How does one cut a triangle?

ISBN 0-940263-01-7
Alexander Soifer is a wonderful problem solver and inspiring teacher. His book will tell young mathematicians what mathematics should be like, and remind older ones who may be in danger of forgetting. This review has the simple aim of persuading as many people as possible to read it . . .

So why am I urging you to read this? Mainly because it is such a refreshing book. Professor Soifer makes the problems fascinating, the methods of attack even more fascinating, and the whole thing is enlivened by anecdotes about the history of the problems, some of their recent solvers, and the very human reactions of the author to some beautiful mathematics. Most of all, the book has charm, somehow enhanced by his slightly eccentric English, sufficient to carry the reader forward without minding being asked to do rather a lot of work.

I am tempted to include several typical quotations but I'll restrain myself to two: From Chapter 8 "Here is an easy problem for your entertainment. Problem 8.1.2. Prove that for any parallelogram P, S(P)=5. Now we have a new problem, therefore we are alive! And the problem is this: what are all possible values of our newly introduced function S(F)? Can the function S(F) help us to classify geometry figures?"

And from an introduction by Cecil Rousseau:

"There is a view, held by many, that mathematics books are dull. This view is not without support. It is reinforced by numerous examples at all levels, from elementary texts with page after page of mind-numbing drill to advanced books written in a relentless Theorem-Proof style. "How does one cut a triangle?" is an entirely different matter. It reads like an adventure story. In fact, it is an adventure story, complete with interesting characters, moments of exhilaration, examples of serendipity, and unanswered questions. It conveys the spirit of mathematical discovery and it celebrates the event as have mathematicians throughout history."

And this isn't just publishers going over the top - it's all true!

-- JOHN Baylis in The Mathematical Gazette

We do not often have possibilities to look into a creative workshop of a mathematician. Usually mathematicians give an account of their results in a ground-out and logically irreproachable form. But their creative pains, methods of investigation and means of obtaining results remain vague, especially for other mathematicians. And therefore every possibility to observe creative methods employed by a mathematician is interesting and useful. In particular, it is important for a beginner, that is, for a schoolboy or a schoolgirl who is interested in mathematics. First of all, it is important because pupils (and even first-year students in mathematical or technical colleges), as a rule, do not imagine what modern mathematics is, its scientific problems and its methods of investigation. School mathematics is far from authentic modern science with respect to its contents and methods.

. . . The beginner, who is interested in the book, not only comprehends a situation in a creative mathematical studio, not only is exposed to good mathematical taste, but also acquires elements of modern mathematical culture. And (not less important) the reader imagines the role and place of intuition and analogy in mathematical investigation; he or she fancies the meaning of generalization in modern mathematics and surprising connections between different parts of this science (that are, as one might think, far from each other) that unite them . . .
This makes the book alive, fresh, and easily readable. Alexander Soifer has produced a good gift for the young lover of mathematics. And not only for youngsters; the book should be interesting even to professional mathematicians.

-- V. G. BOLTYANSKI in SIAM Review

Soifer's work can rightly be called a "mathematical gem."

-- JAMES N. BOYD in Mathematics Teacher

This delightful book considers and solves many problems in dividing triangles into n congruent pieces and also into similar pieces, as well as many extremal problems about placing points in convex figures. The book is primarily meant for clever high school students and college students interested in geometry, but even mature mathematicians will find a lot of new material in it. I very warmly recommend the book and hope the readers will have pleasure in thinking about the unsolved problems and will find new ones.


It is impossible to convey the spirit of the book by merely listing the problems considered or even a number of solutions. The manner of presentation and the gentle guidance toward a solution and hence to generalizations and new problems takes this elementary treatise out of the prosaic and into the stimulating realm of mathematical creativity. Not only young talented people but dedicated secondary teachers and even a few mathematical sophisticates will find this reading both pleasant and profitable.

-- L. M. KELLY in Mathematical Reviews

Since students are not getting enough geometry in school, one of the avenues left is self-study. Any new attractive books in this field are indeed very welcome. How does one cut a triangle? is one of these and is mostly combinatorial geometry that is accessible to bright high school and college students even without too much prior knowledge. Also, there is quite a number of unsolved problems which will challenge professional mathematicians and other scientists. There are various monetary awards for the first received acceptable solutions of some of these latter problems. Like the challenge of climbing mountains because they are, there is a mental challenge of solving the problems not only because they are here (in this book), but also because they are simple to comprehend but still challenging...I like the style of the book. In this regard, I agree with Cecil Rousseau who, in an introduction, says "it is an adventure story, complete with interesting characters, moments of exhilaration, examples of serendipity, and unanswered questions."

-- MURRAY S. KLAMKIN in The American Mathematical Monthly

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