Both hands move smoothly and continuously, the way real clock hands do.
At what exact time will the minute hand next lie directly on top of the hour hand? Give the answer to the fraction of a minute.
💡 Hint
Turn the chase into degrees: find how fast each hand turns, how big the gap is at the start, and divide the gap by the difference in speeds.
Reveal answer
The hands meet at 3:16 and 4/11 minutes, which is about 3:16:22.
Step-by-step solution
- The minute hand travels 360 degrees in 60 minutes, so 6 degrees per minute.
- The hour hand travels 30 degrees (one hour mark) in 60 minutes, so 0.5 degrees per minute.
- At 3:00 the hour hand is 3 × 30 = 90 degrees ahead of the minute hand.
- The minute hand gains 6 − 0.5 = 5.5 degrees per minute. To close a 90-degree gap it needs 90 ÷ 5.5 = 180/11 minutes, which is 16 and 4/11 minutes.
- Check: after 180/11 minutes the minute hand is at 6 × 180/11 = 1080/11 degrees, and the hour hand is at 90 + 0.5 × 180/11 = 1080/11 degrees. They match. Since 4/11 of a minute is about 21.8 seconds, the moment is roughly 3:16:22.
Why it fools people
We read clocks as a set of labeled positions, so our instinct is to think the hands meet at a labeled spot. In reality the hour hand never stops moving. Every minute it slips half a degree further, so the target keeps sliding away as the minute hand chases it. Turning the problem into a race with a head start and a closing speed is the move that cuts through the confusion.
Did you get it?
About This Puzzle
On the town square of a small Midwestern county seat, the old courthouse clock has kept time for longer than anyone in the diner across the street can remember. A retired schoolteacher who eats lunch there every day likes to quiz the regulars. Her favorite question sounds like something a child could answer in a second, and almost nobody gets it right on the first try. Everyone knows the hands of a clock line up at noon. The question is where they line up next after three o’clock, and the honest answer is not a round number. It is a lovely example of how a familiar object hides a real piece of mathematics.
How to Think About It
Stop thinking in hours and start thinking in degrees. A full circle is 360 degrees, so you can work out how many degrees each hand travels per minute. The minute hand makes a full lap every hour. The hour hand moves only one twelfth of a lap in that same hour. At 3:00 the hour hand has a head start, measured in degrees. The question then becomes a chase: one runner is faster, one is slower, and you want to know when the faster one catches up. Use the difference in speed, not the speeds themselves.
Common Traps
The most popular answer is 3:15, because at a quarter past the minute hand points at the 3. But by then the hour hand has not stayed put. It has crept a quarter of the way toward the 4, so the hands are still apart. The second trap is to fix that by adding one more minute and calling it 3:16. That is close, but the puzzle asks for the exact moment, and the true answer is not a whole number of minutes. Guessing and adjusting will never land on it exactly.
Where This Puzzle Comes From
Clock-hand problems have been a staple of arithmetic textbooks and puzzle collections for well over a century, and they still appear in aptitude tests and job interviews today. They belong to the larger family of “pursuit” or “catch-up” problems, which also includes trains leaving stations and runners on a track. The same idea, a gap divided by a difference in speed, solves all of them.
Try This Variation
Same clock, new question. At 7:00 the hour hand points at the 7 and the minute hand at the 12. At what exact time, before 8:00, will the two hands point in exactly opposite directions, forming a straight line through the center?
Show the answer to the variation
At 7:05 and 5/11 minutes. At 7:00 the hour hand is 210 degrees ahead. The hands are opposite when that lead shrinks to 180 degrees, so the minute hand must gain 30 degrees. At 5.5 degrees per minute that takes 30 ÷ 5.5 = 60/11 minutes, or 5 and 5/11 minutes, about 7:05:27.