One of our colleagues used to have a digital alarm clock - well, this was in the days before everyone started using their phones as alarm clocks! In spite of being "steam-age" technology it worked very well; let's face it alarm clocks have only one purpose, and so it either wakes you or it doesn't.

There was one little oddity about it - setting the time after a power cut. If it lost power it displayed 00:00, flashing to draw your attention to it. There were two button to alter the time, one went forward and the other went back. And here is the thing - the forward button advanced the time by ten minutes every second, but the back button decreased the time by one minute every second.

The design was clearly intended to allow you to move quickly to just past the correct time, and then move slowly back to the correct time.

However, you know the type of people who set quizzes and puzzles, always curious and thinking about things! There must be a time of day when it's quicker to just go backwards to the correct time, rather than go forwards and then back.

Obviously if the current time is 23;59 it is far, far quicker to click back one minute than to go forwards and then back. But where is the break even point? At what time of day does it become quicker to go back to the time, rather than forwards?

If x is the number of seconds for which either button is pressed, the clock will advance by 10x minutes or go back x minutes. As there are 1440 minutes in 24 hours we are looking for the solution to

$$ 10x = 1440 - x $$

$$ 11x = 1440 $$

$$ x = \frac{1440}{11} = 130.9 $$

After 130 seconds the clock will have been "wound" back 2 hours 10 minutes (130 minutes) to 21:50. Or it will have been advanced 21 hours 40 minutes (1300 minutes) to 21:40.

So, it is quicker to go back to 21:50, but quicker to go forward to 21:40.

For the minutes in between (21:41 to 21;49) you can only reach these going back from 21:50. We have already seen that the quicker way to 21:50 is backwards, so it follows that the quicker way to these intermediate times is backwards.

If it is 21:41 or later you should go backwards and if it is 21:40 or earlier you should go forwards - provided that you can stop precisely on 21:40, of course!

Just for the record, using the same reasoning and maths, for a twelve-hour clock rather than 24-hour, you should go forward to 10:50 and backward to 10:51.