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'spurely 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 https: github.com motib suprises.
