How many times do the hands of a clock coincide in a day?

In 12 hours, the hour hand completes exactly 1 full revolution while the minute hand completes 12 revolutions, so the minute hand gains 11 full revolutions on the hour hand. This means the minute hand catches up to and overtakes the hour hand 11 distinct times within each aligned 12-hour stretch (e.g., 12:00 to the next 12:00): once at the very start of the stretch, and 10 more times before its end -- the coincidence that would occur exactly at the end of the stretch is the very same instant as the start of the NEXT aligned 12-hour stretch, so it belongs to that next stretch's count, not an extra one added to this stretch.

A full day is two such aligned 12-hour stretches: 12 AM to 12 PM, and 12 PM to 12 AM. Each stretch contributes its own 11 coincidences by the concept above, with no instant shared or double-counted between the two stretches, so the total across the full 24-hour day is 11 + 11 = 22.

This can be verified independently using relative speed: the minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, a relative gain of 5.5 degrees per minute. To lap the hour hand by a full 360 degrees takes 360 / 5.5 = 720/11 minutes, i.e. about 65 minutes 27 seconds. Dividing the full day (24 x 60 = 1440 minutes) by this interval gives 1440 / (720/11) = 1440 x 11 / 720 = 22, confirming the count found above.

Hence, the hands of a clock coincide 22 times in a day.