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Copyright, iv
Preface, xv
Part 1. Social Choice, 1
Chapter 1. The Mathematics of Voting: The Paradoxes of Democracy, 2
1.1. Preference Ballots and Preference Schedules, 4
1.2. The Plurality Method, 6
1.3. The Borda Count Method, 10
1.4. The Plurality-with-Elimination Method (Instant Runoff Voting), 12
1.5. The Method of Pairwise Comparisons, 16
1.6. Rankings, 22
Conclusion Elections, Fairness, and Arrow’s Impossibility Theorem, 27
Profile: Kenneth J. Arrow, 28
Key Concepts, 29
Exercises, 29
Projects and Papers, 40
References and Further Readings, 41
Chapter 2. The Mathematics of Power: Weighted Voting, 42
2.1. An Introduction to Weighted Voting, 44
2.2. The Banzhaf Power Index, 47
2.3. Applications of the Banzhaf Power Index, 55
2.4. The Shapley-Shubik Power Index, 57
2.5. Applications of the Shapley-Shubik Power Index, 63
Conclusion, 65
Profile: Lloyd S. Shapley, 65
Key Concepts, 66
Exercises, 66
Projects and Papers, 74
References and Further Readings, 75
Chapter 3. The Mathematics of Sharing: Fair-Division Games, 76
3.1. Fair-Division Games, 78
3.2. Two Players: The Divider-Chooser Method, 80
3.3. The Lone-Divider Method, 82
3.4. The Lone-Chooser Method, 87
3.5. The Last-Diminisher Method, 90
3.6. The Method of Sealed Bids, 95
3.7. The Method of Markers, 97
Conclusion, 101
Profile: Hugo Steinhaus, 102
Key Concepts, 103
Exercises, 103
Projects and Papers, 121
References and Further Readings, 121
Chapter 4. The Mathematics of Apportionment: Making the Rounds, 122
4.1. Apportionment Problems, 124
4.2. Hamilton’s Method and the Quota Rule, 128
4.3. The Alabama and Other Paradoxes, 130
4.4. Jefferson’s Method, 134
4.5. Adams’s Method, 138
4.6. Webster’s Method, 139
Conclusion, 141
Historical Note: A Brief History of Apportionment in the United States, 142
Key Concepts, 144
Exercises, 144
Projects and Papers, 149
References and Further Readings, 151
Mini-Excursion 1. Apportionment Today: The Huntington-Hill Method, 152
Part 2. Management Science, 165
Chapter 5. The Mathematics of Getting Around: Euler Paths and Circuits, 166
5.1. Euler Circuit Problems, 168
5.2. What Is a Graph?, 171
5.3. Graph Concepts and Terminology, 173
5.4. Graph Models, 176
5.5. Euler’s Theorems, 178
5.6. Fleury’s Algorithm, 180
5.7. Eulerizing Graphs, 185
Conclusion, 189
Profile: Leonhard Euler, 190
Key Concepts, 191
Exercises, 191
Projects and Papers, 199
References and Further Readings, 200
Chapter 6. The Mathematics of Touring: The Traveling Salesman Problem, 202
6.1. Hamilton Paths and Hamilton Circuits, 205
6.2. Complete Graphs, 206
6.3. Traveling Salesman Problems, 209
6.4. Simple Strategies for Solving TSPs, 212
6.5. The Brute-Force and Nearest-Neighbor Algorithms, 215
6.6. Approximate Algorithms, 217
6.7. The Repetitive Nearest-Neighbor Algorithm, 219
6.8. The Cheapest-Link Algorithm, 220
Conclusion, 224
Profile: Sir William Rowan Hamilton, 225
Key Concepts, 226
Exercises, 226
Projects and Papers, 237
References and Further Readings, 239
Chapter 7. The Mathematics of Networks: The Cost of Being Connected, 240
7.1. Trees, 242
7.2. Spanning Trees, 246
7.3. Kruskal’s Algorithm, 249
7.4. The Shortest Network Connecting Three Points, 251
7.5. Shortest Networks for Four or More Points, 256
Conclusion, 263
Profile: Evangelista Torricelli, 264
Key Concepts, 264
Exercises, 265
Projects and Papers, 276
References and Further Readings, 278
Chapter 8. The Mathematics of Scheduling: Chasing the Critical Path, 280
8.1. The Basic Elements of Scheduling, 282
8.2. Directed Graphs (Digraphs), 286
8.3. Scheduling with Priority Lists, 289
8.4. The Decreasing-Time Algorithm, 295
8.5. Critical Paths, 297
8.6. The Critical-Path Algorithm, 300
8.7. Scheduling with Independent Tasks, 302
Conclusion, 305
Profile: Ronald L. Graham, 306
Key Concepts, 306
Exercises, 307
Projects and Papers, 315
References and Further Readings, 316
Mini-Excursion 2. A Touch of Color: Graph Coloring, 318
Part 3. Shape, Growth, and Form, 331
Chapter 9. The Mathematics of Spiral Growth: Fibonacci Numbers and the Golden Ratio, 332
9.1. Fibonacci’s Rabbits, 334
9.2. Fibonnaci Numbers, 335
9.3. The Golden Ratio, 339
9.4. Gnomons, 341
9.5. Spiral Growth in Nature, 346
Conclusion, 349
Profile: Leonardo Fibonacci, 350
Key Concepts, 351
Exercises, 351
Projects and Papers, 358
References and Further Readings, 358
Chapter 10. The Mathematics of Money: Spending It, Saving It, and Growing It, 360
10.1. Percentages, 362
10.2. Simple Interest, 366
10.3. Compound Interest, 370
10.4. Geometric Sequences, 376
10.5. Deferred Annuities: Planned Savings for the Future, 380
10.6. Installment Loans: The Cost of Financing the Present, 383
Conclusion, 390
Key Concepts, 391
Exercises, 392
Projects and Papers, 398
References and Further Readings, 399
Chapter 11. The Mathematics of Symmetry: Beyond Reflection, 400
11.1. Rigid Motions, 402
11.2. Reflections, 403
11.3. Rotations, 405
11.4. Translations, 407
11.5. Glide Reflections, 408
11.6. Symmetry as a Rigid Motion, 410
11.7. Patterns, 415
Conclusion, 420
Profile: Sir Roger Penrose, 421
Key Concepts, 422
Exercises, 422
Projects and Papers, 433
References and Further Readings, 434
Chapter 12. The Geometry of Fractal Shapes: Naturally Irregular, 436
12.1. The Koch Snowflake, 438
12.2. The Sierpinski Gasket, 444
12.3. The Chaos Game, 447
12.4. The Twisted Sierpinski Gasket, 448
12.5. The Mandelbrot Set, 451
Conclusion, 458
Profile: Benoit Mandelbrot, 459
Key Concepts, 459
Exercises, 460
Projects and Papers, 471
References and Further Readings, 472
Mini-Excursion 3. The Mathematics of Population Growth: There Is Strength in Numbers, 474
Part 4. Statistics, 493
Chapter 13. Collecting Statistical Data: Censuses, Surveys, and Clinical Studies, 494
13.1. The Population, 496
13.2. Sampling, 499
13.3. Random Sampling, 505
13.4. Sampling: Terminology and Key Concepts, 507
13.5. The Capture-Recapture Method, 508
13.6. Clinical Studies, 509
Conclusion, 513
Profile: George Gallup, 514
Key Concepts, 514
Exercises, 515
Projects and Papers, 522
References and Further Readings, 523
Chapter 14. Descriptive Statistics: Graphing and Summarizing Data, 524
14.1. Graphical Descriptions of Data, 525
14.2. Variables, 529
14.3. Numerical Summaries of Data, 533
14.4. Measures of Spread, 540
Conclusion, 543
Profile: W. Edwards Deming, 544
Key Concepts, 544
Exercises, 545
Projects and Papers, 554
References and Further Readings, 554
Chapter 15. Chances, Probabilities, and Odds: Measuring Uncertainty, 556
15.1. Random Experiments and Sample Spaces, 558
15.2. Counting Outcomes in Sample Spaces, 560
15.3. Permutations and Combinations, 562
15.4. Probability Spaces, 565
15.5. Equiprobable Spaces, 569
15.6. Odds, 573
Conclusion, 574
Profile: Persi Diaconis, 575
Key Concepts, 576
Exercises, 577
Projects and Papers, 584
References and Further Readings, 584
Chapter 16. The Mathematics of Normal Distributions: The Call of the Bell, 586
16.1. Approximately Normal Distributions of Data, 588
16.2. Normal Curves and Normal Distributions, 590
16.3. Standardizing Normal Data, 592
16.4. The 68-95-99.7 Rule, 594
16.5. Normal Curves as Models of Real-Life Data Sets, 595
16.6. Distributions of Random Events, 596
16.7. Statistical Inference, 598
Conclusion, 602
Profile: Carl Friedrich Gauss, 603
Key Concepts, 603
Exercises, 604
Projects and Papers, 610
References and Further Readings, 611
Mini-Excursion 4. The Mathematics of Managing Risk: Expected Values, 612
Appendices, 623
Answers to Selected Problems, 628
Index, 669
Photo Credits, 679

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