Chessboard Puzzles
Part 4: Other Surfaces
and Variations
Dan Freeman
April 27, 2014
Villanova University
MAT 9000 Graduate Math Seminar
Any Questions from Last Time?
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Introduction
• In the first three presentations, we looked at
the concepts chessboard domination,
chessboard independence and the knight’s tour
• Tonight we will conclude this series of
presentations with a look at these three
concepts on non-regular surfaces
• We will also touch on a few other concepts
associated with chessboard mathematics
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Knight Movement
• Recall that knights move two squares in one
direction (either horizontally or vertically) and
one square in the other direction
• Knights’ moves resemble an L shape
• Knights are the only pieces that are allowed to
jump over other pieces
• In the example below, the white and black
knights can move to squares with circles of the
corresponding color
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Knight’s Tour Revisited
• A knight’s tour is a succession of moves made
by a knight that traverse every square on a
chessboard once and only once
• There are two kinds of knight’s tours, a closed
knight’s tour and an open knight’s tour:
– A closed knight’s tour is one in which the knight’s last
move in the tour places it a single move away from
where it started
– An open knight’s tour is one in which the knight’s last
move in the tour places it on a square that is not a
single move away from where it started
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Toroidal Chessboard
• A torus is a donut-shaped surface in which
both the rows and columns wrap around
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Knight’s Tour on a Torus
• In 1997, John Watkins and his student, Becky
Hoenigman, proved the remarkable result that
every rectangular chessboard has a closed
knight’s tour* on a torus
1
16
7
22
13
4
19
10
20
11
2
17
8
23
14
5
15
6
21
12
3
18
9
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1
11
3
13
5
15
7
17
9
29
19
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35
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33
23
31
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10
2
12
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14
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36
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Knight’s Tour on
3x8 Torus
Knight’s Tour on
4x9 Torus
*Hereafter, a knight’s tour will be used to refer to a closed knight’s tour
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Knight’s Tour on a Cylinder
• Unlike a torus, a cylinder only wraps in one
dimension, not both
• In 2000, John Watkins proved that a knight’s
tour exists on an mxn cylindrical chessboard
unless one of the following two conditions
holds:
1) m = 1 and n > 1; or
2) m = 2 or 4 and n is even
• Here is why the above cases are excluded:
– If m = 1, a knight can’t move at all
– If m = 2 and n is even, then each move would take
the knight left or right by two columns and so then
only at most half of the columns would be visited
– If m = 4 and n is even, then the coloring argument by
Louis Pósa from the last presentation holds
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Klein Bottle
• The Klein bottle operates like a torus, except
when wrapping horizontally, the rows reverse
order
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Knight’s Tour on Klein Bottle
• Just like with the torus, every rectangular
chessboard has a knight’s tour on a Klein bottle
• Examples of knight’s tours on 6x2 and 6x4
Klein bottles are below
Knight’s Tour on
6x2 Klein Bottle
Knight’s Tour on
6x4 Klein Bottle
1
4
1
4
22
19
9
12
15
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12
9
5
2
5
2
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23
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8
17
14
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11
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6
3
6
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10
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10
7
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Möbius Strip
• A Möbius strip is like a cylinder in that it only
wraps in one dimension but is distinguished by
the half-twist it makes when wrapping, like the
Klein bottle
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Knight’s Tour on Möbius Strip
• A knight’s tour exists on a Mobius strip unless
one or more of the following three conditions
hold:
1) m = 1 and n > 1; or n = 1 and m = 3, 4 or 5;
2) m = 2 or 4 and n is even
3) n = 4 and m = 3
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Queens Domination on a Torus
• The queens domination numbers on both a
regular board and a torus for 1 ≤ n ≤ 10 appear
in the table below
• Note that the only case where the two numbers
differ is n = 8.
n
γ(Qnxn)
γtor(Qnxn)
1
2
3
4
5
6
7
8
9
10
1
1
1
2
3
3
4
5
5
5
1
1
1
2
3
3
4
4
5
5
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Knights Domination on a Torus
• The knights domination numbers on both a
regular board and a torus for 1 ≤ n ≤ 8 appear
in the table below
• Note that, shockingly, the value for γtor is lower
for n = 8 than it is for n = 7!
• Also, each value of γtor is unique up to n = 8.
This may or may not be the case in general.
n
γ(Nnxn)
γtor(Nnxn)
1
2
3
4
5
6
7
8
1
4
4
4
5
8
10
12
1
2
3
4
5
6
9
8
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Rooks Domination on a Torus
• Since it doesn’t make any difference whether a
rook is on a regular board or on a torus, it
follows that γ(Rnxn) = γtor(Rnxn) = n
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Bishops Domination on a Torus
• Since the number of distinct diagonals in either
direction drops from 2n – 1 to n on a torus, it is
easy to see that γtor(Bnxn) = n, just like
γ(Bnxn) = n
n Distinct Diagonals on
a Torus Shown in Red
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Kings Domination on a Torus
• γtor(Knxn) = ┌(n / 3)*┌ n / 3 ┐┐
• γtor(Kmxn) = max{┌(m / 3)*┌ n / 3 ┐┐,
┌(n / 3)*┌m / 3 ┐┐}
9 Kings Dominating a
Regular 7x7 Board
7 Kings Dominating a
7x7 Torus
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Kings Independence on a Torus
• The formulas for the kings independence
number on a torus are analogous to those for
the kings domination number
• βtor(Knxn) = └(½*n)*└½*n┘┘
• βtor(Kmxn) = min{└(½*m)*└ ½*n ┘┘,
└(½*n)*└ ½*m ┘┘}
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n-queens Problem on Cylinder
• A formula for βcyl(Qnxn) has not yet been found
• While βcyl(Q5x5) = β(Q5x5) = 5 and βcyl(Q7x7) =
β(Q7x7) = 7, βcyl(Q8x8) = 6 ≠ β(Q8x8) = 8
• The picture below shows why 8 queens fail to
be independent on an 8x8 cylinder
8 Queens Fail to Be
Independent on 8x8 Cylinder
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Independent Domination Number
• The independent domination number for a
given piece P and a given mxn chessboard is
the minimum size of an independent
dominating set, denoted i(Pmxn)
• i(Pmxn) need not equal γ(Pmxn) or β(Pmxn), as
shown in the examples below for queens on a
4x4 board
γ(Q4x4) = 2
i(Q4x4) = 3
β(Q4x4) = 4
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Irredundance Number
• An irredundant set of chess pieces is one in
which each piece in the set either occupies a
square that is not covered by another piece or
else it covers a square that no other piece
covers
• A maximal irredundant set is one that is not a
proper subset of any irredundant set
• The irredundance number for a given piece P
and a given mxn chessboard is the minimum
size of a maximal irredundant set
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Irredundance Number
• Both the set of 9 kings on the left-hand board
and the set of 16 kings on the right-hand board
are maximal irredundant sets
Maximal Irredundant
Set of 9 Kings
Maximal Irredundant
Set of 16 Kings
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Total Domination Number
• W.W. Rouse Ball introduced the concept of
total domination in 1987
• The total domination number is the minimum
number of pieces of a given type P on a given
mxn chessboard that are required to attack
every square on the board, including occupied
ones
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Total Domination Number
• Ball showed the total domination number on an
8x8 chessboard to be 5 for queens, 10 for
bishops, 14 for knights and 8 for rooks
• An arrangement of 5 queens totally dominating
an 8x8 board is given below
Five Queens Totally
Dominating 8x8 Board
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Sources Cited
• J.J. Watkins. Across the Board: The Mathematics of
Chessboard Problems. Princeton, New Jersey:
Princeton University Press, 2004.
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