In this classic puzzle by Henry Ernest Dudeney, a car dealer tells a customer that on a certain journey, the front wheel of a car makes four more revolutions than the back wheel.
The problem asks:
Given that both wheels are rolling on the ground without slipping, what are the relative sizes (circumferences or diameters) of the two wheels?
The puzzle combines geometry and proportional reasoning. Since the number of revolutions equals the distance traveled divided by the wheel’s circumference, the condition can be expressed as:
where is the distance of the journey, and are the circumferences of the two wheels.
Solving for the ratio between the wheels reveals how much smaller one wheel must be so that it makes exactly four more turns than the other over the same distance.
It’s a beautiful illustration of how a simple mechanical observation leads to an elegant mathematical relationship between rotation and distance.
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