Preface 

xv  

part 1 The Mathematics of Social Choice 



The Mathematics of Voting 


2  (46) 

The Paradoxes of Democracy 



Preference Ballots and Preference Schedules 


4  (2) 


6  (4) 


10  (2) 

The PluralitywithElimination Method (Instant Runoff Voting) 


12  (5) 

The Method of Pairwise Comparisons (Copeland's Method) 


17  (6) 


23  (25) 

Conclusion: Elections, Fairness, and Arrow's Impossibility Theorem 


28  (1) 

Profile: Kenneth J. Arrow 


29  (1) 


30  (1) 


30  (11) 


41  (1) 

Appendix 1: Breaking Ties 


42  (1) 

Appendix 2: A Sampler of Elections in the Real World 


43  (3) 

References and Further Readings 


46  (2) 


48  (36) 




50  (3) 


53  (8) 

Applications of Banzhaf Power 


61  (2) 

The ShapleyShubik Power Index 


63  (5) 

Applications of ShapleyShubik Power 


68  (16) 


70  (1) 

Profile: Lloyd S. Shapley 


71  (1) 


72  (1) 


72  (7) 


79  (1) 

Appendix: Power in the Electoral College 


80  (2) 

References and Further Readings 


82  (2) 


84  (44) 

The Mathematics of Sharing 




86  (2) 

Two Players: The DividerChooser Method 


88  (1) 


89  (6) 


95  (3) 

The LastDiminisher Method 


98  (5) 

The Method of Sealed Bids 


103  (3) 


106  (22) 


109  (1) 


110  (1) 


111  (1) 


111  (15) 


126  (1) 

References and Further Readings 


127  (1) 

The Mathematics of Apportionment 


128  (32) 




129  (5) 

Hamilton's Method and the Quota Rule 


134  (2) 

The Alabama and Other Paradoxes 


136  (5) 


141  (3) 


144  (1) 


145  (15) 


147  (2) 

Historical Note: A Brief History of Apportionment in the United States 


149  (1) 


150  (1) 


150  (5) 


155  (2) 

References and Further Readings 


157  (3) 

part 2 Management Science 




160  (36) 

The Circuit Comes to Town 




162  (3) 


165  (3) 

Graph Concepts and Terminology 


168  (2) 


170  (1) 


171  (3) 


174  (5) 


179  (17) 


183  (1) 


184  (1) 


185  (1) 


185  (9) 


194  (1) 

References and Further Readings 


194  (2) 

The Traveling Salesman Problem 


196  (38) 

Hamilton Joins the Circuit 



Hamilton Circuits and Hamilton Paths 


199  (1) 


200  (4) 

Traveling Salesman Problems 


204  (3) 

Simple Strategies for Solving TSPs 


207  (3) 

The BruteForce and NearestNeighbor Algorithms 


210  (2) 


212  (1) 

The Repetitive NearestNeighbor Algorithm 


213  (1) 

The CheapestLink Algorithm 


214  (20) 


218  (2) 

Profile: Sir William Rowan Hamilton 


220  (1) 


221  (1) 


221  (9) 


230  (2) 

References and Further Readings 


232  (2) 

The Mathematics of Networks 


234  (40) 

The Cost of Being Connected 




236  (4) 


240  (2) 


242  (2) 

The Shortest Network Connecting Three Points 


244  (6) 

Shortest Networks for Four or More Points 


250  (24) 


257  (1) 

Profile: Evangelista Torricelli 


258  (1) 


258  (1) 


259  (9) 


268  (3) 

Appendix: The SoapBubble Solution 


271  (1) 

References and Further Readings 


272  (2) 

The Mathematics of Scheduling 


274  (38) 

Directed Graphs and Critical Paths 



The Basic Elements of Scheduling 


276  (5) 

Directed Graphs (Digraphs) 


281  (2) 

Scheduling with Priority Lists 


283  (6) 

The DecreasingTime Algorithm 


289  (2) 


291  (3) 

The CriticalPath Algorithm 


294  (2) 

Scheduling with Independent Tasks 


296  (16) 


299  (1) 

Profile: Ronald L. Graham 


300  (1) 


300  (1) 


301  (8) 


309  (1) 

References and Further Readings 


309  (3) 

part 3 Growth and Symmetry 




312  (26) 

Fibonacci Numbers and the Golden Ratio 




313  (4) 


317  (1) 


318  (6) 


324  (14) 


327  (1) 

Profile: Leonardo Fibonacci 


328  (1) 


329  (1) 


329  (6) 


335  (1) 

References and Further Readings 


336  (2) 

The Mathematics of Population Growth 


338  (34) 

There Is Strength in Numbers 



The Dynamics of Population Growth 


339  (3) 


342  (6) 

The Exponential Growth Model 


348  (9) 

The Logistic Growth Model 


357  (15) 


363  (1) 


363  (1) 


364  (1) 


364  (6) 


370  (1) 

References and Further Readings 


371  (1) 


372  (38) 

Mirror, Mirror Off the Wall... 




373  (2) 


375  (2) 


377  (3) 


380  (1) 


381  (2) 

Symmetry as a Rigid Motion 


383  (5) 


388  (22) 


393  (1) 

Profile: Sir Robert Penrose 


393  (1) 


394  (1) 


394  (11) 


405  (1) 

Appendix: The 17 Wallpaper Symmetry Types 


406  (3) 

References and Further Readings 


409  (1) 

The Geometry of Fractal Shapes 


410  (36) 




412  (6) 


418  (3) 


421  (1) 

The Twisted Sierpinski Gasket 


422  (3) 


425  (21) 


432  (1) 

Profile: Benoit Mandelbrot 


433  (1) 


434  (1) 


434  (8) 


442  (1) 

References and Further Readings 


443  (3) 



Collecting Statistical Data 


446  (30) 

Censuses, Surveys, and Clinical Studies 




448  (4) 


452  (5) 


457  (2) 

Sampling: Terminology and Key Concepts 


459  (1) 

The CaptureRecapture Method 


460  (1) 


461  (15) 


465  (1) 


466  (1) 


467  (1) 


467  (7) 


474  (1) 

References and Further Readings 


475  (1) 


476  (34) 

Graphing and Summarizing Data 



Graphical Descriptions of Data 


477  (4) 


481  (6) 

Numerical Summaries of Data 


487  (8) 


495  (15) 


498  (1) 

Profile: W. Fdwards Deming 


498  (1) 


499  (1) 


499  (9) 


508  (1) 

References and Further Readings 


509  (1) 

Chances, Probabilities, and Odds 


510  (30) 



Random Experiments and Sample Spaces 


511  (3) 


514  (2) 

Permutations and Combinations 


516  (3) 


519  (4) 


523  (3) 


526  (14) 


528  (1) 


529  (1) 


530  (1) 


531  (6) 


537  (1) 

References and Further Readings 


538  (2) 


540  (27) 

Everything Is Back to Normal (Almost) 



Approximately Normal Distributions of Data 


542  (2) 

Normal Curves and Normal Distributions 


544  (2) 

Standardizing Normal Data 


546  (2) 


548  (1) 

Normal Curves as Models of RealLife Data Sets 


549  (2) 

Distributions of Random Events 


551  (2) 


553  (14) 


557  (1) 

Profile: Carl Friedrich Gauss 


557  (1) 


558  (1) 


558  (7) 


565  (1) 

References and Further Readings 


566  (1) 
Answers to Selected Problems 

567  (42) 
Index 

609  (10) 
Photo Credits 

619  