A clock gone mad
A clock's two hands both start at 12. Over the same period, the minute hand makes exactly 5 full turns clockwise and the hour hand exactly 4 full turns anticlockwise. Each hand moves at a varying speed but never stops or reverses, and both end at 12 together.
How many times do the hands pass each other, not counting the start and the end?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Look at the hour hand from the point of view of the minute hand.
Relative to the minute hand, the hour hand always moves the same way. How many full turns does it make?
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Watch the angle between the hands. Since the hands move in opposite directions, this angle only ever increases. The hands meet each time it completes a full turn.
Step 1: relative motion. The minute hand turns clockwise and the hour hand anticlockwise, so the angle from the minute hand to the hour hand, measured anticlockwise, increases all the time.
Step 2: total relative turning. Over the period this angle increases by full turns.
Step 3: the meetings. The hands coincide each time the angle passes a whole number of turns: after 1, 2, ..., 9 turns. The 9th is the end, at 12. So they pass each other 8 times in between.
The point. The varying speeds do not matter at all: only the monotone relative angle and its total change count.