You are here

Pristine Landscapes in Elementary Mathematics

Titu Andreescu, Cristinel Mortici, and Marian Tertiva
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a good book with a terrible title. The mathematics is not elementary, but is a mixture of mostly discrete math with some continuous math (roughly the same level and topics as Graham & Knuth & Patashnik’s Concrete Mathematics). Each chapter covers one “landscape” in mathematics; that is, one particular topic, followed by a large number of ramifications of the topic. For example, the first chapter is about the pigeonhole principle and its many uses. The structure of each chapter is: a few pages expounding its topic and giving examples, and then a large number of exercises (with complete solutions) that use these ideas. These landscapes are “pristine” in the sense that high school and college students have probably never seen them before.

The publisher specializes in Olympiad preparation books, and there is some flavor of that here. But it is much broader and has a mixture of easy, moderate, and difficult problems (a few solutions run to several pages). In general there is no preparation or any hints for the difficult problems, and they are not broken down in steps as in many problem books. The problems come from many sources, including local and global Olympiads, journal problem columns, and the Putnam exam. Many of the solutions depend on making a clever observation before applying systematic methods. I think some of the problems are closer to puzzles and brain teasers (especially those dealing with base-10 digits of numbers) than to serious mathematics, but these are in the minority.

Each chapter is independent of the others, and they cover a wide variety of topics. My two favorite chapters are Chapter 7 on quadratic functions and Chapter 11 on recurrences. The observation that squares are always non-negative is very powerful and comes up unexpectedly in many parts of mathematics; Chapter 7 shows some of these. Chapter 11 does not deal with the common topic of solving recurrences explicitly, but with the uncommon (but very useful) topic of inferring asymptotic and limiting behavior of the sequence from its recurrence.

Bottom line: an idiosyncratic but interesting book that would be useful for enrichment work at the high school or college level. If you like this approach, check out the same authors’ book Mathematical Bridges, that has a similar approach but is more advanced and slanted more to real analysis and abstract algebra.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.