How many times will both needles of a clock overlap each other in a day?

Concept: The minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, so the minute hand gains on the hour hand at a relative rate of 5.5 degrees per minute. The two hands overlap each time this relative gain completes a full 360-degree turn. Because of this relative-speed effect, on any 12-hour clock face the hands overlap only 11 times, not 12 - the last overlap of one 12-hour cycle coincides with the first overlap of the next.

Application:

1. A day has 24 hours, which is exactly two 12-hour periods (00:00-12:00 and 12:00-24:00).

2. Each 12-hour period contributes 11 overlaps, so the day's total is 11 + 11 = 22 overlaps.

Cross-check:

1. Time between consecutive overlaps = 360 / 5.5 = 720/11, which is approximately 65.45 minutes.

2. Number of overlaps in 24 hours = (24 x 60) / (720/11) = 1440 x 11 / 720 = 22, confirming the count above.

So the hands overlap 22 times in a day.