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.
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