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I was asked this question in an entrance exam.

In one day, how many times the hour hand and the minute hand of a clock are at right angles to each other?

My answer was 48. My reasoning was that during any duration of an hour, the minute hand would come at right angle with the hour hand and then it will spin to a direction opposite to its previous direction in a duration less than that of an hour and thus making another right angle with the hour hand. In this way, they got perpendicular twice an hour. Hence 24 hours a day should make them 48 times perpendicular in an entire day.

But many sources suggested an answer of 44. Could someone point out the flaw in my reasoning and justify the correct answer?

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    $\begingroup$ You understand that neither (say) 12:15 nor 12:45 have the hands at a right angle to each other, right? $\endgroup$
    – Sneftel
    Mar 21 at 13:08
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    $\begingroup$ Yes. But from 12:00 to 1:00 shouldn't they be perpendicular at least once since it's about to complete whole revolution? $\endgroup$
    – user81177
    Mar 21 at 13:13
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    $\begingroup$ Have you tried the simpler puzzle about how often in a day the two hands of the clock overlap (i.e. have and angle of zero degrees between them)? $\endgroup$
    – Jaap Scherphuis
    Mar 21 at 13:21
  • $\begingroup$ Yeah. That answer is 22. I get that. $\endgroup$
    – user81177
    Mar 21 at 13:31
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    $\begingroup$ And between each of those 22 times, how many times are the hands perpendicular? $\endgroup$
    – Sneftel
    Mar 21 at 13:32

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Here is a way I like to think about it

During the course of the day, the minute hand goes around the face 24 times, while the hour hand goes around twice.
Now consider a reference frame where the hour hand is fixed (a frame that rotates with the hour hand). In this frame, the minute hand goes around the clock to meet the hour hand 24-2 = 22 times. This means that, in a day, it will make a right angle with the hour hand 22 x 2 = 44 times.

Where does your reasoning fall down

You can work out explicitly what happens. In the hour hand reference frame, the speed of the minute hand is $\frac{11}{12}$ revolutions per hour (because once one hour is up, the minute hand has made it back to where it started, and the hour hand is exactly 5 "minutes" ahead of where it started). This means that the first right angle is seen after $15 \times \frac{12}{11}$ minutes and every other right angle comes exactly $30 \times \frac{12}{11}$ minutes after the previous one. This allows you to work out the times of the right angles.

What you find is that your intuition doesn't quite work out. For example, between 02:00:00 and 04:00:00 there are just three right angles rather than four, as you would expect (at roughly 02:27:16, 03:00:00 and 03:32:44). a similar thing happens between 08:00:00 and 10:00:00.

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