3. A mad constructor created a clock with 150 hands. The first hand rotates at a speed of one revolution per hour, the second hand makes 2 revolutions per hour, ..., the 150th hand makes 150 revolutions per hour. The clock was started from a position where all hands were pointing straight up. When two or more hands meet during the operation of the clock, these hands immediately fall off. After how much time from the start will the hand rotating at a speed of 74 revolutions per hour fall off?
Answer: In 20 minutes. Solution. The first meeting of the hands will occur when the fastest 150th hand catches up with the slowest first hand. After this, they will fall off, and we can forget about them. The second meeting will happen when the fastest of the remaining, the 149th, catches up with the slowest of the remaining second. Reasoning similarly further, we realize that the 74th hand will fall off together with the 77th. Since the 77th hand makes three more revolutions per hour than the 74th, it catches up with the 74th at a speed of 3 revolutions per hour and will first catch up with it after $1 / 3$ of an hour.