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Well done to those of you who sent in the correct answer to this problem. A lot of you explained your reasoning very carefully, but sadly we are unable to name you all here.

Cong from Aberdeen sent in this solution:

In $10$ minutes, the minute hand will sweep $60$ $^\circ$ degrees, because in $5$ minutes the minute hand will sweep $360\div12$ = $30$ $^\circ$.

In $3$ hours, the hour hand will sweep $90$ $^\circ$ degrees, because in $1$ hour the hour hand will sweep $360\div12$ = $30$ $^\circ$ .

If the minute hand goes through $180$ $^\circ$, the hour hand will sweep $15$ $^\circ$. The reason is as follows:

When the minute hand goes through $180$ $^\circ$, it is half an hour. In $1$ hour the hour hand will sweep $360\div12$ = $30$ $^\circ$ and $30$ $^\circ$ $\div2$ = $15$ $^\circ$. So when the minute hand sweeps $180$ $^\circ$, the hour hand will turn $15$ $^\circ$.

Joshua from Sydney Grammar School had a slightly different approach to the second part of the problem:

In three hours, the hour hand will travel a quarter of a full revolution, which is $90$ $^\circ$.

For those of you who misread this question and calculated the number of degrees that the minute hand turned through, better luck next time!