Bicycle Tracks

If you came upon this set of bicycle tracks meandering around in the mud, could you determine which way the bicycle was going simply by examining the tracks? Remember that the front wheel and the rear wheel make separate tracks.

Answer

In “The Adventure of the Priory School,” Sherlock Holmes not only draws a map of the neighborhood of the school but also examines several sets of tyre tracks on the moor. He needed to determine the bicycle’s direction of travel solely from his inspection of the tracks. Holmes could tell from the depth of the wheel impression which track was made by the rear tyre. You don’t have that

information, but a little mathematics reveals the answer.

Note that the rear wheel of a bike always points to the place where the front wheel touches the ground. Therefore, the tangent to the rear wheel track will always cross the front wheel track, while the front wheel track does not exhibit this geometrical property.

Once we identify the rear track, we can pick two random points on it and extend the tangents to where they cross the front track in both directions, measure the segments, and determine which direction yields segments of the same length. Since a bicycle cannot change its length, we learn the direction of travel.