# Hint to Puzzle #35: When the Hands of a Clock First Overlap

35. A regular clock has an hour and minute hand. At 12 midnight the hands are exactly aligned. When is the next time they will exactly align or overlap? How many times a day will they overlap?

I think from the picture it's pretty obvious we are talking about a time just after five minutes past 1. We just need to work out exactly when that is.

We know the speeds of the hands of a clock in units like revolutions per hour, it's part of how we define a clock. The only difference between this 'angular velocity' and the linear speed you might be used to is every time you do a full rotation you start again.

A minute hand for example moves at 360° per hour, that is it's definition. Therefore if 't' is in hours the position of the minute hand, in degrees would be given by 360t.

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