1. Chessboard Puzzles
Part 1: Domination
Dan Freeman
February 20, 2014
Villanova University
MAT 9000 Graduate Math Seminar
2. Brief Overview of Chess
• Chess is a classic board game that has
been played for at least 1,200 years
• Chess is a 2-player turn-based game
played on an 8x8 board
• There are six different types of pieces:
pawn, knight, bishop, rook, queen and king
• The objective of the game is to put the
opponent’s king in a position in which it
cannot escape attack; this position is
known as checkmate
4. Domination Defined
• A dominating set of chess pieces is one
such that every square on the 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” or “cover” the board
• Domination numbers are denoted by
γ(Pmxn) where P represents the type of
chess piece (see legend to the right) and m
and n are the number of rows and columns
of the board, respectively
Domination
Number
Notation
King – K
Queen – Q
Rook – R
Bishop – B
Knight – N
5. Rook Movement
• Rooks move horizontally and vertically
• Rooks 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 the example below, 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
6. Rooks Domination
• Domination among rooks is the simplest of
all chess pieces
• For a square nxn chessboard, the rooks
domination number is simply n
• For a general rectangular mxn chessboard,
γ(Rmxn) = min(m, n)
7. Proof that γ(Rnxn) = n
• Two Russian brothers Akiva and Isaak
Yaglom proved this:
– 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.
Thus, γ(Rnxn) ≥ n.
– Second, if n rooks are placed along a single row
or down a single column, the entire board is
clearly dominated. That is, γ(Rnxn) ≤ n.
– Lastly, since γ(Rnxn) ≥ n and γ(Rnxn) ≤ n, we
conclude that γ(Rnxn) = n.
• The fact that γ(Rmxn) = min(m, n) follows
immediately from the above
8. Bishop Movement
• Bishops move diagonally
• Bishops 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 the example below, 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
9. Bishops Domination
• As with rooks, γ(Bnxn) = n (though in
general, γ(Bmxn) ≠ min(m, n))
• The proof that γ(Bnxn) = n for bishops is
more involved than that for rooks
• The proof starts with rotating the
chessboard 45 degrees, as shown below
45°
5x4 Black Square
10. Proof that γ(Bnxn) = n
• Yaglom and Yaglom proved this:
– Suppose n = 8 (the following argument works for
all even n).
– Clearly, from rotating the board 45 degrees, we
see a 5x4 construction of dark (black) squares in
the middle of the board. Therefore, at least 4
bishops are needed to cover all of the black
squares. By symmetry, at least 4 bishops are
needed to cover all of the light (white) squares.
Therefore, γ(B8x8) ≥ 4 + 4 = 8.
– On the other hand, if we place 8 bishops in the
fourth column of a chessboard, we find that the
entire board is covered. Thus, γ(B8x8) ≤ 8.
– Since γ(B8x8) ≥ 8 and γ(B8x8) ≤ 8, it follows that
γ(B8x8) = 8 and for general even n, γ(Bnxn) = n.
Bishops Domination
on 8x8 Board
11. Proof that γ(Bnxn) = n
– Now suppose n is odd and let n = 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; 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 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, the entire
board is covered.
– In conclusion, γ(Bnxn) = for all n.
12. King Movement
• 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 the example below, the king can move to
any of the squares with a white circle
13. Kings Domination
• Shown are examples of
9 kings dominating 7x7,
8x8 and 9x9 boards
• For 7 ≤ n ≤ 9,
γ(Knxn) = 9
14. Kings Domination
• No matter where one
places a king on any of
the 7x7, 8x8 or 9x9
boards, only one of the
nine dark orange
squares will be covered
15. Kings Domination
• Thus, 32 = 9 is the domination number for
square chessboards where n = 7, 8 or 9
• This triplet pattern continues for larger
boards:
– For n = 10, 11 and 12, γ(Knxn) = 42 = 16.
– For n = 13, 14 and 15, γ(Knxn) = 52 = 25.
• Thus, the general formula for the kings
domination number can be written making
use of the greatest integer or floor function:
– γ(K) = └(n + 2) / 3┘
2.
nxn• Generalizing even further, the formula for
rectangular boards is:
– γ(Kmxn) = └(m + 2) / 3┘* └(n + 2) / 3┘.
16. Knight Movement
• Knights move two squares in one direction
(either horizontally or vertically) and one
square in the other direction as long as they
do not take the place of a friendly piece
• Knights’ moves resemble an L shape
• Knights are the only pieces that are allowed
to jump over other pieces
• In the example below, the white and black
knights can move to squares with circles of
the corresponding color
17. Knights Domination
• 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 the table to the
right.
• As can be seen from the table,
as n increases, γ(Nnxn) increases
in no discernible pattern
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
19. Queen Movement
• Queens move horizontally, vertically and
diagonally
• Queens 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 the example below, the queen can move
to any of the squares with a black circle
20. Queens Domination
• Domination among queens is the most
complicated and interesting of all chess
pieces, as well as the least understood
• No formula is known for the queens
domination number, but lower and upper
bounds have been established
Arrangement of
5 Queens
Dominating
8x8 Board
21. Queens Diagonal Domination
• However, a formula for the queens diagonal
domination number is known
• The queens diagonal domination number,
denoted diag(Qnxn), is the minimum number
of queens all placed along the main
diagonal required to cover the board
• For all n, diag(Qnxn) = n – max(|mid-point
free, all even or all odd, subset of
{1, 2, 3, …, n}|)
• A mid-point free set is a set in which for any
given pair of elements in the set, the
midpoint of those two numbers is not in the
set
22. Upper and Lower Bounds
• Upper bound for queens domination
number: Welch proved that for n = 3m + r,
0 ≤ r < 3, γ(Qnxn) ≤ 2m + r
• Lower bound for queens domination
number: Spencer proved that for any n,
γ(Qnxn) ≥ ½*(n – 1)
• Weakley improved on Spencer’s lower
bound by showing that if the lower bound is
attained, that is, if γ(Qnxn) = ½*(n – 1), then
n ≡ 3 mod 4
• Corollaries of Spencer’s improved lower
bound include:
– γ(Q7x7) = 4.
– For n = 4k + 1, γ(Qnxn) ≥ ½*(n + 1) = 2k + 1.
23. Queens Domination Numbers
• The lower bounds on the previous slide
allow us to narrow the number of
possibilities considerably for the queens
domination number for larger chess
boards, as shown in the table below
n γ(Qnxn)
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
24. Sources Cited
• J.J. Watkins. Across the Board: The
Mathematics of Chessboard Problems.
Princeton, New Jersey: Princeton
University Press, 2004.
• J. Nunn. Learn Chess. London, England:
Gambit Publications, 2000.
• “Chess.” Wikipedia, Wikimedia Foundation.
http://en.wikipedia.org/wiki/Chess
• “A006075 – OEIS.” http://oeis.org/A006075
