How many times a day do the hands of a clock make right angles with each other?

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To start off we are going to imagine what happens to the hands of a clock as time passes starting at 00:00. Hopefully this will help us put some shape on the problem and give us some direction.

Imagine the hands of a clock at 00:00, with both hands pointing to 12. Imagine what happens as time moves forward. The minute hand moves clockwise quickly and the hour hand moves clockwise slowly. As the hands move, the angle between them increases. When the minute hand passes the 3 (i.e. at roughly 00:16), what shape will the hands be? The minute hand will be just past the 3 and the hour hand will be just past the 12 so they will roughly be at right angles to each other.

Now imagine time moves forward and the minute hand is now approaching the 10 (i.e. at about 00:48). What shape are the hands in now? The minute hand is close to the 10 and the hour hand is close to the 1. The hands will be roughly at right angles to each other.

If we continue this process for another hour we can picture the hands making right angles at roughly 01:23 and 01:56 (you can try this yourself by setting the clock on a computer to these times, but don’t forget to change the time back!).

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