Coincidences of clock hands in 24 hours — How many times do the hour and minute hands coincide (overlap) in a day?

Introduction / Context:“Coincide” means the two hands line up exactly. Because the minute hand moves faster, it laps the hour hand slightly less than once per hour; the standard count for 12 hours is 11 overlaps.

Given Data / Assumptions:

• Relative speed between hands: 5.5° per minute.
• Between two successive overlaps, the minute hand must gain 360° on the hour hand.
• We consider a continuous 24-hour period (not just daytime).

Concept / Approach:Overlaps in 12 hours are 11 (not 12) due to the moving reference frame; one “potential” overlap shifts past the 12-hour boundary. Therefore, daily overlaps are exactly double that count.

Step-by-Step Solution:Overlaps in 12 hours = 11.Overlaps in 24 hours = 2 × 11 = 22.

Verification / Alternative check:Explicit formula: times (in minutes after H:00) are t = (60H)/11 for H = 0..10 within each 12-hour window, giving 11 solutions; repeat for the second 12 hours.

Why Other Options Are Wrong:24 assumes one per hour; 21 or 20 undercount; 22 is the established standard result.

Common Pitfalls:Forgetting the boundary effect that drops one overlap per 12-hour span.

Final Answer:22