You are here

Kvant Selecta: Combinatorics I

Serge Tabachnikov
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical World 17
[Reviewed by
Mihaela Poplicher
, on

This book is a collection of translations of articles from Russian Editions of the journal Kvant (meaning "Quantum"). The present volume is the third in a series of translations of articles from Kvant published by the American Mathematical Society; they all appeared in the Mathematical World series, as volumes 14, 15, and 17 respectively.

The articles in this volume are mainly on subjects from combinatorics and discrete mathematics. They are written mainly for very motivated high school students, interested in challenging problems and in getting ready for mathematical competitions. As such, they do not require a substantial mathematical background, but they certainly contain nontrivial material. The book is designed to be used by undergraduate students, as well as high school students and teachers interested in mathematical problems beyond the curriculum, problems that are both fun and challenging. Also, every person who loves mathematics will find this volume very entertaining and interesting.

The Russian magazine Kvant was founded in 1970 and has been enjoyed by many talented high school students ever since. Its every issue was always anticipated with joy not only by Russian students, but by students from other countries as well (as a personal memory: while a high school student in my native Romania, my class had a collective subscription to the magazine).

The articles included in the present volume appeared in Kvant between 1971 and 1987 (what a treat to be able to re-read some paper that I first read as a high school student!). Some of them, like those two on "Economics and Linear Inequalities", published in Kvant in 1971 by A. B. Katok, can be used today in a Linear Programming undergraduate course. Others, like "From the Life of Units", by A. L. Toom (published in Kvant in 1974), "Nonrepeating Sequences", by G. A. Gurevich (published in Kvant in 1975), or "Words with Restrictions", by A. M. Stepin and A. T. Tagi-Zade (original Kvant publication date 1975) are more removed from the regular curriculum.

There are a few articles about "fun and games", but with beautiful mathematical perspectives, included in the present selection: "Two Games with Matchsticks", by I. M. Yaglom (published in Kvant in 1971), or "Best Bet for Simpletons", by P. A. Pevzner (published in Kvant in 1987, with a reference to an 1974 article published by Martin Gardner in Scientific American).

Some articles include rigorous and fun mathematics used in "practical" problems: "How to Detect a Counterfeit Coin" (originally published in Kvant in 1979 by G. Shestopal), "The Generalized Problem of Counterfeit Coins" (published in Kvant in 1980 by M. Mamikon), "Planar Switching Circuits" (an article published initially in Russian in 1976 by S. Ovchinnikov), "Switching Networks" (an article regarding a problem from the Fifth All-Union Mathematical Olympiad, and published in Kvant in 1972 by R. V. Freivald).

A few more articles are included in the present collection, all challenging, fun, informative and interesting. Motivated and interested high school students or undergraduates will find them useful in "training" for mathematical competitions; teachers will find many of these articles (or some parts of them) useful as resource material for linear programming, logic, or combinatorics courses.

In short: the book is a real gem that will reward handsomely a patient, attentive and interested reader, but most of it is not "easy reading". I strongly recommend it to everybody who loves mathematics.

Mihaela Poplicher is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is [email protected].

  • I. M. Yaglom -- Two games with matchsticks
  • A. B. Katok -- Economics and linear inequalities
  • A. B. Katok -- Economics and linear inequalities (Continuation)
  • R. V. Freivald -- Switching networks
  • G. M. Adel'son-Vel'skii, I. N. Bernshtein, and M. L. Gerver -- Who will go to Rio?
  • A. L. Toom -- From the life of units
  • G. A. Gurevich -- Nonrepeating sequences
  • A. M. Stepin and A. T. Tagi-Zade -- Words with restrictions
  • S. Ovchinnikov -- Planar switching circuits
  • P. Bleher and M. Kel'bert -- Classification algorithms
  • G. Shestopal -- How to detect a counterfeit coin
  • M. Mamikon -- The generalized problem of counterfeit coins
  • P. Bleher -- Truthtellers, liars, and deceivers
  • V. A. Uspenskii and A. L. Semenov -- Solvable and unsolvable algorithmic problems
  • P. A. Pevzner -- Best bet for simpletons