The November 2004 issue of the College Mathematics Journal had an article
that may be of some interest to readers of this forum. In "Linearizing
Mile Run Times", authors Ash, Ash, and Catoiu consider the following question:
Runner Steve Elliot set a Michigan High School mile record of 4:07.4 seconds in
1975. Subsequently, the state replaced the mile run with the 1600 meter
run, and soon thereafter one Earl Jones ran the slightly shorter event in
4:07.2. A mile is exactly 1609.344 meters, so it is easy to check that Elliott
had run a slightly longer distance at a higher average speed than had Jones. Is
there any argument that can justify awarding Elliot the 1600m record?
Clearly an argument based on average speed alone is insufficient. By that
reasoning, we would give Michael Johnson the 100m record to go along with
his 200m record, since he ran twice as far in less than twice as much time.
(Montgomery's 100m record is 9.78s vs Johnson's 19.32.)
On the other hand, suppose it could be argued that Elliot must have run SOME
1600m segment of his mile race in less than the time it took Jones. Would
it then be reasonable to give him the 1600m record?
Unfortunately, as the authors show, there does not have to be any such 1600m
segment. Depending on how Elliot actually paced himself, it is possible that
either (i) he ran EVERY 1600m segment of his race faster than Jones, or (ii)
he ran EVERY 1600m segment slower than Jones. It is true that Elliot ran
some 1600 distance in less time than Jones, but that distance might be in
two disconnected pieces.
Of course, even if could be proved that Elliot had run some connected 1600m
interval faster than 4:07.2, one might still object that he may have had
a running start, or that the effort was slightly wind-aided.
In the last section of the paper the authors discuss some oddities based
upon metric to English conversion. Particularly striking is an approximate
conversion method based upon the Fibonacci sequence. Recall that Fibonacci's
eponymous sequence begins 1,1,2,3,5,8,13, ..., each number being obtained as
the sum of the two that came before. Suppose a distance in km happens to be
the sum of two Fibonacci numbers, say 18km = 5km + 13km. Then a good
approximation to the same distance in miles is gotten by adding the two
Fibonacci numbers that came immediately before, in this case 11mi = 3mi + 8mi.
This works because the ratio of successive Fibonacci numbers
rapidly approaches the "golden ratio," an irrational quantity that is
approximately 1.618.... It just so happens that this value is very close to
the conversion factor from miles to kilometers (1.609344.)
For those who wish to consult the article, the exact reference is:
Garrett I. Ash, J. Marshall Ash, and Stefan Catoiu, "Linearizing Mile Run
Times", College Math. J. 35(2004), 370-374.