The crazy clock with hands going opposite ways

Question

In a dream, Alice sees a clock whose two hands start together at 12. Over the course of the dream the minute hand goes clockwise round the dial exactly 5 times, and the hour hand goes anticlockwise exactly 4 times, both returning to 12 at the end. Their speeds vary unpredictably, but each hand always moves in its own direction and never stops for long.

How many times do the hands cross during the dream? A crossing means they meet and pass through each other; do not count the start or the end.

Accepted Answer

Eight times. The relative angle. Measure the angle from the hour hand to the minute hand, going clockwise. The minute hand adds to it by turning clockwise, and the hour hand also adds to it by turning anticlockwise (moving the reference hand backwards increases the gap). So the relative angle only ever increases, and over the whole dream it increases by 5 + 4 = 9 full turns. Counting crossings. The hands coincide whenever the relative angle is a whole number of turns: 0, 1, 2, ..., 9 turns. The values 0 (start) and 9 (end) are excluded, leaving 1 through 8: eight crossings. Because the relati…

The crazy clock with hands going opposite ways

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