Brainteasers & Puzzles

When the Hands Overlap

After 3:00 the minute hand gains 5.5 degrees per minute on a 90 degree deficit, so the hands first coincide 180/11 minutes past the hour, at 3:16:21 and 9/11 seconds.

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What is the first time after 3pm when the hour and minute hands of a clock are exactly on top of each other?

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#Set the two angles equal

Measure each hand in degrees clockwise from 12, using mm for the minutes past 3. The minute hand sweeps 6∘6^\circ per minute, so it sits at 6m6m. The hour hand begins at the 3, which is 90∘90^\circ, and creeps on at 0.5∘0.5^\circ per minute, so it sits at 90+0.5m90 + 0.5m. They coincide when

6m=90+0.5m.(1)6m = 90 + 0.5m. \tag{1}

#Solve for the minute

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Starting at 3:00 the hour hand leads by 90 degrees, and the minute hand closes that gap at 5.5 degrees a minute. They meet after 180 over 11 minutes, so the first overlap past 3pm lands at 3:16 and 4/11 minutes, near 3:16:22.

Collecting terms leaves the minute hand gaining 5.5∘5.5^\circ each minute on a 90∘90^\circ lead,

5.5 m=90  ⟹  m=18011=16411.(2)5.5\,m = 90 \;\Longrightarrow\; m = \frac{180}{11} = 16\tfrac{4}{11}. \tag{2}

So the hands first align at 3:16 and 411\tfrac{4}{11} of a minute. Since 411\tfrac{4}{11} of a minute is 2191121\tfrac{9}{11} seconds, the exact moment is 3:16:21 and 911\tfrac{9}{11}, near 3:16:22. The same 1211\tfrac{12}{11} hour spacing separates every consecutive overlap around the dial.