This book presents surprising results in mathematics that usually do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics.
The repository contains LaTeX / TikZ source code for five languages: English, French, Spanish, German, Hebrew. (Compile Hebrew with XeLaTeX.)
English
Published by Springer. Copyright 2022 by Mordechai Ben-Ari, Creative Commons Attribution. Download PDF and order the printed book at SpringerLink.
French
Translation by Nicolas Bacaër. Copyright 2023 by Nicolas Bacaër. Download PDF and order the printed book from CoolLibri or Cassini.
Spanish
This translation was prepared using DeepL and edited by Abraham Arcavi.
German
Published by Springer. Copyright 2025 by Mordechai Ben-Ari, Creative Commons Attribution. Download PDF and order the printed book at SpringerLink. The published book has some editing differences the LaTeX source code in the repository.
Hebrew
I wrote the translation which was edited by Rachel Zaks.
The collapsing compass Euclid proved that any construction that can be done using a fixed compass can performed with a collapsing compass.
Trisection of an angle Trisecting an angle is impossible but it can be done using instruments somewhat more complex that the straightedge and compass. Many claimed constructions are actually good approximations.
Squaring the circle Adam Kochansy and Ramanujan gave constructions with straightedge and compass for lengths that are very good approximations to pi.
The five-color theorem The proof that any graph can be colored with four colors is extremely difficult, but it is relatively simple to prove that it can be done with five or six colors. Alfred Kempe gave a proof of the four-color theorem that was later shown to be incorrect.
How to guard a museum A clever argument based on graph coloring shows that n/3 guards are sufficient to guard any museum with n walls.
Induction This chapter brings little-known inductive proofs in number theory, as well as a proof of McCarthy's 91 function and a presentation of the Josephus problem.
Solving quadratic equations Poh-Shen Loh's method of solving quadratic equations based on properties of their roots. The chapter also presents geometric solutions of the equations.
Ramsey theory This chapter presents simple cases of Pythagorean triples, van der Waarden’s problem and Ramsey’s theorem. The Pythagorean triples problem was recently solved with the aid of computer program for SAT solving. Pythagorean triples were known to the Babylonians four thousand years ago.
Langford's problem The problem is to arrange pairs of colored blocks in a row such that there is an arithmetical progression of blocks between each pair.
Origami Three chapters on the mathematics of origami: the seven axioms, Lill's method and Beloch's fold for finding roots of polynomials and constructions impossible by straightedge and compass trisecting an angle, squaring a circle and construcing a regular nonagon.
A compass is sufficient Mohr and Mascheroni showed that any construction that can be done with straightedge and compass can done with only a compass.
A straightedge and one circle is sufficient Steiner showed that any construction that can be done with straightedge and compass can done with only a straightedge provided that a single circle already exists somewhere in the plane.
Are triangles with the equal areas and perimeters congruent? No, non-congruent triangles can be constructed with the aid of elliptic functions.
Construction of a regular heptadecagon Gauss proved using only algebraic methods that a regular heptadecagon (17-sided polygon) can be constructed with a straightedge and compass.
Appendix with proofs of trigonometric identities and lesser-known theorems of geometry.