In the lecture I will talk about some mathematical puzzles that have preoccupied me over the years, and I will also reveal to young people in the audience some of the secrets of our trade. The first puzzle is about high dimensional trees: what they are and how to count them. The second puzzle deals with high dimensional geometric bodies, and a question of Borsuk. The third puzzle is about errors in counting votes in elections, and the forth puzzle is: are computationally superior quantum computers feasible?