## 264. Math Puzzle $5 / 87$

An analog clock is showing exactly the sixth hour. The hour and minute hands are at an angle of $180^{\circ}$. During the next hour, the clock hands form an angle of $110^{\circ}$ exactly twice.

How many minutes elapse between the two angles of $110^{\circ}$?
At 6 o'clock, the angle between the minute and hour hands is $180^{\circ}$. After that, the angle becomes smaller, and between 6 and 7 o'clock, the hour hand is caught up. Then the angle increases again, and at 7 o'clock, the hands form an angle of $150^{\circ}$. We observe a total angle change of exactly $330^{\circ}$ in 60 minutes.

We are looking for the time points at which the hands form an angle of $110^{\circ}$, i.e., for an angle change of $220^{\circ}$.

If the hands take exactly 60 minutes for $330^{\circ}$, then for $220^{\circ}$, exactly 40 minutes will pass.