Correctly proving this without assistance is one thing, but explaining it to non-mathematicians via a YouTube video sounds so difficult that some I.M.O candidates may struggle with this. Even so, I think the author is perhaps a professional/skilled mathematician or both which greatly helps explain this proof in a concise fashion.
On the other hand, I find that problems like this may be (ab)used in the future for technical interviews at financial/asset/investment management institutions for software engineering roles. Over the top indeed, but I think it would very difficult to justify using mathematical proof questions in interviews.
https://professorpositions.com/szego-assistant-professor-at-...
I'm sure she will get a tt job when she wants however.
Quite a big difference
I'm not really worried. In general, the trend for hiring software engineers has been away from silly puzzles, not towards. Microsoft and Google both used to use them and now they don't, and other companies have been following along. In general, hiring fads and follow-the-leader are not great, but in this case it's for the best that other companies have taken their lead.
To be sure, there are still lots of other problems with how developers are hired, but the stupid puzzles at least have mostly faded away.
Programming is literally isomorphic to finding proofs (Curry-Howard FTW!). From the other perspective on proofs, they're about communicating technical concepts in a clear way, which is a vital skill for a developer in an organization. So no, I don't think it would be that hard to justify. I was kind of joking at first, but that's actually pretty compelling...
It's mainly about showcasing a problem that is objectively hard (based on IMO competition results) then revealing the simple thought process that will lead you to an "obvious" answer in a few short minutes. The main point is that there are problems where once you know the "trick", you can't accurately judge how hard the problem is anymore.
I think people would appreciate that point a lot more if they struggle with the problem for a bit first!
Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line l going through a point P ∈ S. Rotate l clockwise around the pivot P until the line contains another point Q of S. The point Q now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from S.
Show that for a suitable P ∈ S and a suitable starting line l containing P, the resulting windmill will visit each point of S as a pivot infinitely often.
It is a kind of puzzle to try out on your own, knowing that a rather simple mathematical reasoning solves it. It doesn't rely on some obscure mathematical knowledge as might be suspected from an Olympiad problem.
If you want to see Grant Sanderson, instead of only hearing his voice, he uploaded a Q&A where he explains his motivations. Mainly, creating educational math videos that other people would not be able to create.
The cell with the maximum depth on the arrangement contains the points in the "middle", meaning they have as many points on one side, as they have on the other.
Then you can prove that a line starting on any such point will visit every other point an infinite number of times.
"Knowing when the math is hard is way harder than the math itself"
But then maybe the math is hard only because it's not being explained well? (I hope it's uncontroversial to suggest that our current methods of teaching math are not the best of all possible worlds.)
I get that this problem came up in the context of of a math puzzle contest, and that some people enjoy solving puzzles. I am questioning their utility as an educational device.
I kinda think that we should teach math as fast as we can so that we can concentrate on the stuff that's really hard, not just apparently hard because someone is being coy with the easy routes.
> I am questioning their utility as an educational device.
Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?
Kind of, although I don't think they do it deliberately.
Things like teaching logarithms without a slide rule.
> Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?
You're right. Let me try again.
Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.
I think that most math seems hard to most people only because we are not creative in the ways that it is presented. We should use science to figure out how to present math so that people get it as fast as they can, in part so that we can find and concentrate on the actually hard math problems.
E.g. Alan Kay using Smalltalk to teach calculus to little kids in the context of modelling falling objects, to me kinda proves that it shouldn't take a whole semester to teach calculus to teenagers.
Any formal proof along those lines?
