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  • Book
  • Open Access
  • © 2022

Mathematical Surprises


  • This book is open access, which means that you have free and unlimited access

  • Describes little known and surprising results in classical elementary Mathematics

  • Perfectly suitable for an extra-curriculum high-school Mathematics circle

  • Contains detailed explanations and high-quality illustrations

Buying options

Softcover Book USD 49.99
Price excludes VAT (USA)

Table of contents (16 chapters)

  1. Front Matter

    Pages i-xvi
  2. The Collapsing Compass

    • Mordechai Ben-Ari
    Pages 1-9Open Access
  3. Trisection of an Angle

    • Mordechai Ben-Ari
    Pages 11-28Open Access
  4. Squaring the Circle

    • Mordechai Ben-Ari
    Pages 29-39Open Access
  5. The Five-Color Theorem

    • Mordechai Ben-Ari
    Pages 41-52Open Access
  6. How to Guard a Museum

    • Mordechai Ben-Ari
    Pages 53-59Open Access
  7. Induction

    • Mordechai Ben-Ari
    Pages 61-72Open Access
  8. Solving Quadratic Equations

    • Mordechai Ben-Ari
    Pages 73-87Open Access
  9. Ramsey Theory

    • Mordechai Ben-Ari
    Pages 89-103Open Access
  10. Langford’s Problem

    • Mordechai Ben-Ari
    Pages 105-111Open Access
  11. The Axioms of Origami

    • Mordechai Ben-Ari
    Pages 113-130Open Access
  12. Lill’s Method and the Beloch Fold

    • Mordechai Ben-Ari
    Pages 131-140Open Access
  13. Geometric Constructions Using Origami

    • Mordechai Ben-Ari
    Pages 141-150Open Access
  14. A Compass Is Sufficient

    • Mordechai Ben-Ari
    Pages 151-162Open Access
  15. A Straightedge and One Circle is Sufficient

    • Mordechai Ben-Ari
    Pages 163-173Open Access
  16. Are Triangles with Equal Areas and Perimeters Congruent?

    • Mordechai Ben-Ari
    Pages 175-181Open Access
  17. Construction of a Regular Heptadecagon

    • Mordechai Ben-Ari
    Pages 183-198Open Access
  18. Back Matter

    Pages 199-226

About this book

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction.

Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass.

Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems.

Supplementary material to the book can be found at


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Authors and Affiliations

  • Department of Science Teaching, Weizmann Institute of Science, Rehovot, Israel

    Mordechai Ben-Ari

About the author

Mordechai (Moti) Ben-Ari is professor emeritus at the Department of Science Teaching of the Weizmann Institute of Science. He has written numerous textbooks including Mathematical Logic for Computer Science and Elements of Robotics (with Francesco Mondada), both published by Springer. He has received the ACM SIGCSE Award for Outstanding Contributions to Computer Science Education and the ACM Karl V. Karlstrom Outstanding Educator Award.

Bibliographic Information

  • Book Title: Mathematical Surprises

  • Authors: Mordechai Ben-Ari

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s) 2022

  • License: CC BY

  • Softcover ISBN: 978-3-031-13565-1Published: 08 September 2022

  • eBook ISBN: 978-3-031-13566-8Published: 07 September 2022

  • Edition Number: 1

  • Number of Pages: XVI, 226

  • Number of Illustrations: 147 b/w illustrations, 23 illustrations in colour

  • Topics: Mathematics, Computer Science

Buying options

Softcover Book USD 49.99
Price excludes VAT (USA)