Time-travelling ant
Imagine an ant sitting on the minute hand of a clock at midnight.
It travels with the minute hand for a little bit over an hour and when the minute hand overlaps with the hour hand, the ant switches to the hour hand. It then waits for the minute hand and when this comes, the ant again switches to the minute hand. It keeps doing this switching for 12 hours.
How many full rotations around the clock will the ant cover in 24 hours?
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1 solution
Over 2 4 hours, the minute hand makes 2 4 rotations around the clock and the hour hand makes 2 rotations around the clock.
Because the ant alternates between the minute hand and hour hand at equal time intervals, it spends exactly half the time on the minute hand and the other half on the hour hand.
Therefore, the ant covers a total of 2 1 ⋅ 2 4 = 1 2 rotations on the minute hand and a total of 2 1 ⋅ 2 = 1 rotation on the hour hand, for a total of 1 2 + 1 = 1 3 rotations.