Question 150
Saba is trying to get from the ground floor of an infinitely tall skyscraper to the $n$th floor using a special elevator. The elevator has buttons $0$, $1$, $2$, ... and it moves alternately up/down (first it moves up). Moreover, upon pressing button $k$, the elevator moves by $2^k$ floors, and each subsequent button pressed must have a smaller number than the previous button. Prove that Saba can reach each floor $n \ge 1$ in exactly two different ways. (Morteza Saghafian)