If you pull straight back on the lower pedal of your bicycle, will the bike move forward or backward? This classic puzzle has a surprising twist, so be sure to think about it carefully before watching the answer in this video. The puzzle is discussed in many publications, but none of them present the whole story shown here, as far as I know.

For partial solutions, see Martin Gardner, *Mathematical Carnival* (Random House, 1975), p. 182, or D. E. Daykin, “The Bicycle Problem,” *Mathematics Magazine* 45, no. 1 (Jan. 1972), p. 1.

Thank you to Jamie Swan and Rick Jones for providing the bicycle.

Related:

More videos from the Mathematical Impressions series.

i am wondering how the results of this experiment might shed some insight on the efficency of SPD debate?

Wow, very nicely done! I did “predict” the results correctly (including the gear dependence) and even drew the trochoids on my board when thinking about it. Still, I’m a university physics professor, and I have to admit I was only 90% sure of my answer! An excellent puzzle and demonstration.

Very good video!

The trochoid curves remind me of another surprising fact: that, at any instant, some parts of a train actually move bakwards WITH RESPECT TO THE GROUND! Yes, any point at the edge of the flange of a train wheel follows a trochoid with a loop at the bottom (like the bottom one in the video) and at that moment it is going BACKWARDS.

Of course, not *all* bicycles work the same way as *typical* bicycles. Certain mountain bikes with “granny gears” will move forward — all that’s needed is for the effective rear wheel to be smaller than the actual crank size.

This problem has been a favorite of mine since I got it wrong in high school physics class, and I have used it throughout my 40 years of teaching.

For the past 20 years, my daughter’s tricycle has sat under my desk, ready when my explanations based on drawing cycloids were (almost always) unconvincing. (The tricycle in an important easier case.)

I have a number of times attempted to collect on bets made by (usually) over confident engineers and physicists with wrong answers, but have found that no money will be forthcoming unless a bicycle or tricycle can be found and the demonstration made. (And then some of them would try to weasel out of it, saying that I had not properly explained the situation! It reminds me of the car and goats problem.) A trip to to K-Mart was necessary one time.

A related question which requires actual estimation is whether in a front wheel drive car which is moving there is any part of the engine that actually moves backward with respect to the road.