1 The Mathematics of Elections

1.1 The Basic Elements of an Election

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality-with-Elimination Method

1.5 The Method of Pairwise Comparisons

1.6 Fairness Criteria and Arrow’s Impossibility Theorem

2 The Mathematics of Power

2.1 An Introduction to Weighted Voting

2.2 Banzhaf Power

2.3 Shapley-Shubik Power

2.4 Subsets and Permutations

3 The Mathematics of Sharing

3.1 Fair-Division Games

3.2 The Divider-Chooser Method

3.3 The Lone-Divider Method

3.4 The Lone-Chooser Method

3.5 The Method of Sealed Bids

3.6 The Method of Markers

4 The Mathematics of Apportionment

4.1 Apportionment Problems and Apportionment Methods

4.2 Hamilton’s Method

4.3 Jefferson’s Method

4.4 Adams’s and Webster’s Methods

4.5 The Huntington-Hill Method

4.6 The Quota Rule and Apportionment Paradoxes

5 The Mathematics of Getting Around

5.1 Street-Routing Problems

5.2 An Introduction to Graphs

5.3 Euler’s Theorems and Fleury’s Algorithm

5.4 Eulerizing and Semi-Eulerizing Graphs

6 The Mathematics of Touring

6.1 What Is a Traveling Salesman Problem?

6.2 Hamilton Paths and Circuits

6.3 The Brute-Force Algorithm

6.4 The Nearest-Neighbor and Repetitive Nearest-Neighbor Algorithms

6.5 The Cheapest-Link Algorithm

7 The Mathematics of Networks

7.1 Networks and Trees

7.2 Spanning Trees, MSTs, and MaxSTs

7.3 Kruskal’s Algorithm

8 The Mathematics of Scheduling

8.1 An Introduction to Scheduling

8.2 Directed Graphs

8.3 Priority-List Scheduling

8.4 The Decreasing-Time Algorithm

8.5 Critical Paths and the Critical-Path Algorithm

9 Population Growth Models

9.1 Sequences and Population Sequences

9.2 The Linear Growth Model

9.3 The Exponential Growth Model

9.4 The Logistic Growth Model

10 Financial Mathematics

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Retirement Savings

10.5 Consumer Debt

11 The Mathematics of Symmetry

11.1 Rigid Motions

11.2 Reflections

11.3 Rotations

11.4 Translations

11.5 Glide Reflections

11.6 Symmetries and Symmetry Types

11.7 Patterns

12 Fractal Geometry

12.1 The Koch Snowflake and Self-Similarity

12.2 The Sierpinski Gasket and the Chaos Game

12.3 The Twisted Sierpinski Gasket

12.4 The Mandelbrot Set

13 Fibonacci Numbers and the Golden Ratio

13.1 Fibonacci Numbers

13.2 The Golden Ratio

13.3 Gnomons

13.4 Spiral Growth in Nature

14 Censuses, Surveys, Polls, and Studies

14.1 Enumeration

14.2 Measurement

14.3 Cause and Effect

15 Graphs, Charts, and Numbers

15.1 Graphs and Charts

15.2 Means, Medians, and Percentiles

15.3 Ranges and Standard Deviations

16 Probabilities, Odds, and Expectations

16.1 Sample Spaces and Events

16.2 The Multiplication Rule, Permutations, and Combinations

16.3 Probabilities and Odds

16.4 Expectations

16.5 Measuring Risk

17 The Mathematics of Normality

17.1 Approximately Normal Data Sets

17.2 Normal Curves and Normal Distributions

17.3 Modeling Approximately Normal Distributions

17.4 Normality in Random Events

Answers to Selected Exercises

Index

Photo Credits

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