Chessboard Puzzles Part 1 - Domination

Chessboard Puzzles: Domination

Part 1 of a 4-part Series of Papers on the Mathematics of the Chessboard

by Dan Freeman

March 24, 2014

Dan Freeman Chessboard Puzzles: DominationMAT 9000 Graduate Math Seminar

Table of Contents

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

Motivation........................................................................................................................................4

Overview of Chess...........................................................................................................................4

Definition of Domination................................................................................................................6

Rooks Domination...........................................................................................................................6

Bishops Domination........................................................................................................................8

Kings Domination..........................................................................................................................11

Knights Domination.......................................................................................................................14

Queens Domination.......................................................................................................................18

The Spencer-Cockayne Construction.........................................................................................20

Upper and Lower Bounds for γ(Qnxn).........................................................................................24

Queens Diagonal Domination....................................................................................................27

Conclusion.....................................................................................................................................29

Sources Cited.................................................................................................................................31

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Table of Figures

Image 1: Chess Piece Symbols........................................................................................................5Image 2: Starting Chessboard Arrangement....................................................................................5Image 3: Rook Movement...............................................................................................................7Image 4: Uncovered Square on 8x8 Board with 7 Rooks...............................................................7Image 5: Bishop Movement.............................................................................................................8Image 6: Chessboard Rotated 45°....................................................................................................9Image 7: Bishops Domination on 8x8 Board................................................................................10Image 8: 5x5 White Square Inside 9x9 Board...............................................................................10Image 9: 4x4 Black Square Inside 9x9 Board...............................................................................11Image 10: King Movement............................................................................................................11Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards..........................................................12Image 12: Each King Can Over Only One of the Orange Squares...............................................13Image 13: Knight Movement.........................................................................................................15Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards.......................................17Image 15: Knights Domination on 11x11 Board...........................................................................18Image 16: Queen Movement.........................................................................................................19Image 17: Five Queens Dominating an 8x8 Board......................................................................19Image 18: Queen in Center of 5x5 Board......................................................................................20Image 19: Five Queens Inside a 5x5 Square Dominating a 9x9 Board.........................................21Image 20: Five Queens on 11x11 Board.......................................................................................22Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 Board..............................23Image 22: Upper Bound of 8 Queens Covering 11x11 Board.......................................................25Image 23: In Diagonal Domination, a Queen Lies Halfway Between Two Empty Columns.......29

Table 1: Domination Number Notation...........................................................................................6Table 2: Knights Domination Numbers for 1 ≤ n ≤ 20..................................................................15Table 3: Queens Domination Numbers for 1 ≤ n ≤ 25..................................................................26Table 4: Domination Number Formulas by Piece.........................................................................29

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Motivation

In the past few years, I have become quite interested in the game of chess and have begun

to play it fairly regularly. Though I am by no means an expert in chess nor can I even be

considered a good player, I have noticed the undeniable relationship between the game and

several branches of mathematics, most notably number theory, one of my favorite areas of the

discipline. As a lifelong student of mathematics, in this and my subsequent three papers in this

series, I wish to survey most of the well-known problems and concepts associated with the

mathematics of the chessboard. The fact that chess is not only a fun game to play but also a

game with a long and rich history makes it that much more enjoyable to study the math behind it.

Overview of Chess

Chess is a classic board game that has been played for at least 1,200 years. Historical

evidence indicates that chess was being played back in A.D. 800, though a few earlier references

suggest that the game existed in India circa A.D. 600. Chess may have been played earlier than

that, but this is unclear because the ubiquitous 8x8, 64-square board on which it is played is used

for numerous other games as well [2, p. 6].

Chess is a 2-player turn-based game played on the aforementioned 8x8 board. The game

includes six different types of pieces: pawn, knight, bishop, rook, queen and king (see Image 1

for symbols representing each piece). To distinguish the pieces of the two players, one player’s

pieces are lighter in color than the other player’s; the former player is called “white” while the

latter player is called “black.” A game of chess always begins with the white player moving

first. Each player begins with eight pawns, two knights, two bishops, two rooks, one queen and

one king in the arrangement depicted on the board in Image 2 [5].

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Image 2: Starting Chessboard Arrangement

While the objective of the game won’t directly tie into this paper, for the reader who may

be less familiar with chess, it is worth pointing out how a game of chess is won and lost. A

player wins by putting his or her opponent’s king in a position such that it cannot escape attack

from the winning player’s pieces. This position is known as checkmate. A game does not have

to end this way; it can also end in a draw or a stalemate, the details of which are outside the

scope of this paper.

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Pawn

Knight

Bishop

Rook

Queen

King

Image 1: Chess Piece Symbols


Definition of Domination

A dominating set of chess pieces is one such that every square on an mxn1 chessboard is

either occupied by a piece in the set or under attack by a piece in the set. The domination

number for a certain piece and certain size chessboard is the minimum number of such pieces

required to “dominate” the board. The term “cover” is frequently used as a synonym for

“dominate” in the study of chessboard domination [1, pp. 95-97]. Domination numbers are

denoted by γ(Pmxn) where P represents the type of chess piece, as denoted in Table 1.

Table 1 : Domination Number Notation

Piece Abbreviation

Knight NBishop BRook RQueen QKing K

Rooks Domination

Before exploring domination among rooks, we first need to establish how rooks move on

the chessboard. Rooks are permitted to move any number of squares either horizontally or

vertically, as long as they do not take the place of a friendly piece or pass through any piece

(own or opponent’s) currently on the board. As with any piece, rooks are allowed to move to a

square occupied by an enemy piece, thereby removing the enemy piece from the board (such a

move is known as a capture). In Image 3, the white rook can move to any of the squares with a

white circle and the black rook can move to any of the squares with a black circle [5].

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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Image 3: Rook Movement

Domination among rooks is the simplest of all chess pieces. For a square nxn

chessboard, the rooks domination number is simply n [1, p. 99]. Moreover, for a general

rectangular mxn chessboard, γ(Rmxn) = min(m, n).

In 1964, two Russian brothers Akiva and Isaak Yaglom proved that γ(Rnxn) = n, as

follows. First, suppose there are fewer than n rooks placed on an nxn board. Then there must be

at least one row and at least one column that contain no rooks. Hence, the square where this

empty row and column intersect is uncovered, that is, it is not under attack by any of the rooks

(see Image 4). Thus, γ(Rnxn) ≥ n. Second, if n rooks are placed along a single row or down a

single column, the entire board is clearly covered. That is, γ(Rnxn) ≤ n. In conclusion, since

γ(Rnxn) ≥ n and γ(Rnxn) ≤ n, it follows that γ(Rnxn) = n [1, p. 99].

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Image 4: Uncovered Square on 8x8 Board with 7 Rooks


The fact that γ(Rmxn) = min(m, n) is immediately apparent from the Yaglom brothers’

proof above. Clearly, if m < n, then one need only place m rooks down a single column on the

board to cover all of the squares in each row. Likewise, if n < m, then one need only place n

rooks along a single row to cover all of the squares in each column. In either case, the rooks

domination number is the minimum of the number of rows m and the number of columns n.

Bishops Domination

Unlike rooks, bishops move diagonally, not horizontally and vertically. Bishops are

allowed to move any number of squares in one diagonal direction as long as they do not take the

place of a friendly piece or pass through any piece (own or opponent’s) currently on the board.

In Image 5, the white bishop can move to any of the squares with a white circle and the black

bishop can move to any of the squares with a black circle [5].

Image 5: Bishop Movement

As is the case with rooks, the domination number for bishops on a square nxn chessboard

is n (though in general, γ(Bmxn) ≠ min(m, n); in fact, no formula is known for γ(Bmxn) [4, p.

13]). However, the proof that this is the case requires a little more creativity than the proof for

rooks. As with the proof for the rooks domination number, the one for bishops was published by

the Yaglom brothers in 1964. The proof starts with rotating an 8x8 chessboard 45 degrees, as

shown in Image 6 [1, p. 100].

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We are fixing n to be 8, but the following argument works for all even positive integers.

From the rotated chessboard at the right of Image 5, we see a 5x4 square formed by the dark

orange squares inside the black-bordered rectangle (for general even n, an (½*n)x(½*n + 1)

square will emerge) . Therefore, at least 4 bishops (in general, ½*n) are needed to cover all of

the dark squares (from here on, the dark orange squares will be referred to as black and the light

orange squares will be referred to as white). By symmetry, at least 4 bishops (in general, ½*n)

are needed to cover all of the white squares as well. Thus, γ(B8x8) ≥ 4 + 4 = 8 (in general, γ(Bnxn)

≥ ½*n + ½*n = n). On the other hand, if we place 8 bishops on the fourth column of an 8x8

board as in Image 7, the entire board is covered [5]. Likewise, in general, if we place n bishops

on the (½*n)th column of an nxn board, then the board is dominated. Since γ(B8x8) ≥ 8 and γ(B8x8)

≤ 8, it follows that γ(B8x8) = 8, and, similarly, for general even n, γ(Bnxn) = n [1, pp. 100-101].

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Image 6: Chessboard Rotated 45°

5x4 Black Square


Image 7: Bishops Domination on 8x8 Board

Now suppose that n is odd, that is, n is of the form 2k + 1. The board corresponding to

squares of one color (without loss of generality, suppose this color is white) will contain a (k +

1)x(k + 1) group of squares (see Image 8 for white 5x5 square inside 9x9 board); hence, at least

k + 1 bishops are needed to cover the white squares. Likewise, the board corresponding to black

squares will contain a kxk group of squares (see Image 9 for a black 4x4 square inside a 9x9

board) and hence at least k bishops are needed to cover the black squares. Thus, at least (k + 1) +

k = 2k + 1 = n bishops are needed to dominate the entire nxn board. To see that γ(Bnxn) ≤ n,

observe that if n bishops are placed down the center column (more precisely, the (k + 1)st

column), the entire board is covered. Therefore, γ(Bnxn) = n for all odd n, and since we already

showed it to be true for even n, the result is proven for all n [1, p. 101].

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Image 8: 5x5 White Square Inside 9x9 Board


Kings Domination

Kings are allowed to move exactly one square in any direction as long as they do not take

the place of a friendly piece. In Image 10, the king can move to any of the squares with a white

circle [5].

Image 10: King Movement

Domination among kings is a little bit more complicated than that of rooks and bishops,

yet is still completely determined formulaically. To arrive at a formula for the kings domination

number, it is first helpful to look at a set of kings dominating 7x7, 8x8 and 9x9 boards. In Image

11, nine kings each are covering a 7x7, 8x8 and 9x9 board. Therefore, for 7 ≤ n ≤ 9,

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Image 9: 4x4 Black Square Inside 9x9 Board


γ(Knxn) ≤ 9 [1, p. 102].

Furthermore, no matter where one places a king on any of the 7x7, 8x8 or 9x9 boards,

only one of the nine dark orange squares on each board in Image 12 will be covered. Therefore,

γ(Knxn) ≥ 9. So in fact γ(Knxn) = 9 for 7 ≤ n ≤ 9 [1, p. 102].

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Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards


Note that 9 is the square of 3, the number of rows and columns of kings needed to cover a

7x7, 8x8 and 9x9 chessboard. For n = 10, 11 and 12, an additional row and column of kings is

needed to cover the board, so γ(Knxn) = 42 = 16 [1, p. 103]. Observe that this makes kings

domination very inefficient, as an additional 7 kings are required to cover the board when n

increases by just one from 9 to 10. In fact, it becomes increasingly inefficient as n increases,

since 9 more kings are needed to dominate a 13x13 board than what is required for a 12x12

board (25 total kings as compared to 16), 11 more kings are needed for n = 16 than for n = 15 (36

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Image 12: Each King Can Over Only One of the Orange Squares


as compared to 25), and so on. Thus, the kings domination function, γ(Knxn), certainly behaves in

a non-linear fashion, unlike the rooks and bishops domination functions, which are simply equal

to n.

We can now see that γ(Knxn) is constant for each number n within a certain triplet of

successive of positive integers (1, 2 and 3; 4, 5 and 6; 7, 8 and 9, etc.). The kings domination

number only jumps every 3 values of n. Thus, the formula for γ(Knxn) can be expressed as three

separate equations as below (where k is a non-negative integer):

k2 = (n / 3)2 if n = 3k

γ(Knxn) = (k + 1)2 = ((n + 2) / 3)2if n = 3k + 1

(k + 1)2 = ((n + 1) / 3)2if n = 3k + 2

The above formula can be compressed into a single equation making use of the handy

greatest integer or floor function, as follows: γ(Knxn) = └(n + 2) / 3┘2 [1, p. 103]. For rectangular

chessboards, this formula can be generalized to γ(Kmxn) = └(m + 2) / 3┘*└(n + 2) / 3┘, since the

kings domination number is directly related to the number of rows and columns of kings on the

board.

Knights Domination

Knights are allowed to move two squares in one direction (either horizontally or

vertically) and one square in the other direction as long as they don’t take the place of a friendly

piece. The full move resembles the letter L. Knights are unique in that they are the only pieces

allowed to jump over other pieces (both friendly and enemy). In Image 13, the white and black

knights can move to squares with circles of the corresponding color [5].

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Image 13: Knight Movement

No explicit formula is known for the knights domination number. However, several

values of γ(Nnxn) have been verified; the first 20 knights domination numbers appear in Table 2

[6]. As can be seen from the table, as n increases, γ(Nnxn) increases in no discernible pattern.

Table 2: Knights Domination Numbers for 1 ≤ n ≤ 20

n γ ( N nxn)

1 12 43 44 45 56 87 108 129 1410 1611 2112 2413 2814 3215 3616 4017 4618 5219 5720 62

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The differences in domination numbers between successive values of n are not

monotonically increasing. For example, γ(N6x6) – γ(N5x5) = 8 – 5 = 3, while γ(N7x7) – γ(N6x6) = 10

– 8 = 2. In other words, the difference between the 6th and 5th knights domination numbers is 3

while the difference between the 7th and 6th knights domination numbers is only 2. In addition,

γ(N18x18) – γ(N17x17) = 52 – 46 = 6, while γ(N19x19) – γ(N18x18) = 57 – 52 = 5. This lack of

monotonicity makes it difficult to tell how quickly γ(Nnxn) grows as n becomes larger and larger.

In Image 14, a minimum number of dominating knights are placed on 4x4, 5x5, 6x6, 7x7

and 8x8 boards [1, p. 97]. Note that there appears to be much symmetry with respect to the

placement of these knights on each board. The four knights on the 4x4 board are placed in a

square in the center. The five knights on the 5x5 board are arranged in a plus sign sort of shape.

On the 6x6 board, four knights are arranged in a square in the center just like with the 4x4 board,

with an additional four knights occupying the corners. On the 7x7 board, two groups of five

knights are placed in horizontal lines on the rows just above and below the middle row. Lastly,

on the 8x8 board, four groups of three knights are arranged in right-angle patterns at symmetric

locations on the board.

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However, Image 15 shows that the symmetry displayed among the knights on the boards

in Image 14 fails to hold in the 11x11 case. The 21 knights that dominate this board exhibit no

observable symmetry or pattern whatsoever. This breakdown in symmetry for larger values of n

gives a visual explanation of why domination among knights is not all that well understood. In

1971, Bernard Lemaire devised the arrangement of 21 knights in Image 15, and Alice McRae

showed that 21 was the minimum number of knights needed to cover the 11x11 board, that is,

γ(N11x11) = 21. Further developments were made in 1987 when Eleanor Hare and Stephen

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Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards


Hedetniemi developed a linear-time algorithm for computing knights domination numbers on

rectangular mxn chessboards [1, p. 98].

Queens Domination

Queens are the most powerful chess piece and move horizontally, vertically and

diagonally. Similar to rooks and bishops, they are allowed to move any number of squares in

one direction as long as they do not take the place of a friendly piece or pass through any piece

(own or opponent’s) currently on the board. In Image 16, the queen can move to any of the

squares with a black circle [5].

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Image 15: Knights Domination on 11x11 Board


Image 16: Queen Movement

Domination among queens is the most complicated and interesting of all chess pieces, as

well as the least understood. As with knights, no formula is known for the queens domination

number. Simply analyzing the standard 8x8 chessboard, one can already start to see the

complexities associated with domination among queens. Yaglom and Yaglom proved that are a

whopping 4,860 different ways to cover an 8x8 board with five queens [1, p. 113]. One such

arrangement is shown in Image 17.

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Image 17: Five Queens Dominating an 8x8 Board


The Spencer-Cockayne Construction

Additional evidence of the convoluted nature of queens domination is due to Spencer-

Cockayne in 1990 [1, pp. 116-117]. Starting with a 5x5 board and placing a queen in the center

square, we see that this queen clearly covers the 3x3 square in the center of the board, but leaves

open eight squares symmetrically placed along the four edges of the 5x5 board (for what it’s

worth, these eight squares all happen to be a knight’s move away from the queen). This 5x5

board is displayed in Image 18 with the eight uncovered squares colored in orange.

Now if we place four queens symmetrically spaced apart on previously uncovered

squares on the 5x5 board, the five queens in total dominate a 9x9 board! See Image 19 for this

construction.

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Image 18: Queen in Center of 5x5 Board


Now consider an 11x11 board that surrounds this 9x9 board. The same five queens as

before now control all of the squares on the 11x11 board, except for eight symmetrically located

squares, as was the case with the lone queen on the 5x5 board [1, pp. 117-118]. These eight

uncovered squares are highlighted in orange in Image 20.

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Image 19: Five Queens Inside a 5x5 Square Dominating a 9x9 Board


By placing four additional queens in a symmetric fashion on squares that were previously

uncovered on the 11x11 board, the nine queens in total on the board now completely control a

15x15 board (see Image 21)!

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Image 20: Five Queens on 11x11 Board


A natural question to ask at this point is whether the pattern associated with this

construction continues ad infinitum. That is, will 13 queens cover a 21x21 board, 17 queens

cover a 27x27 board, 21 queens cover a 33x33 board, etc.? Unfortunately, the answer is no.

While 13 queens do control a 21x21 board, 17 queens only dominate a 25x25 board, not a 27x27

board [1, p. 136]. This is an instance of why queens domination is so difficult. In addition, as of

today, we still do not know whether 9 is the minimum number of queens needed to cover a

15x15 board (see Table 3 for possible domination numbers) [1, p. 119]. Furthermore, only 11

queens are required to cover a 21x21 board, not 13 [1, p. 132]. Thus, in essence, the Spencer-

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Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 Board


Cockayne construction provides us little information about what values γ(Qnxn) might be for

arbitrary values of n.

Upper and Lower Bounds for γ ( Q nxn)

While there is still much to be discovered about queens domination, both upper and lower

bounds have been established for γ(Qnxn). L. Welch showed that for n = 3m + r, 0 ≤ r ≤ 3, γ(Qnxn)

≤ 2m + r [3, p. 3]. He went about showing this by dividing a 3nx3n chessboard into nine nxn

blocks. He then placed n queens in the upper right-hand block and n queens in the lower right-

hand block such that the entire 3nx3n board was covered. Thus, it takes at most 2n queens to

cover a 3nx3n board. If n is not divisible by 3, then one can simply perform the same block

construction on only k rows and columns where k is the greatest multiple of 3 less than or equal

to n. Then one could take care of the remaining one or two rows and columns by placing one or

two queens, respectively, such that the remaining squares are covered. Therefore, we have just

proved exactly what Welch’s result suggests, that is, one needs at most (2/3)*k + n mod 3 queens

to cover a 3nx3n board. For example, for n = 11, the greatest multiple of 3 less than or equal to n

is 9, so k = 9. Also, 11 mod 3 ≡ 2. So γ(Qnxn) ≤ (2/3)*k + n mod 3 = (2/3)*9 + 11 mod 3 = 6 + 2

= 8 [1, p. 119]. This concept is illustrated in Image 22.

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Spencer proved the following remarkably simple lower bound: γ(Qnxn) ≥ ½*(n – 1) [1, p.

121]. Weakley expanded on this lower bound by showing that if γ(Qnxn) = ½*(n – 1), then n ≡ 3

mod 4 [1, p. 124]. Both proofs are fairly involved so I will omit them. A couple of corollaries

that emerge from Spencer’s and Weakley’s theorems are as follows:

1. γ(Q7x7) = 4 [1, p. 128]

2. For n = 4k + 1, γ(Qnxn) ≥ ½*(n + 1) = 2k + 1 [1, p. 129]

I will omit the proofs of these corollaries but show how Spencer’s lower bound and the lower

bound from Corollary 2 can be used to narrow down the possibilities for γ(Qnxn), if not outright

determine γ(Qnxn).

For n = 9, by Corollary 2, γ(Q9x9) = ½*(9 + 1) = 5. Since there exists an arrangement of

five queens that dominate a 9x9 chessboard2, we conclude that γ(Q9x9) = 5. For n = 10, Corollary 2 In this and the examples that follow, none of the arrangements of queens dominating a chessboard will be shown. The fact that such dominating arrangements exist is to be taken as given.

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Image 22: Upper Bound of 8 Queens Covering 11x11 Board


2 can’t be used so we are forced to use Spencer’s lower bound. So γ(Q10x10) ≥ ½*(10 – 1) = 4.5.

Since 4.5 is not an integer, we can simply take the least integer greater than or equal to 4.5 (the

ceiling), which is 5, as the lower bound. Five queens can be arranged so as to dominate a 10x10

board. Therefore, γ(Q10x10) = 5. For n = 11, Spencer’s lower bound is also 5 and there exists an

arrangement of 5 queens that dominate an 11x11 board. Therefore, γ(Q11x11) = 5 as well. For n =

12, Spencer’s lower bound gives ½*(12 – 1) = 5.5, which rounds up to 6. One can arrange six

queens so as to cover a 12x12 board, so γ(Q12x12) = 6. For n = 13, we can use Corollary 2 since

13 ≡ 1 mod 4. Therefore, γ(Q13x13) ≥ ½*(13 + 1) = 7. In 1994, Burger, Mynhardt and Cockayne

produced a covering of a 13x13 board with 7 queens. Thus, γ(Q13x13) = 7 [1, p. 129-130].

The first value of n for which γ(Qnxn) is not known is 14. Spencer’s lower bound tells us

that γ(Q14x14) ≥ 7. However, no one has been able to devise a placement of seven queens that

dominate a 14x14 board; the best known arrangements of seven queens leave just two squares

uncovered. An eighth queen can be placed so as to cover those two squares. Therefore, γ(Q14x14)

is either 7 or 8 [1, p. 130].

The same sort of reasoning shown above for n = 9 through 14 can be used to deduce

possible values of γ(Qnxn) for larger values of n. Possible γ values for 1 ≤ n ≤ 25 are shown in

Table 3 [1, pp. 124, 128-132].

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Table 3: Queens Domination Numbers for 1 ≤ n ≤ 25

n γ ( Q nxn)

1 12 13 14 25 36 37 48 59 510 511 512 613 714 7 or 815 7, 8 or 916 8 or 917 918 919 9 or 1020 10 or 1121 1122 11 or 1223 11, 12 or 1324 12 or 1325 13

Queens Diagonal Domination

Before concluding my paper, I would like to touch on one last idea related to queens

domination. While queens domination itself has several complications, queens diagonal

domination is much easier to solve. The queens diagonal domination number, denoted

diag(Qnxn), is defined to be the minimum number of queens all placed along the main diagonal

such that the nxn board is dominated. Obviously, diag(Qnxn) ≥ γ(Qnxn) for all n because of the

limitation that queens must be placed on the main diagonal with diagonal domination as opposed

to just anywhere any the board with regular domination [1, pp. 114-115].

Unlike with γ(Qnxn), a formula for diag(Qnxn) is known. The formula is diag(Qnxn) = n –

max(|mid-point free, all even or all odd, subset of {1, 2, 3, …, n}|). Before we can proceed with

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proving this formula, a couple of things need to be defined. First, the vertical bars | in the

definition denote the number of elements in the set in question. Second, a mid-point free set is a

set in which for any given pair of elements in the set, the midpoint or average of those two

numbers is not in the set [1, pp. 115-116].

To begin the proof of the formula for the queens diagonal domination number, suppose

that there are a minimum number of queens, all placed along the main diagonal that dominate an

nxn board. Without loss of generality, suppose the squares along the diagonal are white in color.

Now let C be the set of all columns that do not contain any queens. For any two columns i and j

in C, the corresponding square in the ith column and in the jth row (call it the (i, j) square for

short) is not under attack by any queen vertically or horizontally (since there are no queens in

columns i and j). Therefore, (i, j) must be under attack by a queen diagonally and hence the

square must be white. Thus, i + j is even, which implies that both i and j are odd or they are both

even. Since i and j are arbitrary, all numbers in C must be even or all of them must be odd.

Additionally, since the queen attacking the (i, j) square is along the diagonal, it must be on some

square of the form (k, k). Also, i + j = k + k, which means that k = ½*(i + j). In other words, the

queen in column k is exactly halfway between the unoccupied columns i and j. Therefore, given

any two unoccupied columns, the column midway between the two must be occupied, which

implies that C – the set of all unoccupied columns – is a mid-point free set. Consequently, in

order to minimize the number of queens placed along the diagonal needed to dominate a

chessboard, one must maximize the set of empty columns such that the columns are all of the

same parity and the set is mid-point free. Hence, the size of this maximum mid-point free set is

subtracted from n, which gives us the desired formula: diag(Qnxn) = n – max(|mid-point free, all

even or all odd, subset of {1, 2, 3, …, n}|) [1, pp. 115-116]. Image 23 illustrates the argument

laid out in this proof for an 11x11 chessboard.

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Image 23: In Diagonal Domination, a Queen Lies

Halfway Between Two Empty Columns

If we let n = 11, we see that diag(Qnxn) can be different from γ(Qnxn). Let C = {2, 4, 8,

10}, which one can check to see that it is mid-point free. Therefore, if we place queens in the

columns not contained in C, that is, columns 1, 3, 5, 6, 7, 9 and 11, we have an arrangement of 7

queens on the diagonal that cover the board. Therefore, diag(Qnxn) = 7. However, as we already

observed earlier, γ(Q11x11) = 5 [1, p. 116].

Conclusion

Chessboard domination remains an unsolved problem in recreational mathematics today.

While domination among rooks, bishops and kings on square nxn chessboards has more or less

been completely characterized, knights and queens domination is still largely an enigma. These

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facts are summarized in Table 4, which gives a compact view of what is known and unknown

about the domination numbers for the five chess pieces analyzed in this paper.

Table 4: Domination Number Formulas by Piece

Piece (P) γ ( P nxn) (Square) γ ( P mxn) (Rectangular)

Rook n min(n, m)Bishop n Unknown

King └(n + 2) / 3┘2 └(m + 2) / 3┘*└(n + 2) /

3┘Knight Unknown Unknown

QueenUnknown, though upper and lower bounds exist

Unknown

It is no doubt that the irregular L-shaped movement of knights and the versatility of

queens with their vertical, horizontal and diagonal movement has caused domination among

these pieces to be difficult to analyze. I am not confident that a formula will be discovered in the

near future for the domination numbers for either of these two pieces. However, I believe that

the mathematical community is closer to solving the queens domination problem than the knights

domination problem by virtue of the fact that several upper and lower bounds have already been

established for the former. Computer analysis of large chessboards will certainly be key to

uncovering new information and patterns. In addition, I believe that further analysis of

rectangular boards may prove helpful in understanding how the domination functions γ(Pmxn)

behave in a broader sense (note that γ(Bmxn) is unknown so there is still considerable work to do

here).

In my next paper in this series, I will examine the notion of chessboard independence.

Exploring this idea and making the link between it and domination will give us a greater

understanding and appreciation of the mathematical dynamics at play with chess pieces and the

chessboard.

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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] J. Nunn. Learn Chess. London, England: Gambit Publications, 2000.

[3] E.J. Cockayne. Chessboard Domination Problems. Discrete Math, Volume 86, 1990.

[4] J. DeMaio, W.P. Faust. Domination on the mxn Toroidal Chessboard by Rooks and Bishops.

Department of Mathematics and Statistics, Kennesaw State University.

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

[6] “A006075 – OEIS.” http://oeis.org/A006075

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http://oeis.org/A006075

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

Chessboard Puzzles Part 1 - Domination

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