A nice mathematical puzzle, with a solution anyone can understand, is to determine the direction a bicycle went when you come upon its tracks. The answer involves thinking about tangent lines, geometric constraints and the bicycle’s steering mechanism. Once you learn the trick, you’ll find yourself using it every time you happen upon a bike trail.

The question goes back to the early days of the bicycle age and a 1903 Sherlock Holmes story by Sir Arthur Conan Doyle called “The Adventure of the Priory School.” Surprisingly, Holmes did not analyze the tangents as we do in the video and his reasoning in the story was incorrect. A geometric approach to the question was included in a 1990s “Geometry and the Imagination” course that John Conway, Peter Doyle, Jane Gilman and William Thurston taught at Princeton and the Geometry Center at the University of Minnesota. Holmes’ error is discussed in the book “Which Way Did the Bicycle Go?” by Joseph D. E. Konhauser, Dan Velleman and Stan Wagon.

Related:

More videos from the Mathematical Impressions series.

Wow thanks! That was a really nice and well explained video. Very educating.

Best,

The viewers might be interested in current research on “bicycle mathematics”:

http://www.tphys.uni-heidelberg.de/~wegner/Fl2mvs/Movies.html,

(where one finds other examples, besides the concentric circles); and

http://www.math.psu.edu/tabachni/prints/bike7.pdf or math/0405445

(Israel J. Math., 151 (2006), 1-28)

http://www.math.psu.edu/tabachni/prints/bicycle9.pdf or arXiv:0801.4396

(Experimental Math., 18 (2009), 173-186),

http://www.math.psu.edu/tabachni/prints/FLT8.pdf or arXiv:1207.0834

(March 2013 special issue of the American Mathematical Monthly),

http://www.math.psu.edu/tabachni/prints/Discrbike3.pdf or arXiv:1211.2345

(preprint)

This still does not solve the problem fully. There are bikes that go in reverse gear when you pedal backwards, and in that case one cannot say which way the bike came from because the tracks are exactly the same for the two bikes. It would be analogous to riding a bike on which the back wheel stirs. To complicate the problem further, when the vertical axis of the bike is tilted, the bike starts to turn towards the tilting side, which is a result of a torque on tire-ground intersection, generated be the gravity. On a bike that only turns by that mechanism (operated by a very skilled biker!), it would be again, impossible to say which way the bike came from.