Chessboard PuzzlesPart 4: Other Surfaces

and Variations

Dan Freeman

April 27, 2014

Villanova UniversityMAT 9000 Graduate Math Seminar

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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 24

Knight’s Tour on 3x8 Torus

*Hereafter, a knight’s tour will be used to refer to a closed knight’s tour

Knight’s Tour on 4x9 Torus

1 11 3 13 5 15 7 17 9

29 19 27 35 25 33 23 31 21

10 2 12 4 14 6 16 8 18

20 28 36 26 34 24 32 22 30

1

1

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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

1 4

9 12

5 2

11 8

3 6

7 10

1 4 22 19

15 18 12 9

5 2 20 23

17 14 8 11

3 6 24 21

13 16 10 7

Knight’s Tour on 6x2 Klein Bottle

Knight’s Tour on 6x4 Klein Bottle

1 1

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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 1 12 1 13 1 14 2 25 3 36 3 37 4 48 5 49 5 5

10 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 1 12 4 23 4 34 4 45 5 56 8 67 10 98 12 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.