How to Solve Clock Coincidence (Hands Overlap) Problems
Solve clock coincidence problems: why hands overlap 11 times in 12 hours, the 5.5 degrees/minute rate, and a full worked example.
Expected Interview Answer
Clock hands coincide (overlap exactly) 11 times every 12 hours, not 12, because the minute hand must lap the hour hand — cover one extra full circle relative to it — and it does so at a relative speed of 5.5 degrees per minute, giving a coincidence every 720/11 ≈ 65.45 minutes.
In 12 hours the minute hand completes 12 full revolutions while the hour hand completes 1, so the minute hand gains exactly 11 full revolutions on the hour hand, producing 11 coincidences (the first at 12:00 counts, and the 11th lands back at 12:00, so there are 11 distinct overlaps, not 12). The relative angular speed between the hands is 6 − 0.5 = 5.5 degrees per minute, so a full 360-degree lap relative to the hour hand takes 360/5.5 = 720/11 minutes ≈ 65 minutes 27.27 seconds. Starting from any known coincidence (such as 12:00), successive coincidences occur every 720/11 minutes, so the nth coincidence after 12:00 is at (n × 720/11) minutes past 12. The same relative-speed idea, with different rates, solves “hands opposite” (180 degrees apart) or “hands at right angles” (90 degrees apart) problems.
- The 11-times-in-12-hours fact avoids the common "12 times" trap
- One relative-speed value, 5.5 degrees/minute, drives every coincidence timing
- The same method extends to opposite-hands and right-angle variants
AI Mentor Explanation
Two runners on a circular boundary track, one running exactly 12 laps while the other completes only 1 lap in the same time, meet each other 11 times, not 12, because the faster runner must lap the slower one — an extra full circle — to create each meeting after the first. Clock hands work identically: the minute hand’s 12 laps against the hour hand’s 1 lap in 12 hours produces exactly 11 coincidences, the same lapping logic that governs any two objects moving at different fixed circular speeds.
Worked example
Hour hand lead at 3:00
- 90 degrees
Time to close gap
- 90 ÷ 5.5 = 16 4/11 min
Coincidence time
- 3:16 and 4/11 minutes
Step-by-Step Explanation
Step 1
Find the hour hand’s head start
At H:00, the hour hand leads the minute hand by 30×H degrees.
Step 2
Apply the relative speed
The minute hand closes the gap at 5.5 degrees per minute.
Step 3
Solve for time to close the gap
Time = head-start degrees ÷ 5.5, giving minutes past H:00.
Step 4
Verify against the 11-per-12-hours rule
Coincidences recur every 720/11 minutes; check the answer fits that spacing.
What Interviewer Expects
- Correct statement that hands coincide 11 times, not 12, in 12 hours
- Correct use of the 5.5 degrees/minute relative speed to time a specific coincidence
- Ability to extend the method to opposite-hands (180°) and right-angle (90°) variants
- Correct handling of the fractional minute/second in the final answer
Common Mistakes
- Assuming the hands coincide 12 times in 12 hours instead of 11
- Using the minute hand’s absolute speed (6°/min) instead of the relative speed (5.5°/min)
- Forgetting the hour hand’s head start degrees when starting the calculation from H:00
- Rounding the fractional minutes incorrectly instead of keeping it as a precise fraction
Best Answer (HR Friendly)
“The key fact people often get wrong is that clock hands coincide 11 times in 12 hours, not 12, because the minute hand has to lap the hour hand an extra time to create each new coincidence. From there, I find how many degrees the hour hand is ahead at the starting hour, and divide by the 5.5-degrees-per-minute relative speed to find exactly when the minute hand catches up — the same relative-speed idea also answers when the hands are opposite or at right angles, just with a different target angle.”
Follow-up Questions
- How many times do the hands form a right angle in a 12-hour period, and why?
- How many times are the hands exactly opposite each other in 24 hours?
- How would you find the time between 7 and 8 o’clock when the hands are exactly opposite?
- How does the second hand change the analysis if you need all three hands to align?
MCQ Practice
1. How many times do the hour and minute hands coincide in a 24-hour period?
The hands coincide 11 times every 12 hours, so over 24 hours that is 11 × 2 = 22 times.
2. What is the time gap between two consecutive coincidences of the hour and minute hands?
720/11 minutes = 65 and 5/11 minutes, the fixed interval between consecutive coincidences.
3. Between 8:00 and 9:00, how many degrees ahead of the minute hand does the hour hand start?
At 8:00, the hour hand is at 30×8 = 240 degrees from 12, while the minute hand is at 0, so the gap is 240 degrees.
Flash Cards
How many times do hands coincide in 12 hours? — 11 times, not 12, because the minute hand must lap the hour hand.
Interval between consecutive coincidences? — 720/11 minutes ≈ 65 minutes 27.27 seconds.
Relative speed used in coincidence problems? — 5.5 degrees per minute (minute hand minus hour hand speed).
How to time a coincidence starting from H:00? — Divide the hour hand’s head-start degrees (30×H) by 5.5.