A correct clock is watched from 12 o'clock right round to 12 o'clock again, that is for 12 hours. Again and again the minute hand catches the hour hand up and passes it. Not counting the starting moment, how many times does the minute hand overtake the hour hand in those 12 hours?

In 12 hours the minute hand goes round 12 times and the hour hand goes round once, so the minute hand gains exactly 12 − 1 = 11 whole laps on it, and each gained lap is one overtaking. So there are 11 overtakings, spread out about every 65 and a half minutes. The traps: 12 assumes one overtaking in every hour, which breaks down in the hour before 12 o'clock — there the meeting happens exactly at 12, the moment already counted as the last of the eleven; 24 uses two meetings an hour, which is the pattern for right angles, not for overtaking; 22 is the count for a whole day of 24 hours, twice the stretch asked about.