Geometric Etudes in Combinatorial Mathematics

1. Front Matter

   Pages i-xxxv

   PDF

2. Original Etudes

   1. Front Matter

      Pages 1-1

      PDF

3. ORIGINAL ETUDES

   1. Tiling a Checker Rectangle

      • Alexander Soifer

      Pages 3-60

   2. Proofs of Existence

      • Alexander Soifer

      Pages 61-83

   3. A Word About Graphs

      • Alexander Soifer

      Pages 85-127

   4. Ideas of Combinatorial Geometry

      • Alexander Soifer

      Pages 129-212

4. New Landscape, or The View 18 Years Later

   1. Front Matter

      Pages 214-214

      PDF

5. NEW LANDSCAPE, OR THE VIEW 18 YEARS LATER

   1. Mitya Karabash and a Tiling Conjecture

      • Alexander Soifer

      Pages 215-219

   2. Norton Starr’s 3-Dimensional Tromino Tiling

      • Alexander Soifer

      Pages 221-225

   3. Large Progress in Small Ramsey Numbers

      • Alexander Soifer

      Pages 227-230

   4. The Borsuk Problem Conquered

      • Alexander Soifer

      Pages 231-234

   5. Etude on the Chromatic Number of the Plane

      • Alexander Soifer

      Pages 235-246

6. Back Matter

   Pages 251-264

   PDF

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words. A painter may embody an ‘idea,’ but the idea is usually commonplace and unimportant. In poetry, ideas count for a great deal more; but as Housman insisted, the importance of ideas in poetry is habitually exaggerated... A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must ?t together in a harmonious way. Beauty is the ?rst test: there is no permanent place in the world for ugly mathematics. —G.H.Hardy, A Mathematician’s Apology, 1940 [Har, pp. 24–25] I grew up on books by Isaac M. Yaglom and Vladimir Bolty- ski. I read their books as a middle and high school student in Moscow. During my college years, I got to know Isaak Moiseevich Yaglom personally and treasured his passion for and expertise in geometry and ?ne art. In the midst of my xxv xxvi Preface college years, a group of Moscow mathematicians, including Isaak Yaglom, signed a letter protesting the psychiatric - prisonment of the famous dissident Alexander Esenin-Volpin.

• Combinatorics
• Pigeonhole principle
• algebra
• combinatorial geometry
• graph theory
• polyominoes

From the book reviews:

“This book itself has also a good chance to occupy a permanent place in the mathematical literature. Among its virtues is the lively and fluent style, in which it introduces and explains the problems. … In summing up, we warmly recommend this book to any interested reader: take and read, and dip into the exercises and the problems … .” (Gábor Gévay, Acta Scientiarum Mathematicarum (Szeged), Vol. 77 (3-4), 2011)

Characteristically, each of the topics included in the book require very little in the way of preparation and evolve fast into open questions and research level conjectures...This is a delightful book that will be welcomed enthusiastically by students and organizers of mathematical circles and mathematics fans.---Alexander Bogomolny

Boltyanski and Soifer have titled their monograph aptly, inviting talented students to develop their technique and understanding by grappling with a challenging array of elegant combinatorial problems having a distinct geometric tone. The etudes presented here are not simply thoese of Czerny, but are better compared to the etudes of Chopin, not only technically demanding and addressed to a variety of specific skills, but at the same time possessing an exceptional beauty that characterizes the best of art...Keep this book at hand as you plan your next problem solving seminar. ---The American Mathematical Monthly

 

• at Colorado Springs, College of Letters, Arts, and Sciences, University of Colorado, Colorado Springs, USA

  Alexander Soifer

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