It turned out to be a challenge to write a book like this one, keeping about the same level of difficulty through the entire text. In the end, here is a text which poses somewhat increasing difficulty, as one turns its pages. The topics come from a variety of fields of mathematics and even physics, and much impact is given to computation with mathematica. Some basic important theorem which are used in mathematics again and again are stated without proof. The intention is not to give any textbook. Instead, it is my hope to keep alive the interest of a broader group of young and older readers, who are not mainly concerned to dig themselves into a single field of speciality. Too, the reader will realize that I was not able to solve all problems. The intention is to show a somewhat realistic and honest picture of the working mathematician. This intention implies not to hide all awkward looking explicit calculations, and even more important, not to hide all failures. In the end the main purpose of the book is to sharpen the mathematical abilities of the reader, and my own ones, too, -while keeping an equilibrium of entertainment and challenge.