# How to Guard an Art Gallery and Other Discrete Mathematical Adventures

Publication Year: 2009

Published by: The Johns Hopkins University Press

#### Cover

#### Title Page, Copyright

#### Contents

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pp. v-vii

#### Preface

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pp. ix-xi

The adventures in this book are launched by easily understood
questions from the realm of *discrete mathematics*, a
wide-ranging subject that studies fundamental properties of
the counting numbers 1, 2, 3, . . . and arrangements of finite
sets
The book grew from talks for mathematically inclined ...

#### 1. How to Count Pizza Pieces

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pp. 1-32

little experimentation should convince you that seven pieces
is the most you can make with three cuts. But how many
pizza pieces can you make with more cuts?
The pizza-cutter’s problem. What is the largest number
of pieces of pizza we can make with *n* straight cuts through
a circular pizza? ...

#### 2. Count on Pick’s Formula

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pp. 33-72

Counting questions. How many ways are there to make
change for a dollar from a supply of quarters, dimes, and
nickels? What about change for *D* dollars?
Although the questions we have posed appear entirely
unrelated, a remarkable formula discovered by the Austrian ...

#### 3. How to Guard an Art Gallery

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pp. 73-112

Figure 3.1 shows the unusual floor plan of the Sunflower Art Gallery and the locations of four guards. Each guard is stationary but can rotate in place to scan the surroundings in all directions. Guards cannot see through walls or around corners. Every point in the gallery is visible to at least one ...

#### 4. Pixels, Lines, and Leap Years

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pp. 113-138

A computer monitor has a rectangular array of thousands of
tiny square cells called

#### 5. Measure Water with a Vengeance

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pp. 139-168

In the 1995 action movie *Die Hard: With a Vengeance*, Bruce
Willis plays a maverick law enforcement officer who must
solve a series of fiendish puzzles posed by Simon Gruber,
the mastermind of a diabolical bank heist. In one memorable
scene, Simon directs Willis and his reluctant sidekick
to a fountain in a public park, where they find two empty, ...

#### 6. From Stamps to Sylver Coins

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pp. 169-206

Question. We have a large supply of 5- and 8-cent stamps.
Can we make exact postage for a 27-cent postcard?
A bit of trial and error should convince you the answer
is no. A methodical approach involves some algebra. The
question asks whether there is a solution (*x, y*) to the equation ...

#### 7. Primes and Squares:Quadratic Residues

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pp. 207-244

The blocks of eleven repeat, and the only nonzero remainders
are 1, 3, 4, 5, and 9. These five numbers are the *quadratic
residues* modulo 11.
Our goal in this chapter is to discover and explain properties
of quadratic residues. Our discussion culminates in
the beautiful Law of Quadratic Reciprocity. We will also find ...

#### References

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pp. 245-250

#### Index

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pp. 251-257

E-ISBN-13: 9780801897047

E-ISBN-10: 0801897041

Print-ISBN-13: 9780801892998

Print-ISBN-10: 0801892996

Page Count: 272

Illustrations: 103 line drawings

Publication Year: 2009