It’s not often that the same problem makes it into a math paper and a running blog. But if any mathematician can make it happen, it’s Diana Davis. Davis is a postdoc at Northwestern University, an avid runner, and a creative communicator of mathematics. In 2012, her excellent video about cutting sequences on the double pentagon won the physics and math category in the Dance Your PhD contest. (I’ve watched the video several times, sometimes on my own computer and sometimes in lectures by Fields Medalists.)
In July, Davis and her coauthors Keith Burns and Orit Davidovich posted a paper called “Average Pace and Horizontal Chords” inspired by two other runners, Molly Huddle and Mary Keitany. They write,
On November 16, 2013, Molly Huddle ran 37:49 for 12 kilometers, a world record for that distance. People applauded this performance, but some pointed out that Mary Keitany’s world record of 65:50 for the half marathon, which is 21.1 kilometers, is actually faster than Huddle’s record: Keitany averaged 3:07 per kilometer, while Huddle averaged 3:09 per kilometer. Therefore, Keitany must have run some 12 km subset of the race faster than Huddle—right?
I’m willing to bet I’m not the only mathematician who would blithely invoke the intermediate value theorem and call it a day. (Incidentally, this reaction is part of why mathematicians are so focused on rigorous proofs. Intuition is great, but it has to be backed up by a watertight argument.) The intermediate value theorem is a powerful fact from calculus that states that if a function is continuous (it doesn’t jump around abruptly), it must take all possible values between its maximum and minimum.
An example is helpful: the humble parabola, modeled by the function y=x2, has the y-value 0 at x=0 and 9 at x=3. That means somewhere in between 0 and 3, y has to take the values 1, 2, 3, π, and any other number between 0 and 9. The intermediate value theorem has some surprising implications. For instance, at any given time, there are two points exactly opposite each other on the earth that have exactly the same temperature!
The 12K question just feels like an intermediate value problem. I remember doing problems that seemed just like it in my first semester of calculus: if I go 4 miles in exactly an hour (I’m not quite as fast as Huddle and Keitany), there must be some mile in there that took me exactly 15 minutes. How do we know? Chop up the distance into one-mile chunks. If they all took 15 minutes, we’re done. If not, then one of them must have been under 15:00 and one must have been over. The intermediate value theorem says that some mile in there, maybe from 1.5 to 2.5, took exactly 15 minutes.
It stands to reason, then, that if Keitany ran the race at an average pace of 3:07 per km (37:24 per 12 km), she must have run some 12K stretch in exactly 37:24. Somewhat shockingly, the answer is no. Davis and her coauthors dispatch this question right off the bat with an example:
Suppose that Keitany ran 27:00 for the first and last 9.1 km, and 11:50 for the middle 2.9 km. Then her total time for the race would still be 2×27:00+11:50=65:50, but her time for each 12 km subinterval would have been 27:00+11:50=38:50, much slower than Huddle’s record.
What went wrong with the intermediate value theorem? In the example I used, the distance, 4 miles, was exactly 4 times the length I was interested in. This relationship doesn’t hold for the 12K and half marathon. At 21.1 km, the half marathon isn’t an exact whole number multiple of 12 km. The argument I used breaks down if we can’t divide the distance into chunks that are all 12 km. It’s hard for me to believe there’s no way to salvage the argument, but Davis et al’s example shows that I can’t.
Davis says their paper is probably the first application of a statement called the universal chord theorem to running. The universal chord theorem, which has been studied in different guises for about 200 years, deals with functions where f(0)=f(1)=0 and investigates the lengths of horizontal chords of those functions. (A horizontal chord is just a line segment that connects a point on a graph to another point that has the same height.*) In the 30s, Heinz Hopf figured out how to cook up a function with a given set of horizontal chord lengths, provided the set met a few criteria.
Burns, Davis, and Davidovich expand on previous work by streamlining the proof of the theorem and devising a simpler recipe for cooking up nice, smooth functions that have the desired chord lengths. (For all the details, check out their paper.) In running terms, they explain when a runner must run a mile (or other length of interest) at exactly their average pace and come up with a race plan for avoiding the average pace if possible. Actually running the race is left as an exercise for the reader.
*This sentence was changed after publication to correct an error. I had initially written that a horizontal chord connects a point on a graph to the next point at the same height, but it can be to any point at the same height. Thanks to Diana Davis for pointing out the error.