In the lecture, Gil Kalai will talk about some mathematical puzzles that have preoccupied him over the years, and he will also reveal to young people in the audience some of the secrets of the 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 fourth puzzle is: are computationally superior quantum computers feasible?