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: Domination
MAT 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
2
The Spencer-Cockayne Construction ........................................................................................ 20
Upper and Lower Bounds for γ(Qnxn)........................................................................................ 24
Queens Diagonal Domination ................................................................................................... 27
Conclusion .................................................................................................................................... 29
Sources Cited ................................................................................................................................ 31

Table of Figures
Image 1: Chess Piece Symbols ....................................................................................................... 5
Image 2: Starting Chessboard Arrangement ................................................................................... 5
Image 3: Rook Movement............................................................................................................... 7
Image 4: Uncovered Square on 8x8 Board with 7 Rooks............................................................... 7
Image 5: Bishop Movement ............................................................................................................ 8
Image 6: Chessboard Rotated 45° ................................................................................................... 9
Image 7: Bishops Domination on 8x8 Board................................................................................ 10
Image 8: 5x5 White Square Inside 9x9 Board .............................................................................. 10
Image 9: 4x4 Black Square Inside 9x9 Board .............................................................................. 11
Image 10: King Movement ........................................................................................................... 11
Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards.......................................................... 12
Image 12: Each King Can Over Only One of the Orange Squares............................................... 13
Image 13: Knight Movement ........................................................................................................ 15
Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards....................................... 17
Image 15: Knights Domination on 11x11 Board .......................................................................... 18
Image 16: Queen Movement......................................................................................................... 19
Image 17: Five Queens Dominating an 8x8 Board...................................................................... 19
Image 18: Queen in Center of 5x5 Board ..................................................................................... 20
Image 19: Five Queens Inside a 5x5 Square Dominating a 9x9 Board ........................................ 21
Image 20: Five Queens on 11x11 Board....................................................................................... 22
Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 Board ............................. 23
Image 22: Upper Bound of 8 Queens Covering 11x11 Board ...................................................... 25
Image 23: In Diagonal Domination, a Queen Lies Halfway Between Two Empty Columns ...... 29
Table 1: Domination Number Notation .......................................................................................... 6
Table 2: Knights Domination N umbers for 1 ≤ n ≤ 20 ................................................................. 15
Table 3: Queens Domination N umbers for 1 ≤ n ≤ 25.................................................................. 27
Table 4: Domination Number Formulas by Piece ........................................................................ 30
3

Motivation
4
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].

5
Image 1: Chess Piece Symbols
King
Queen
Rook
Bishop
Knight
Pawn
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.

Definition of Domination
6
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 N
Bishop B
Rook R
Queen Q
King 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.

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

8
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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Image 6: Chessboard Rotated 45°
5x4 Black Square
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].

10
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].
Image 8 : 5x5 White Square Inside 9x9 Board

Kings Domination
11
Image 9: 4x4 Black Square Inside 9x9 Board
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,

γ(Knxn) ≤ 9 [1, p. 102].
12
Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards
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].

13
Image 12: Each King Can Over Only One of the Orange Squares
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

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.
14
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)2 if n = 3k + 1
(k + 1)2 = ((n + 1) / 3)2 if 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].

15
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(Nnxn)
1 1
2 4
3 4
4 4
5 5
6 8
7 10
8 12
9 14
10 16
11 21
12 24
13 28
14 32
15 36
16 40
17 46
18 52
19 57
20 62

16
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.

17
Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards
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

Hedetniemi developed a linear-time algorithm for computing knights domination numbers on
rectangular mxn chessboards [1, p. 98].
Image 15: Knights Domination on 11x11 Board
Queens Domination
18
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].

19
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.
Image 17: Five Queens Dominating
an 8x8 Board

The Spencer-Cockayne Construction
20
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.
Image 18: Queen in
Center of 5x5 Board
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.

21
Image 19: Five Queens Inside a 5x5 Square
Dominating a 9x9 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.

22
Image 20: Five Queens on 11x11 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)!

23
Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 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-

Cockayne construction provides us little information about what values γ(Qnxn) might be for
arbitrary values of n.
Upper and Lower Bounds for γ(Qnxn)
24
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.

25
Image 22: Upper Bound of 8 Queens
Covering 11x11 Board
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).

26
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 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].
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.

27
Table 3: Queens Domination Numbers for 1 ≤ n ≤ 25
n γ(Qnxn)
1 1
2 1
3 1
4 2
5 3
6 3
7 4
8 5
9 5
10 5
11 5
12 6
13 7
14 7 or 8
15 7, 8 or 9
16 8 or 9
17 9
18 9
19 9 or 10
20 10 or 11
21 11
22 11 or 12
23 11, 12 or 13
24 12 or 13
25 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

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].
28
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.

29
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

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.
30
Table 4: Domination Number Formulas by Piece
Piece (P) γ(Pnxn) (Square) γ(Pmxn) (Rectangular)
Rook n min(n, m)
Bishop n Unknown
King └(n + 2) / 3┘
2 └(m + 2) / 3┘*└(n + 2) / 3┘
Knight Unknown Unknown
Queen
Unknown, 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.

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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