As Sherlock Holmes aficionados will know, in the 1903 story ‘The Adventure of the Priory School’, Holmes determined the direction in which a bicycle was travelling simply by observing the tyre tracks which it had made – asserting that the deeper of the two wheel marks must have come from the heavier rear wheel …

[Scroll on 93 years]

But in 1996 the essay Which Way Did the Bicycle Go?…and Other Intriguing Mathematical Mysteries. (*Dolciani Mathematical Expositions Series of the Mathematical Association of America*, No. 18.) authors Joseph D. E. Konhauser, Daniel J. Velleman, and Stan Wagon questioned the absolute validity of Sherlock’s methodology. They described what they saw as a better method of bicycle-track-direction-detection based purely on calculus – and independent of the depth of track.

To clarify, their diagram above shows a notional tyre track in which the heavier track was made by the *front* wheel.

[Scroll on 6 years]

But then, in 2002, a new twist was exposed. David L Finn, (Associate Professor of Mathematics at the Rose-Hulman Institute of Technology) determined (by the use of differential geometry) that it is theoretically possible to construct a *unicycle* track with a bicycle. *See:* Can a Bicycle Create a Unicycle Track? (*College Mathematics Journal*, 2002). Thus the tracks which Holmes observed could have come from *two* bicycles instead of one. Considerably complicating the chances of solving what was an already tricky-enough case.

Professor Finn’s diagram above shows a mathematically generated version of a single tire track that can be created by a bicycle.

NOTE : Professor Stan Wagon, one of the co-authors of *‘Which Way Did the Bicycle Go?’* has designed, constructed and ridden a bicycle with square wheels. Here’s video of a tricycle built on Wagon’s principle:

ALSO SEE: There are two new (2013) books about the science of Sherlock Holmes, reviewed here by Jonathon Keats at *New Scientist*.

UPDATE Sep. 2020

Prof. Wagon and Prof. Velleman have launched a book of mathematical puzzles – including a chapter on bicycle/unicycle tracks, entitled Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles. (*MAA Press:* An Imprint of the American Mathematical Society)

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