, 2014. As with closed tours, the number of open tours for square chessboards larger than 8x8

remains unknown [6].

Table 1: Number of Permutations of Open Knight’s

Tours for 1 ≤ n ≤ 8

n Number of Permutations

1 12 03 04 05 1,7286 6,637,9207 165,575,218,320 8 19,591,828,170,979,904

Magic Square Construction from Knight’s Tours

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. For example, a very old and famous 3x3 magic

square appears in Image 10; each row, column and main diagonal sums to 15. This magic

square is known as the Lo-shu magic square because of a legend that over 4,000 years ago a

turtle in the Yellow (Lo) River in China had this 3x3 magic square inscribed on its shell [1, pp.

54-55].

Image 10: Lo-shu Magic Square

In 1732, African mathematician Muhammad ibn Muhammad wrote a manuscript about

the construction of magic squares of odd order, that is, squares of size 3x3, 5x5, 7x7, and so on.

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He imagined that the chessboard was on a torus, that is, a surface that wraps around from the

right edge back to the left and from the bottom edge back to the top. By way of example, for the

Lo-shu magic square, Muhammad ibn Muhammad would start by placing a 1 in the bottom

middle square, then place a 2 in the square diagonally down and to the right, which is the top

right-hand corner square. Then he would place a 3 in the square diagonally down and to the

right from the square with a 2; this is the middle left square. Noting that a third consecutive

diagonally down and to the right move would land him back to where he started at 1,

Muhammad ibn Muhammad instead moves two squares straight down to land at the upper right-

hand corner square, placing a 4 here. For the next two moves, he would revert to the diagonal

movement used in the first two moves, thereby placing a 5 and 6 on the center and bottom right-

hand corner squares, respectively. Then, once again, instead of making a third straight move

diagonally down and to the right, Muhammad ibn Muhammad places a 7 on the middle right

square, two rows directly below the square with the 6. Lastly, he would finish out the magic

square construction by placing an 8 and 9 on the bottom left-hand square and the top middle

square, respectively [1, pp. 53-54]. This construction is illustrated in Image 11 [1, p. 54].

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Image 11: Muhammad ibn Muhammad’s

Construction of Lo-shu Magic Square


The same construction depicted in Image 11 can also be used to create a 5x5 magic

square, as shown in Image 12 [1, p. 62]. Each row, column and main diagonal sums to 65.

Image 12: Muhammad ibn Muhammad’s

5x5 Magic Square Using Diagonal Move

Muhammad ibn Muhammad also used a knight’s move to build magic squares. The

pattern is similar to the one described above, but instead of making diagonal moves, he would

use a knight’s move. In addition, when he would come across a square he had already visited,

instead of moving straight down two squares, he would move two squares to the left.

Muhammad ibn Muhammad constructed a 5x5 magic square by starting with a 1 in the upper

right-hand corner and then making knight’s moves, one square to the left and two squares down,

as shown in Image 13. His knight move construction actually yields a more special form of

magic square in that the sum of all of the positive and negative diagonals, not just the two main

ones, equate to the same value (65) [1, pp. 55-56]. As one can see from Image 12, this extra

condition fails with the 5x5 magic square that is constructed using Muhammad ibn Muhammad’s

diagonal move.

Image 13: Muhammad ibn Muhammad’s5x5 Magic Square Using Knight’s Move

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Completely ignorant of Muhammad ibn Muhammad’s work, in 1996, John Watkins and a

student of his, Barry Balof, constructed magic squares using not just knight’s moves, but

knight’s tours. The only difference between their method and that of Muhammad ibn

Muhammad is that Balof and Watkins used a knight’s move to avoid traveling to a square that

had already been visited, instead of moving two squares to the left. Balof and Watkins

constructed a 7x7 magic square by starting with a 1 in the upper left-hand corner (as opposed to

the upper right-hand corner that Muhammaad ibn Muhammad started with) and then making

knight’s moves, one square down and two squares to the right (as opposed to one square to the

left and two squares down as used in Muhammaad ibn Muhammad’s construction). When

blocked by a square that had already been visited, Balof and Watkins would move up two

squares and to the right one square, as is the case when moving from square 7 to square 8 in the

7x7 magic square shown in Image 14 [1, pp. 56-57].

Image 14: Balof and Watkins’s7x7 Magic Square Using Knight’s Tour

Balof and Watkins proved that their knight’s tour method of constructing magic squares

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 one can use this method to construct what is

known as a semi-magic square, in which the sums of the rows and columns equal the same

number, but the two main diagonals fail to match this value [1, pp. 56-57].

Euler produced an 8x8 semi-magic square using an open knight’s tour, as in Image 15 [1,

p. 58]. Each row and each column sum to 260, but the positive main diagonal sums to 210 and

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the negative main diagonal sums to 282. Whether or not Euler intended this, it turns that each

4x4 quadrant of this semi-magic square are themselves semi-magic squares, in which each row

and each column sum up to 130. Furthermore, incredibly enough, the four numbers that lie

within each 2x2 quadrant within the 4x4 quadrants also add up to 130 (take, for example, the 2x2

block in the upper left-hand corner of the 8x8 semi-magic square, in which the numbers 1, 48, 30

and 51 sum to 130) [1, p. 57].

Image 15: Euler’s 8x8 Semi-magicOpen Knight’s Tour

While the 8x8 semi-magic square that Euler constructed using an open knight’s tour is

definitely an impressive feat, a long-standing problem had been until recently to find an 8x8

magic square using a closed knight’s tour. Euler’s square above failed to achieve this on two

counts: 1) His was a semi-magic square, not a magic square, and 2) He used an open knight’s

tour, not a closed knight’s tour [1, p. 57]. On August 5, 2003, Guenter Stertenbrink announced

that no closed knight’s tour can be used to produce an 8x8 magic square, after a computer

program written by J.C. Meyrignac exhaustively searched all possibilities [1, p. 59]. However,

as a result of this computer analysis, 140 semi-magic closed knight’s tours were found to exist on

the 8x8 board [7]. Two examples of semi-magic squares constructed from closed knight’s tours

are shown in Image 16, by Jaenisch [1, p. 58], and Image 17, by Wenzelides [1, p. 58].

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1


Image 16: Jaenisch’s 8x8 Semi-magic

Closed Knight’s Tour

Image 17: Wenzelides’ 8x8 Semi-magic

Closed Knight’s Tour

In order for a square nxn chessboard to have a magic closed knight’s tour, n must be

divisible by 4 [1, p. 58]. We already know by Pósa’s coloring proof that a closed knight’s tour

does not exist on a 4x4 board (in fact, neither does an open tour [1, pp. 51-52]) and we have just

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1

1


seen that no magic closed tour exists on an 8x8 board. However, magic closed tours have been

shown to exist on 12x12, 16x16, 20x20, 24x24, 32x32, 48x48 and 64x64 boards [1, p. 59]!

Awani Kuma discovered the 12x12 magic closed tour shown in Image 18 in 2003; each column,

row and main diagonal add up to 870 [8]. In December 2005, Dan Thomasson found two 16x16

magic closed tours that were 180° rotation-symmetric (neither shown here); each column, row

and main diagonal sum to 2,056 [9].

Image 18: Kuma’s 12x12 Magic Closed Knight’s Tour

Knight’s Tour Latin Squares

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. In 2005, Dan Thomasson showcased an odd relationship between Latin

squares and knight’s tours in his intriguing website http://www.borderschess.org/LatinKT-

Problem.htm. Thomasson uses four “mini” knight’s tours to combine to form a Latin square. In

each of the four 16-move knight’s tours shown in Image 19, the numbers 1 through 8 mark the

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http://www.borderschess.org/LatinKT-Problem.htm

http://www.borderschess.org/LatinKT-Problem.htm


first 8 squares in the tour and then the numbers 1 through 8 are used again to indicate squares 9-

16 [9].

When the four mini knight’s tours in Image 19 above are superimposed onto one board,

the result is a Latin square because each of the numbers 1 through 8 appear in each row and each

column exactly once. This is illustrated in Image 20 [9].

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Image 19: Four Mini Knight’s Tours Used

to Construct Knight’s Tour Latin Square


Image 20: Thomasson’s Knight’s Tour

Latin Square

There are many numerical oddities associated with this knight’s tour Latin square. Each

1x2 block of numbers shaded in white or gray in Image 21 below sum to 9. It then follows that

when each horizontal pair of numbers is taken as a single number (e.g. 2 and 7 as 27 in the upper

left-hand corner block), the result is a number that is divisible by 9 [9].

Image 21: Each Horizontal Pair of Numbers in the

Knight’s Tour Latin Square Sums to 9

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Thomasson’s same knight’s tour Latin square is divided into alternating 1x4 blocks of

four numbers in Image 22. Obviously, since each 1x2 block of numbers juxtaposed as a single

4-digit number is divisible by 9, then each 1x4 block taken as a single number is also divisible by

9. If we divide each 4-digit number that appears in the 1x4 blocks on the left side of the square

by 9, and then sum each of the resulting numbers, we get 4,444. Likewise, if we do the same to

the 1x4 blocks of numbers that appear on the right side of the square, we also end up with 4,444.

These calculations are shown in Tables 5 and 6 for the left and right side of the knight’s tour

Latin square, respectively [9].

Image 22: Knight’s Tour Latin Square

Divided into 1x4 Blocks

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Table 2: 1x4 Number Blocks on Left Side of

Knight’s Tour Latin Square

1x4 Number Block1x4 Number Block

Divided by 9

2,781 3091,836 2048,163 9077,218 8024,527 5033,654 4066,345 7055,472 608

Sum of 1x4 Number Blocks Divided by 9

4,444

Table 3: 1x4 Number Blocks on Right Side of

Knight’s Tour Latin Square

1x4 Number Block1x4 Number Block

Divided by 9

4,563 5077,254 8062,745 3055,436 6046,381 7091,872 2088,127 9033,618 402

Sum of 1x4 Number Blocks Divided by 9

4,444

As can be seen from the tables above, when all the 4-digit numbers in the knight’s tour

Latin square are divided by 9, the results are 3-digit numbers with a zero in the tens’ place. Also,

each of the digits 2-9 appear exactly once in the hundreds’ place of the eight 3-digit numbers

corresponding to the left side of the board, and again each of the digits 2-9 appear exactly once

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in the units’ place of these same 3-digit numbers. The same phenomenon occurs with the 3-digit

numbers that correspond to the right side of the board [9].

Conclusion

While the knight’s tour problem has been completely settled in that we know precisely

which size chessboards have closed or open knight’s tours and which ones do not, much remains

unanswered surrounding this fascinating chessboard puzzle. Mathematicians are just beginning

to scratch the surface on knight’s tour combinatorics, as exemplified by the fact that the number

of distinct tours (both open and closed) is only known for chessboards as large as the standard

8x8 board. The coincidental fact that the number of distinct open tours on an 8x8 board has

literally just been confirmed only three days ago as of the writing of this paper exemplifies both

that progress is being made in this area as well as that there is still an immense amount of work

to be done. It may take a few more years, but heavy-duty computer analysis will eventually

reveal the number of knight’s tour permutations on 10x10, 12x12 and even larger chessboards.

In addition, magic squares, and in particular their relationship with knight’s tours,

remains an active area of research in recreational mathematics. There are many questions that

still need to be answered here. We know that in order for a magic closed knight’s tour to exist,

the size of the chessboard must be divisible by 4, but among these boards, which ones do in fact

have a magic closed tour? Are there (4k)x(4k) boards other than the 4x4 board that do not have a

magic closed tour? How about semi-magic closed tours? What criteria must a chessboard meet

in order to exhibit a semi-magic closed tour? The same questions can and should also be asked

regarding open knight’s tours. Furthermore, what is needed to construct a semi-magic tour with

each of the quadrants of the board also semi-magic, as is the case with Euler’s semi-magic open

knight’s tour in Image 15? Is it possible for a magic or semi-magic knight’s tour to have the

property in which the sum of each diagonal, not just the two main ones, all share the same value

(as is the case with Muhammad ibn Muhammad’s 5x5 magic square in Image 12)? These are

just a snippet of the questions that can and should drive further research into the subject of magic

and semi-magic knight’s tours.

Equally if not more intriguing is the connection knight’s tours have with Latin squares.

Dan Thomasson examined some unexpected numerical patterns that result when knight’s tours

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are used to build Latin squares, but certainly this is only a fraction of what can be learned here.

Further research down this road may lead to even more interesting, unexpected results that may

prove useful in answering other open chessboard puzzles or more general questions regarding

recreational mathematics. All in all, while knight’s tours have been studied by some of the

greatest mathematicians for centuries, there are enough unanswered questions and underexplored

avenues in this topic to keep mathematicians busy studying knight’s tours for centuries to come.

In my next and final paper in this series, I will analyze the three major topics in

domination, independence and knight’s tour that I examined in my first three papers on irregular

surfaces such as the torus, cylinder, Klein bottle and Möbius strip. I will conclude this fourth

and final paper by looking at ideas similar to domination and independence such as the

independent domination number, upper domination number, irredundance number, upper

irredundance number and total domination number. In examining these different variations, I

hope to demonstrate just how diverse all of the problems in chessboard mathematics truly are.

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Sources Cited

[1] J.J. Watkins. Across the Board: The Mathematics of Chessboard Problems. Princeton, New

Jersey: Princeton University Press, 2004.

[2] P. Cull, J. De Curtins. Knight’s Tour Revisited. Department of Computer Science, Oregon

State University.

[3] “Chess.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Chess

[4] "Learn How to Perform the Knight's Tour.” YouTube. https://www.youtube.com/watch?

v=Ma1C6wcR0Jg

[5] “A001230 – OEIS.” http://oeis.org/A001230

[6] “A165134 – OEIS.” http://oeis.org/A165134

[7] “MathWorld News: There Are No Magic Knight's Tours on the Chessboard.” Wolfram

MathWorld. http://mathworld.wolfram.com/news/2003-08-06/magictours/

[8] “Knight Tours.” http://www.magic-squares.net/knighttours.htm

[9] “The Knight’s Tour.” Borders Chess Club. http://www.borderschess.org/KnightTour.htm

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http://www.borderschess.org/KnightTour.htm

http://www.magic-squares.net/knighttours.htm

http://mathworld.wolfram.com/news/2003-08-06/magictours/

http://oeis.org/A165134

http://oeis.org/A001230

http://en.wikipedia.org/wiki/Chess

chessboard puzzles part 3 - knight's tour

Documents

knights tour problem

knights move15image

knights tourmat

knights tour16image

knights tourpart

moivres open knights

mini knights tours

knights tour latin square