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Clock Hands Overlap: Puzzle and Step-by-Step Solution

  • Mental Math
  • Medium
  • Reported at Optiver, SIG, IMC, Akuna

The puzzle

How many times a day do the hour and minute hands of a clock overlap? At what time is the first overlap after 12:00?

Short answer: Twenty-two times in 24 hours, once every 12/11 hours (65 5/11 minutes). The first overlap after 12:00 is at about 1:05:27.

Step 1: Work in relative speed

The minute hand turns 360 degrees an hour, 6 degrees a minute. The hour hand turns 30 degrees an hour, 0.5 degrees a minute. So the minute hand gains 5.5 degrees a minute on the hour hand.

The hands overlap whenever the minute hand has gained a full lap of 360 degrees. That takes 360 / 5.5 = 720 / 11 minutes, which is 65 and 5/11 minutes, or 12/11 hours.

Step 2: Count the overlaps

In 12 hours the minute hand turns 12 times and the hour hand once, so the minute hand laps the hour hand 12 - 1 = 11 times. Equivalently, 12 hours divided by 12/11 hours is 11. In 24 hours, that is 22 overlaps.

The intuitive answer of 24, one per hour, is wrong because there is no overlap between 11:00 and 1:00 other than at 12:00 itself. The overlap that would be 'in the 11 o'clock hour' is the one at 12:00.

Step 3: The first overlap after noon

Start at 12:00 with both hands together. The next overlap is 720/11 minutes later, which is 65.4545... minutes, or 1 hour, 5 minutes and 5/11 of a minute. Five elevenths of a minute is 300/11 seconds, about 27.27 seconds. So the first overlap is at about 1:05:27.

The k-th overlap after 12:00 is at 12k/11 hours, so the overlaps fall at 1:05:27, 2:10:55, 3:16:22 and so on, each about 65 minutes and 27 seconds apart.

Step 4: Common follow-ups

How many times are the hands at right angles? They are at 90 degrees twice per lap, once at +90 and once at -90, so 2 × 22 = 44 times a day. How many times are they exactly opposite? Once per lap: 22 times a day.

What is the angle at 3:15? The minute hand is at 90 degrees. The hour hand is at 90 + 15 × 0.5 = 97.5 degrees. The angle is 7.5 degrees, not 0, which is the trap the question sets.

Why interviewers ask it

Lead with relative speed, 5.5 degrees a minute; every variant follows from it. Saying why the answer is 22 and not 24 is the part interviewers care about.

Knowing the proof is half of it. Work Clock Hands Overlap interactively, then take it under a clock with the rest of the Outcry simulators.

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