Customer Reviews

Old and New Unsolved Problems in Plane Geometry and Number Theory (Dolciani Mathematical Expositions)

5 star:		(2)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

If progress towards the solution of the unsolved problems is to be continuous, then those in the field must be reminded on a regular basis. Furthermore, it is pretty clear that many of these problems will not be solved using "traditional" approaches, but by some sort of back or side door method. Therefore, those in other areas of mathematics need to see the unsolved problems presented. One never knows when someone working in an area that appears to be unrelated will read a problem and realize that what they are doing can be used to obtain a solution.
This work not only presents the problems, but also gives enough background to allow the non-specialist to understand them. Several leading theorems and their proofs are given, all in a clear, concise style. Many exercises are given at the end of sections, with hints toward solution at the back of the book.
The methods used would also allow this book to be used as a text in an advanced undergraduate colloquium or beginning graduate seminar in the golden oldies of mathematics. The competent undergraduate will have no difficulty comprehending the majority of the material.
An excellent addition to any collection, this book is a key. For it is clear that many of the unsolved problems will be defeated by one who never started with that intent, but saw the problem and had the correct leap of insight.

Published in Journal of Recreational Mathematics, reprinted with permission.

This is a very cool book. It's designed to be used by people with a range of mathematical knowledge and talent. There are easier problems given along with the very hard unsolved ones, and every topic is discussed in two separate places, once at an elementary level and once at a deeper level for people with more training.

My big misgiving about this book is that it's way out of date. They produced a new edition in 1996, but it hasn't been updated since then. For example, problem 1.2 (a conjecture about illuminable polygons) was solved in 1995, but that fact didn't make it into the 1996 edition. Fermat's last theorem, problem 13 in the book, was also been solved in 1995, but that was also too late to make it into the book. Normality of pi, problem 21, has not been cracked, but significant progress has been made in the last decade. Other problems, like the various conjectures surrounding different versions of Conway's angel problem, would presumably have been good ones to include in the book if it had been updated since 1996.

So if you're going to use this book, you'd better realize that you're missing quite a lot of information, and you'll need to supplement it with more information gleaned from the internet, e.g., from Wikipedia's article on "Unsolved problems in mathematics."

I admit I didn't read the entire book. But don't think that discredits my review. You see this is a special math book... One that you don't read cover to cover. You simply skim through the book and pick a unsolved problem that interest you. So you see it isn't read cover to cover like a textbook. And that is what makes this such an excellent math book. The book isn't about remembering rules it is about problem solving. And the organization of the book helps in gathering facts and understanding how others have approached the problem.

Unsolved problems is part of what mathematics are based on. Most of the content is easy to understand at undergraduate level. For fun I recommend only reading the problem's description and do your own research and later compare it to the second section of the book. I have worked on Prime numbers and have made some progress. (Just check my profile.) Math work does not get done without math problems. So if you are looking for a learning experience this is an excellent place to start.