In a 12-hour clock, the hour hand and the minute hand are initially at 12:00 (both pointing at 12). How many times do the hour and minute hands overlap in a 24-hour period?

In a 12-hour period, the hour and minute hands will overlap 11 times. This is because, every time the minute hand catches up to the hour hand, it happens a bit later than the previous overlap. The first overlap occurs at exactly 12:00, and the subsequent overlaps occur roughly every 1 hour and 5 minutes, but not exactly at the hour mark each time. After 12 hours, the hands overlap 11 times.

Since the pattern repeats every 12 hours, the hands will overlap 11 more times in the second 12-hour period.

Thus, in a 24-hour period, the hour and minute hands overlap 22 times (11 times in the first 12 hours, and 11 more times in the second 12 hours).