by **Diana Davis**

Suppose you did a 1.5 mile tempo run in 9:00 (a 6:00 pace). You must have covered some mile of the race in exactly 6:00, right?

Amazingly, the answer is no! My recent math paper, written with some colleagues at Northwestern, explores and answers this question and some related questions about functions. Let’s see why there need not be a mile at the average pace:

For the 1.5 miles in 9:00, if you ran at a *constant* 6:00 pace, then certainly there would be a mile in exactly 6:00. But if you ran the first and last half miles in 2:00 each, and the middle half mile in 5:00, then you covered every mile in 7:00!

**Molly Huddle**.

In November 2013, Huddle ran 37:49 for 12k, defeating **Shalane Flanagan** and setting a world best for this distance. However, after the race, LetsRun reported:

Please realize women have run faster for 12k before. Mary Keitany‘s half-marathon world record (21.1 km) of 65:50 would put her going through 12k if evenly paced in 37:26.7.

*every*12k subset was slower than 37:49. See the paper for example with precise times, and a graph.

**Kenenisa Bekele** holds the world records for 5k and 10k, but he never ran a mile race in under 4:00. Or did he? This message board thread explains:

Person 1: Bekele ran a 7:26 3000. That averages under a 4 minute mile for the entire race.

Person 2: Yes, but were there 1760 yards that were consecutively run in under 4 minutes? Otherwise the answer is no, Bekele has never gone sub-4.

Person 3: How would someone run 3000m in 7:26 without ever breaking sub 4-mile pace for a mile portion of the race? Explain that.

Person 4: For an easy example, imagine he ran 1 second for the first 200m, 3:00 for the next 3 laps, 59 seconds for the middle 200m, 3:00 for the next 3 laps, then 1 second for the final 200m. No matter how you slice that, no 1609m is faster than 4:00. That gives a 7:01 3k. Adding time would not create any faster intermediate splits. Realistically, say 25, 60, 60, 60, 36, 60, 60, 60, 25, run evenly for each split.

- If your race distance is a whole number of miles, then you
*must*have done some mile in exactly the average pace. (It might be from mile 0.48 to 1.48, for example.) - If your race distance is not a whole number of miles, then it’s possible to construct a “race plan” example where
*no*mile is at exactly the average pace.

Note that the units “miles” can be changed to anything else — say, 12km in Huddle’s example — and then the rule is based on whether the race distance is a whole number multiple of that distance.

*exactly*6:36.