# ESRogs comments on What are the best elementary math problems you know?

• Gotcha, in that case:

Pick the paratrooper nearest to centroid of all paratroopers and go directly towards their location in a straight line.

As you move, the location of the centroid will change, but which paratrooper is nearest won’t change. This is because your contribution to the average distance to that paratrooper will decrease by at least as much as your contribution to the average distance to any other paratrooper (because you’re moving in a straight line towards them).

(For n equals two, you have to pick one of you to move and one to stay fixed ahead of time. For n is three or more, except for in measure zero cases, there will be exactly one paratrooper who is nearest to the centroid, with no ties.)

EDIT: Actually I think there’s a flaw in this, but I don’t see which part is wrong. The reason I think there’s a flaw is that 1) I think the centroid moves away from you as you move towards it, but 2) it seems like my argument about the delta in your contribution to the average distance to a point applies to the location of the centroid itself, in which case the location of the centroid shouldn’t move as you move towards it. So there’s a contradiction...

EDIT2: Oh, I see the flaw. The centroid moves away from you in your direction of travel. The magnitude of the decrease in average distance to the centroid is maximized along the line connecting you and the centroid, but it’s negative for points in between you two. So, if you move towards a point that point’s distance from the center of mass is actually increasing the most, not the least.

EDIT3: Wait, actually, I think we can rescue this. What if we ignore the centroid altogether, and just choose the paratrooper that has the lowest average distance to each of the other paratroopers. Then I think maybe my original line of reasoning works?

EDIT4: I figured out what I was missing here — when you move towards a point, it’s true that the average distance of that point to all other points is shrinking at least as much as for any other point, as I was saying above, except for yourself. You might be moving closer to multiple points, meaning that you might become the point with the least average distance to other points!

• So a general bit of advice for this problem is that the solution I have in mind is not trivial at all—it involves plenty of cleverness to come up with and some careful accounting of all of the edge cases to prove that it works. My solution is most likely not the simplest possible but my suspicion is that no solution is going to be as simple as the ones people have proposed thus far.

If you think you have a simple solution, your solution is probably wrong.