MATHEMATICAL LABYRINTHS. PATHFINDING

by Boris Pritsker

Mathematical Labyrinths. Pathfinding  provides an overview of various non-standard problems and the approaches to their solutions. The essential idea is a framework laid upon the reader on how to solve nonconventional problems - particularly in the realm of mathematics and logic. It goes over the key steps in approaching a difficult problem, contemplating a plan for its solution, and discusses set of mental models to solve math problems. The book is not a routine set of problems. It is rather an entertaining and educational journey into the fascinating world of mathematical reasoning and logic. It is about finding the best path to a solution depending on the information given, asking and answering the right questions, analyzing and comparing alternative approaches to problem solving, searching for generalizations and inventing new problems. It also considers as an important pedagogical tool playing mathematical and logical games, deciphering mathematical sophisms, and interpreting mathematical paradoxes. It is suitable for mathematically talented and curious students in the age range 10-20. There are many 'Eureka'- type, out of the ordinary, fun problems that require bright idea and insight. These intriguing and thought-provoking brainteasers and logic puzzles should be enjoyable by the audience of almost any age group, from 6-year-old children to 80-year-old and older adults.

324 pages, Hardcover

Published January 4, 2021

About the author

Boris Pritsker was born in Kiev, Ukraine, and studied mathematics at Kiev State Pedagogical University, where he graduated summa cum laude with a degree in Mathematics and Education (the equivalent of a U.S. Bachelor of Science). Pritsker served as a mathematics teacher in two high schools, including a specialized school for gifted and talented students featuring advanced curricula in algebra, geometry, trigonometry, and calculus. During this time, he developed a series of educational seminars exploring classic mathematical problems and advanced topics that extend beyond the standard secondary school curriculum. In algebra and trigonometry, Pritsker examined strategies for solving various types of equations and inequalities, and systems of equations and inequalities. Additionally, he applied non-traditional techniques to an in-depth analysis of Fibonacci numbers. The seminars highlighted several sophisticated topics rarely addressed in mathematical literature. These included utilizing Diophantine equations to identify common or extraneous solutions within trigonometric equations and applying non-traditional techniques to solve multivariable equations. Through his work with the Fibonacci sequence, Pritsker established the necessary and sufficient conditions for a natural number to be identified as a Fibonacci number.
In the realm of geometry, the seminars covered non-trivial construction problems involving inaccessible vertices, as well as Steiner’s and Mascheroni’s challenges—constructions using a straightedge only or a compass only. Along with traditional synthetic techniques, the coursework integrated isometric and homothetic transformation properties. The seminars prompted students to derive alternative proofs for fundamental theorems, such as Fagnano’s Problem, Apollonius’s Theorem, and the Nine-Point Circle. Beyond pure geometry, the coursework examined the relationship between the inversion in a circle and mechanical engineering applications. Additionally, students investigated the intersection of geometry and physics, utilizing Archimedes’ Law of the Lever to discover new geometric properties.
Pritsker also trained school math teams for participation in math competitions; several students from these math teams won the prestigious local and regional math Olympiad contests.

After immigrating to the United States, Pritsker earned an MBA magna cum laude from the Zicklin School of Business at Baruch College, City University of New York. A licensed CPA in New York State, he has spent the past twenty-seven years at CBIZ Inc., a national accounting, insurance, and financial consulting firm, where he rose to the position of Director.

Driven by a lifelong devotion to mathematical research and education, Pritsker has spent years exploring challenging topics in Euclidean geometry, algebra, trigonometry, and pre-calculus. His findings and original problems have been featured in educational journals across the Soviet Union, the United States, Singapore, and Australia.

Pritsker authored internationally acclaimed mathematics books “Geometrical Kaleidoscope”, Dover Publications, 2017, “The Equations World”, Dover Publications, 2019, “Mathematical Labyrinths. Pathfinding”, World Scientific, 2021, "Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions", CRC Press, Taylor & Francis Group, 2023, “Geometrical Kaleidoscope” Second Edition, World Scientific, 2024, and "Learning Geometry Through Problem Solving", World Scientific, 2026. The books have been featured with detailed positive reviews in the Mathematical Association of America and ZBmath web-sites. “Geometrical Kaleidoscope” was assigned a BLL rating and the MAA Basic Library List Committee suggested that undergraduate mathematics libraries consider the book for acquisition. “Mathematical Labyrinths. Pathfinding” was translated into Turkish language and published in 2025 year, “Matematikte Problem Çözme” - Zihin Fırtınaları Ve Sıra Dışı Yolculukl

Reviews

[Volodymyr Andrusenko · Rating 5 out of 5]

This title clearly proves that solving math and logical problems can be fun. One of the best chapters in this regard is “Thinking outside the box” written with sense of humor and giving many cute examples of unexpected approaches to solving logical gems.
The book talks about general strategies for solving problems and provides a very helpful guidance and excellent self-contained resource for developing problem-solving skills.
The author managed to put together not just a great mix of entertaining logical puzzles, sophisms, and paradoxes but serious math topics as well. He walks you through the problem-solving process step-by-step-by-step.
Take, for example, this cute brainteaser from chapter 1:
In one of the tourists kiosks in Paris the front page written in English listed the languages in which the suggested city tours for foreign visitors are run during a day. The list has German, Italian, Russian, and many other languages. What language concludes the list?
The problem looks illogical, not solvable, very confusing. But you get an amazing simple solution based just on the problem’s conditions analysis. The author explains the basics how to properly understand what a problem is about, what its question is, how to use it in selecting your problem-solving strategy. Going through the subsequent chapters you see that similar approaches can be used for any advanced math challenges as well, including very interesting techniques for rarely discussed in math literature geometric constructions with the restricted elements.
The author also walks you through helpful techniques for dealing with converse problems and inversion problems, gives numerous examples finding easy solutions just by restating a problem. He also shows how to invent new problems and encourages searching for alternative solutions; two chapters are devoted to these topics. Speaking about alternative solutions and logical thinking, there is a separate chapter devoted to card game preferans containing several nice compositions. I did not come across preferans mentioning in any math literature I read. This one is a cherry on top.
In my mind, among many books of this type, this book is a stand-out. It serves a dual purpose of logical thinking development and math problem-solving skills strengthening at the same time.

[Gregory Tabachnik · Rating 5 out of 5]

This book is a very nice book for understanding how to approach challenging problems by applying common sense and logic. It is designed for math lovers. But even if you are not one of those math enthusiasts, it should be still fun to read. Just concentrate on cute logic riddles and follow the author’s advice how to understand what the problem is about, what are the most important of the given conditions, what has to be found or proved. As you study the concepts and ideas from the first chapters, you proceed to more complicated stuff. You learn not just how to solve a problem, but why it is solved in such a way. There are several really difficult math problems aimed at more advanced readers with serious math background. I think one of the strengths of this book is in its multipurpose. It can be used by those who do not do well in math in school and is willing to learn to effectively study topics within math disciplines. It can be used by advanced students to hone their problems skills getting ready to Math Olympiads as well. The book is listed on Amazon as belonging to “Mathematics Recreation and games” and “Logic”. I don’t think this is a proper classification for the book. There are many serious math topics and very interesting ideas as well. For instance, I enjoyed two chapters in geometrical constructions with inaccessible points and isometric and homothetic plane transformations. You not just get multiple very efficient techniques simplifying the problems’ solutions but learn the necessity of rigorous proofs of all your steps in the constructions process. Interesting insights in restating a problem, utilizing related problems and converse problems, inventing new problems, and examining alternative solutions are considered in chapters 8-10. Very useful and rewarding.

[Viki · Rating 5 out of 5]

I love this book. The author did a marvelous job delivering his ideas on how to approach any logical (including intermediary and complex math challenges) problems conceptually.
For any similar math book, you would expect some unrelated topics presented in several chapters independently. Surprisingly, the author managed to put together entertaining and highly educational issues, making it a very nice book. Despite the variety of seemingly unrelated topics, going through book exercises, you constantly utilize already covered techniques, just applying them under different circumstances!
Working with kids, you need to skip the advance stuff and focus on fun logical puzzles, explain the goal of understanding what the problem is about and what is required in it. As you proceed with more serious math, you have plenty of problems to work on.
Very enjoyable and useful!

[Tatyanak · Rating 5 out of 5]

I really enjoyed reading this book. Very interesting and logical.

[Gary Ravinsky · Rating 5 out of 5]

Very thought provoking, challenging, and intriguing math and logic book. Awesome.