''Problem Solving Through Recreational Mathematics'' is a textbook 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 modern mathematics ...

problem solving Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...

techniques and their application to problems in recreational mathematics, intended as a textbook for general education courses in mathematics for

liberal arts education Liberal arts education (from Latin "free" and "art or principled practice") is the traditional academic course in Western higher education. ''Liberal arts'' takes the term ''art'' in the sense of a learned skill rather than specifically the ...

students. It was written by

Bonnie Averbach Bonnie Averbach (née Rudolph; 1933 – October 16, 2019) was an American mathematics and actuarial science educator who worked for many years on the faculty of Temple University and was known for her books in mathematics. Life and career Aver ...

and Orin Chein, published in 1980 by

W. H. Freeman and Company W. H. Freeman and Company is an imprint of Macmillan Higher Education, a division of Macmillan Publishers. Macmillan publishes monographs and textbooks for the sciences under the imprint. History The company was founded in 1946 by William H. ...

, and reprinted in 2000 by Dover Publications.

Audience and reception

''Problem Solving Through Recreational Mathematics'' is based on mathematics courses taught by the authors, who were both mathematics professors at Temple University. It follows a principle in mathematics education popularized by George Pólya, of focusing on techniques for mathematical problem solving, motivated by the idea that by doing mathematics rather than being told about its "history, culture, or applications",

students (for whom this might be their only college-level mathematics course) can gain a better idea of the nature of mathematics. By concentrating on problems in recreational mathematics, Averbach and Chein hope to motivate students by the fun aspect of these problems. However, this approach may also lead the students to lose sight of the important applications of the mathematics they learn, and contains little to no material on mathematical proof. The book's exercises include some with detailed solutions, some with less-detailed answers, and some that provide only hints to the solution, providing flexibility to instructors in using this book as a textbook. Cartoons and other illustrations of the concepts help make the material more inviting to students. As well as for general education at the college level, this book could also be used to help prepare students going into mathematics education, and for mathematics appreciation for secondary school students. It could also be used as a reference by secondary school mathematics teachers in providing additional examples for their students, or as personal reading for anyone teenaged or older who is interested in mathematics. Alternatively, reviewer Murray Klamkin suggests using the books of Polyá for these purposes, but adding ''Problem Solving Through Recreational Mathematics'' as a supplement to these books.

Topics

The book begins with an introductory chapter on problem-solving techniques in general, including six problems to motivate these techniques. The rest of the book is organized into eight thematic chapters, each of which can stand alone or be read in an arbitrary order. The topics of these chapters are: * Logic puzzles, especially focusing on " Knights and Knaves" types of puzzles in which some characters are truthful while others answer only falsely. * Word problems involving time and motion, with

continuous variable In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on all re ...

s and with solutions using algebra. * Number theory, particularly focusing on

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s, continuing the theme of word problems but with discrete variables for numbers of people, goods, or costs, and also including material on divisibility, prime numbers, and the

Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...

. * Numeral systems and cryptarithms. * Graph theory, including Euler tours and Hamiltonian cycles. *

Game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

and combinatorial game theory, including material on games of

perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...

and on the games of tic tac toe,

nim Nim is a mathematical two player game. Nim or NIM may also refer to: * Nim (programming language) * Nim Chimpsky, a signing chimpanzee Acronyms * Network Installation Manager, an IBM framework * Nuclear Instrumentation Module * Negative index met ...

, and

hex Hex or HEX may refer to: Magic * Hex, a curse or supposed real and potentially supernaturally realized malicious wish * Hex sign, a barn decoration originating in Pennsylvania Dutch regions of the United States * Hex work, a Pennsylvania Dutch ...

. *

Solitaire Solitaire is any tabletop game which one can play by oneself, usually with cards, but also with dominoes. The term "solitaire" is also used for single-player games of concentration and skill using a set layout tiles, pegs or stones. These game ...

games and puzzles, including

polyominoes A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in po ...

, peg solitaire, and the

15 puzzle The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles wide, leaving one unoccupied tile position ...

. *A collection of leftover problems which did not fit into any of the other chapters.

References

{{reflist, refs= {{citation , last = Austin , first = Keith , date = March 1982 , doi = 10.2307/3617323 , issue = 435 , journal = The Mathematical Gazette , jstor = 3617323 , pages = 71–72 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 66 {{citation , last = Dees , first = Roberta L. , date = November 1981 , issue = 3 , journal = The Arithmetic Teacher , jstor = 41190004 , pages = 54–55 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 29 {{citation , last = King , first = Richard H. , date = April 1981 , issue = 4 , journal = The Mathematics Teacher , jstor = 27962454 , pages = 301–302 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 74 {{citation , last = Klamkin , first = Murray S. , date = March 1983 , doi = 10.2307/2975564 , issue = 3 , journal =

The American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...

, jstor = 2975564 , pages = 216–218 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 90 {{citation , last = Mortimer , first = Mike , date = January 1982 , issue = 1 , journal = Mathematics in School , jstor = 30213691 , page = 35 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 11 {{citation , last = Wolfe , first = Hugh C. , bibcode = 1981PhT....34e..84A , date = May 1981 , doi = 10.1063/1.2914582 , issue = 5 , journal = Physics Today , pages = 84–84 , title = Review of ''Problem Solving Through Recreational Mathematics'' , volume = 34 Mathematics textbooks 1980 non-fiction books Recreational mathematics Problem books in mathematics