Math on the Run

ddnbb wrote:

Sure. We give an example where Keitany runs 27:00 for the first and last 9.1km and 11:50 for the middle 2.9km, so the whole 21.1km is 65:50 as required, but every 12km subset is 38:50, slower than Huddle's 37:49. But within that 9.1km, Keitany could have wildly erratic paces, as long as her total time comes out to 27:00; her pace need not be symmetric.

The idea behind this example is that she runs fast at the beginning and end, and slow in the middle, and every 12km includes that slow middle part. Any example has to be something like this, so it has to be "symmetric" in that sense. You could change the 9.1 and 2.9 a little bit to break the symmetry. (But then it wouldn't be as elegant of an example!)

Yes, it's the overlapping middle part that forces the paradox. A more visual example, run 3 half mile segments back to back. The first @4m/m and takes 2 mins, the second at 6m/m (3 minutes) and the last at 4 m/m (2 minutes). In terms of time, it looks like this:

| 2 | 3 | 2 |

It's not possible to form any continuous mile within that isn't comprised of a 2 minute half mile and a 3 minute half mile, simply because any mile within must contain the 3 minute half mile portion.

The average pace for the full 1.5 miles is 7 min / 1.5miles, or 4.66... min/mile. Yet no mile within is run in less than 5 minutes.

Bad Wigins wrote:

and only now do I notice the link to the actual paper... it seemed at first like they were relying on only the example in the SA article.

But I would still like to see an example where non-divisor segments less than half the distance fail.

Maybe the following would be a good example?

Instead of the HM longer distance, suppose the distance was 30K, which is more than twice the distance of the 12K. The 30K can be split into five - 6K segments. Most of the slow running pace could be concentrated into the second and fourth segments, assuring slow 12K running pace for all possible 12K segments, but allowing for a fast running pace overall. I think it is necessary that there are more fast running pace segments than slow ones over the whole run for the argument to work.