Preface 

xiii  
PART 1 The Mathematics of Social Choice 


The Mathematics of Voting 



The Paradoxes of Democracy 


2  (2) 

Preference Ballots and Preference Schedules 


4  (1) 


5  (3) 


8  (1) 

The PluralitywithElimination Method 


9  (5) 

The Method of Pairwise Comparisons 


14  (5) 


19  (27) 

Conclusion: Fairness and Arrow's Impossibility Theorem 


24  (1) 


25  (1) 


25  (13) 

Appendix 1: Breaking Ties 


38  (2) 

Appendix 2: A Sampler of Elections in the Real World 


40  (3) 

References and Further Readings 


43  (1) 




44  (2) 


46  (2) 


48  (6) 

Applications of the Banzhaf Power Index 


54  (2) 

The ShapleyShubik Power Index 


56  (5) 

Applications of the ShapleyShubik Power Index 


61  (17) 


62  (1) 


63  (1) 


63  (11) 

Appendix: Power in the Electoral College 


74  (1) 

References and Further Readings 


75  (1) 



The Mathematics of Sharing 


76  (2) 

FairDivision Problems and FairDivision Schemes 


78  (2) 

Two Players: The DividerChooser Method 


80  (1) 


81  (4) 


85  (3) 

The LastDiminisher Method 


88  (5) 

The Method of Sealed Bids 


93  (2) 


95  (27) 


98  (1) 


99  (1) 


99  (20) 

References and Further Readings 


119  (1) 

The Mathematics of Apportionment 




120  (2) 


122  (2) 

A Little Bit of U.S. History 


124  (2) 

The Mathematics of Apportionment: Basic Concepts 


126  (2) 


128  (1) 


129  (1) 


129  (2) 

More Problems with Hamilton's Method 


131  (2) 


133  (2) 

Jefferson's Method and the Quota Rule 


135  (1) 


136  (1) 


137  (21) 

Conclusion: Balinski and Young's Impossibility Theorem 


139  (1) 


140  (1) 


140  (9) 

Appendix 1: The HuntingtonHill Method 


149  (2) 

Appendix 2: A Brief History of Apportionment in the United States 


151  (3) 

References and Further Readings 


154  (2) 
PART 2 Management Science 




The Circuit Comes to Town 


156  (2) 


158  (4) 


162  (2) 

Graph Concepts and Terminology 


164  (2) 


166  (2) 


168  (2) 


170  (3) 


173  (24) 


177  (1) 


178  (1) 


178  (15) 

References and Further Readings 


193  (1) 

The Traveling Salesman Problem 



Hamilton Joins the Circuit 


194  (3) 

Hamilton Circuits and Hamilton Paths 


197  (2) 


199  (2) 

TravelingSalesman Problems 


201  (3) 

Simple Strategies for Solving TSPs 


204  (3) 

The BruteForce and NearestNeighbor Algorithms 


207  (2) 


209  (1) 

The Repetitive NearestNeighbor Algorithm 


210  (2) 

The CheapestLink Algorithm 


212  (28) 


216  (1) 


217  (1) 


218  (18) 

References and Further Readings 


236  (2) 

The Mathematics of Networks 




238  (2) 


240  (3) 


243  (1) 


244  (2) 

The Shortest Distance Between Three Points 


246  (5) 

The Shortest Network Linking More Than Three Points 


251  (27) 


255  (2) 


257  (1) 


258  (16) 

Appendix: The SoapBubble Solution 


274  (1) 

References and Further Readings 


275  (1) 

The Mathematics of Scheduling 



Directed Graphs and Critical Paths 


276  (2) 

The Basic Elements of Scheduling 


278  (6) 


284  (1) 

The Priority List Model for Scheduling 


285  (6) 

The DecreasingTime Algorith 


291  (1) 


292  (4) 

The CriticalPath Algorithm 


296  (1) 

Scheduling with Independent Tasks 


297  (21) 


300  (1) 


301  (1) 


301  (13) 

References and Further Readings 


314  (2) 
PART 3 Growth and Symmetry 




Fibonacci Numbers and the Golden Ratio 


316  (2) 


318  (3) 

The Equation x2 = x + 1 and the Golden Ratio 


321  (2) 


323  (7) 


330  (16) 


331  (1) 


332  (1) 


332  (11) 

References and Further Readings 


343  (1) 

The Mathematics of Population Growth 



There Is Strength in Numbers 


344  (2) 

The Dynamics of Population Growth 


346  (3) 


349  (5) 

The Exponential Growth Model 


354  (7) 

The Logistic Growth Model 


361  (17) 


365  (1) 


366  (1) 


366  (9) 

References and Further Readings 


375  (1) 



Mirror, Mirror, Off the Wall... 


376  (2) 


378  (1) 


379  (1) 


380  (2) 


382  (1) 


383  (1) 


384  (1) 


385  (5) 


390  (24) 


393  (1) 


394  (1) 


394  (14) 

Appendix: The Seventeen Wallpaper Symmetry Types 


408  (3) 

References and Further Readings 


411  (1) 




412  (2) 


414  (5) 


419  (2) 


421  (1) 

The Twisted Sierpinski Gasket 


422  (2) 

Symmetry of Scale in Art and Literature 


424  (2) 


426  (22) 


432  (2) 


434  (1) 


434  (10) 

References and Further Readings 


444  (2) 
PART 4 Statistics 


Collecting Statistical Data 



Censuses, Surveys, and Clinical Studies 


446  (2) 


448  (3) 


451  (5) 


456  (3) 

Sampling: Terminology and Key Concepts 


459  (2) 


461  (15) 


465  (1) 


465  (1) 


466  (7) 

References and Further Readings 


473  (1) 



Graphing and Summarizing Data 


474  (2) 

Graphical Descriptions of Data 


476  (3) 

Variables: Quantitative and Qualitative; Continuous and Discrete 


479  (5) 

Numerical Summaries of Data 


484  (9) 


493  (21) 


496  (1) 


497  (1) 


497  (14) 

References and Further Readings 


511  (1) 

Chances, Probabilities, and Odds 




512  (2) 

Random Experiments and Sample Spaces 


514  (2) 

Counting: The Multiplication Rule 


516  (2) 

Permutations and Combinations 


518  (4) 


522  (1) 


523  (2) 

Probability Spaces with Equally Likely Outcomes 


525  (3) 


528  (16) 


530  (1) 


530  (1) 


530  (11) 

References and Further Readings 


541  (1) 



Everything Is Back to Normal (Almost) 


542  (2) 

Approximately Normal Distributions of Data 


544  (2) 

Normal Curves and Normal Distributions of Data 


546  (2) 

Standardizing Normal Data Sets 


548  (2) 


550  (1) 

Normal Curves as Models of RealLife Data Sets 


551  (1) 

Normal Distributions of Random Events 


552  (2) 


554  (17) 


558  (1) 


559  (1) 


559  (10) 

References and Further Readings 


569  (2) 
Answers to Selected Problems 

571  (58) 
Index 

629  (6) 
Photo Credits 

635  