A clock is broken and is stuck at 3:15. If it stays stuck at this time for 24 hours, how many times will the hands of the clock form a right angle during that period?
The Broken Clock Riddle
Hint:
To solve this, focus on the right angles between the clock’s hands. A right angle occurs when the hands are 90 degrees apart.
Think about how often the hands align at a 90-degree angle in an hour. It happens twice: once when the minute hand moves ahead and then again when it moves behind the hour hand.
Now, if the clock were working, the right angle would happen twice every hour. But since it’s stuck at 3:15, you need to figure out how often this configuration repeats itself within 24 hours.
The key to solving this riddle is understanding how often the hands of a clock form a right angle. When the clock is stuck at 3:15, we need to see how often the hands are 90 degrees apart during a 24-hour period.
How Right Angles Occur on a Clock:
The minute hand moves 360° every hour (or 6° per minute), and the hour hand moves 30° every hour (or 0.5° per minute).
A right angle occurs when the angle between the hour and minute hands is 90°.
At 3:15, the hands form a right angle, as the minute hand is on the 3, and the hour hand is slightly past the 3 (since it’s 15 minutes into the hour).
Frequency of Right Angles:
Normally, two right angles occur every hour: one when the minute hand is ahead of the hour hand, and one when the minute hand is behind it.
However, the situation changes when the clock is stuck at 3:15. Every time the clock is at 3:15 (and the hands are at a 90° angle), the cycle repeats after 12 hours.
Conclusion:
Since the clock stays stuck at 3:15 for 24 hours, it forms a right angle twice during that time, once in the first 12 hours and once in the second 12 hours.
So, the total number of right angles formed during the 24-hour period is 2.
