Monday’s puzzle was open to various interpretations, but under what seems to me to be the most straightforward interpretation, if the number of runners you pass is the same as the number who pass you, you’re the mean runner, not the median.
You can find plenty of correct analysis in Monday’s comment section (see in particular Harold’s perfect comment #39), but here’s a more longwinded explanation:
First, suppose you randomly sample a large number of other runners and discover that half of them are faster than you and half are slower. Then you’re entitled to conclude that you’re the median runner (or, if we’re being careful, you’re entitled to conclude that you’re probably close to the median, since there’s always a chance your sample was unrepresentative).
Now in the problem as given it’s certainly true that half the runners you encounter are faster than you and half are slower. So you might be tempted to use the above reasoning and conclude that you’re the median runner. But that won’t work, because the runners you encounter are not a random sample.
So let’s start over. We might as well assume that you’re the center of the universe, so you’re completely motionless. Everyone who’s faster than you is running forward and everyone who’s slower than you is running backward. People “pass” you when they run past you in the forward direction, and you “pass” them when they run past you in the backward direction.
If there are, say, 32 10-mile-an-hour forward runners per mile, how many will pass you in an hour? Answer: 320. (Think about it). If there are 19 5-mile-an-hour forward runners per mile, how many will pass you in an hour? Answer: 95. (Think about it again.) In total, how many pass you in a given hour? Answer: 320+95=415.
And what’s the total speed of all the runners in a given mile? Answer: 32 at 10mph plus 19 at 5mph = 415.
In other words: The total speed of all the forward runners in a given mile is equal to the number who pass you per hour. Likeiwse, the total speed of all the backward runners in a given mile is equal to the number who you pass (i.e. who pass you going backward) per hour.
Now if we count forward speeds as positive and backward speeds as negative, then the total speed of all the runners in a given mile is equal to
which in turn is equal (by the arguments we made above) to
Since we’ve assumed that the number who pass you going forward is equal to the number going backward, this total comes to exactly zero.
And finally, the average speed of the other runners is equal to their total speed divided by the number of runners. Since their total speed is zero, so is their average speed. On average, the other runners are standing still — just like you!
You, in other words, are the average runner.
Lame hat tip: This was inspired by a question on MathOverflow which I no longer seem able to find. When I find it, I’ll replace this sentence with a proper link.