This book was based on lectures and conversations that I held several times with schoolchildren of either VII-VIII or IX-X classes in the school mathematical lecture hall at Moscow State University named after M.V. Lomonosov. I hope that this book will awaken from the reader not only curiosity, but also mathematical activity. The latter may manifest itself in the fact that the reader will go the way recommended to the listeners of my lectures-conversations: first, he will get acquainted with examples of erroneous reasoning set forth in chapters I (for schoolchildren, starting from the 7th grade of high school), and III (for IX-X grades); then in each case it will try to open the error on its own; finally, he will read chapters II and IV, where he will find explanations, respectively, to chapters I and III, as well as some additions. The fine print and a significant part of the footnotes can be skipped: they are designed for readers, the most prepared, as well as for leaders of mathematical circles.