''Chases and Escapes: The Mathematics of Pursuit and Evasion'' is a mathematics book on continuous

pursuit–evasion Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early ...

problems. It was written by Paul J. Nahin, and published by the

Princeton University Press Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, ...

in 2007. It was reissued as a paperback reprint in 2012. The Basic Library List Committee of the Mathematical Association of America has rated this book as essential for inclusion in undergraduate mathematics libraries.

Topics

The book has four chapters, covering the solutions to 21 continuous pursuit–evasion problems, with an additional 10 "challenge problems" left for readers to solve, with solutions given in an appendix. The problems are presented as entertaining stories that "breathe life into the mathematics and invite wider engagement", and their solutions use varied methods, including the computer calculation of numerical solutions for differential equations whose solutions have no closed form. Most of the material was previously known, but is collected here for the first time. The book also provides background material on the history of the problems it describes, although this is not its main focus. Even before beginning its main content, the preface of the book begins with an example of pure evasion from known pursuit, the path used by the

Enola Gay The ''Enola Gay'' () is a Boeing B-29 Superfortress bomber, named after Enola Gay Tibbets, the mother of the pilot, Colonel Paul Tibbets. On 6 August 1945, piloted by Tibbets and Robert A. Lewis during the final stages of World War II, it be ...

to escape the blast of the nuclear bomb it dropped on Hiroshima. The first chapter of the book concerns the opposite situation of "pure pursuit" without evasion, including the initial work in this area by

Pierre Bouguer Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture". Career Bouguer's father, Jean Bouguer, one ...

in 1732. Bouger studied a problem of pirates chasing a merchant ship, in which the merchant ship (unaware of the pirates) travels on a straight line while the pirate ship always travels towards the current position of the merchant ship. The resulting

pursuit curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterize ...

is called a radiodrome, and this chapter studies several similar problems and stories involving a linearly moving target, including variations where the pursuer may aim ahead of the target and the

tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angl ...

curve generated by a pursuer that follows the target at constant distance. Chapter 2 considers targets moving to evade their pursuers, beginning with an example of circular evasive motion described in terms of a dog chasing a duck in a pond, with the dog beginning at the center and the duck moving circularly around the bank. Other variants considered in this chapter include cases where the target is hidden from view, and moving on an unknown trajectory. Chapter 3 considers "cyclic pursuit" problems in which multiple agents pursue each other, as in the

mice problem In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

. The fourth and final chapter is entitled "Seven classic evasion problems". It begins with a problem from

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of L ...

's ''

Mathematical Games A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...

'', the reverse of the dog-and-duck problem, in which a person on a raft in a circular lake tries to reach the shore before a pursuer on land reaches the same point. It also includes hide-and-seek problems and their formulation using game theory, and the work of

Richard Rado Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from th ...

and

Abram Samoilovitch Besicovitch Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian mathematician, who worked mainly in England. He was born in Berdyansk ...

on a man and lion of equal speed trapped in a circular arena, with the lion trying to catch the man, first popularized in ''

A Mathematician's Miscellany ''A Mathematician's Miscellany'' is an autobiography and collection of anecdotes by John Edensor Littlewood. It is now out of print but ''Littlewood's Miscellany'' is its successor, published by Cambridge University Press and edited by Béla Bo ...

'' by J. E. Littlewood.

Audience and reception

The book assumes an undergraduate-level understanding of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

and

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s. It also uses some game theory but its coverage of the necessary material in this area is self-contained. It is not a textbook, but could be used to provide motivating examples for courses in calculus and differential equations, or as the basis of an undergraduate research project to a student who has completed this material. As well, the book may be of interest to any reader with the requisite background who enjoys mathematics. Game theorist Gerald A. Heuer writes that "The treatment in general is very good, and readers are likely to appreciate the author's friendly and lively writing style." On the other hand,

Mark Colyvan Mark Colyvan is an Australian philosopher and Professor of Philosophy at the University of Sydney. He is a former president of the Australasian Association of Philosophy. Colyvan is known for his research on philosophy of mathematics The ...

, a philosopher, would have preferred to see heavier coverage of the game-theoretic aspects of the subject, and notes that the mathematical idealizations used here can lead to inaccurate conclusions for real-world problems. Despite these quibbles, Colyvan writes that "this book provides an excellent vehicle to pursue the mathematics in question, and the mathematics in question is most certainly worth pursuing". Reviewer Bill Satzer calls the book "highly readable", and reviewer Justin Mullins writes that author Paul Nahin "guides us masterfully through the maths".

References

{{reflist, refs= {{citation , last = Colyvan , first = Mark , author-link = Mark Colyvan , date = December 2007 , journal = Australian Review of Public Affairs , title = The calculus of cat and mouse (review of ''Chases and Escapes'') , url = https://www.australianreview.net/digest/2007/12/colyvan.html {{citation , last = Dartnell , first = Lewis , date = December 1, 2007 , journal = Plus Magazine , title = Review of ''Chases and Escapes'' , url = https://plus.maths.org/content/chases-and-escapes {{citation , last = Heuer , first = G. A. , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ...

, mr = 2319182 , title = Review of ''Chases and Escapes'' , year = 2008 {{citation , last = Mahanti , first = Prabhat Kumar , journal =

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

, title = Review of ''Chases and Escapes'' , zbl = 1154.91006 {{citation , last = Mullins , first = Justin , date = June 27, 2007 , journal =

New Scientist ''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publish ...

, title = The hunter and the hunted (review of ''Chases and Escapes'') , url = https://www.newscientist.com/article/mg19426102-200-the-hunter-and-the-hunted/ {{citation , last = Puharic , first = Douglas , date = December 2013 – January 2014 , doi = 10.5951/mathteacher.107.5.0394 , issue = 5 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor = 10.5951/mathteacher.107.5.0394 , pages = 395 , title = Review of ''Chases and Escapes'' , volume = 107 {{citation , last = Satzer , first = William J. , date = June 2007 , journal = MAA Reviews , publisher = Mathematical Association of America , title = Review of ''Chases and Escapes'' , url = https://www.maa.org/press/maa-reviews/chases-and-escapes-the-mathematics-of-pursuit-and-evasion {{citation , last = Sonnabend , first = Thomas , date = March 2008 , issue = 7 , journal =

, jstor = 20876207 , page = 558 , title = Review of ''Chases and Escapes'' , volume = 101 {{citation , last = Tabachnikov , first = Serge , authorlink = Sergei Tabachnikov , date = March 2009 , doi = 10.1007/s00283-009-9036-z , issue = 2 , journal =

The Mathematical Intelligencer ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released qua ...

, pages = 78–79 , title = Review of ''Chases and Escapes'' , volume = 31 Pursuit–evasion Mathematics books 2007 non-fiction books