Given a bicycle track - can you tell which way the bike went?

Sherlock Holmes goofed on this problem in "The Adventure of the Priory School". He argued that the rear wheel would make more of a depression than the front wheel, but that does not resolve the issue of direction.

There is a nice geometrical solution to this problem. It uses two aspects of a normal bicycle:

  1. The front and back wheel are separated by a constant distance.
  2. The rear wheel always points to the point where the front wheel touches the ground.
Item 2 implies that the tangent of the rear wheel track should intersect the front wheel track at a given distance that is unknown but constant for any given bicycle (as noted in item 1).

It is therefore helpful to know which track is which, but trial and error would sort out which of the four possible combinations is correct (front/back track, and one of two directions) after we have looked at a number of tangents from either track. A unique identification should be possible when we see that the tangent in one direction from one of the tracks will intersect the other track consistently at the same distance.


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