Enigma 1714

Spoiler for New Scientist Enigma 1714:

The penny-farthing had one very large wheel with pedals, over which was the saddle, and one much smaller trailing wheel to enable steering. We have some early film of my grandfather riding one. It had a large wheel 6 metres in circumference with 48 spokes, and a small one of one-third that diameter having 20 spokes. It had no brakes.

He was hurtling downhill at less than 50 km/h – and just before he fell off, the film appears to show both wheels not rotating.

If the film is showing 16 frames per second how fast, in kilometres per hour, was he then travelling?

Not much puzzling about this.

One quibble: It’s not how many frames per second the film is showing that matters, it’s how many frames per second were filmed.

Anyway, for both wheels to appear not to rotate, it must be that in 1/16 second both wheels rotate by an amount equal to an integer number of spoke spacings. Now, when the bicycle moves 6 meters the front wheel makes a full revolution, which means 48 spokes go by; the back wheel makes 3 revolutions, which means 60 spokes go by. Or in lowest terms, when the bicycle moves 50 cm, 4 spokes go by on the front and 5 on the back. That’s the smallest distance that gives an even number of spoke spacings rotation on both wheels. So the bike must be moving some multiple of 50 cm in 1/16 second. Converting to km/h, the speed is 28.8 km/h — or a multiple thereof, but this is the only possibility under 50 km/h.

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