The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

The approach to solving this problem looks very elegant to viewers with&#x2F;without a mathematical background and the author&#x27;s use of visual explanations towards solving it step-by-step helps untangle the ambiguities in this puzzle.<p>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&#x2F;skilled mathematician or both which greatly helps explain this proof in a concise fashion.<p>On the other hand, I find that problems like this may be (ab)used in the future for technical interviews at financial&#x2F;asset&#x2F;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.

This phenomenon, whereby a person who knows the &quot;trick&quot; to solve a puzzle cannot accurate gauge its difficulty, seems to extend beyond mathematics. Adventure games (including text-based, parser-driven, and point-and-click) suffer badly from this problem. They are chock-full of puzzles that only make sense in hindsight (if at all). They can be really fun though!

This can be formally solved by constructing the arrangement of lines passing through each set of two points, then computing the dual of the arrangement.<p>The cell with the maximum depth on the arrangement contains the points in the &quot;middle&quot;, meaning they have as many points on one side, as they have on the other.<p>Then you can prove that a line starting on any such point will visit every other point an infinite number of times.

This guy makes excellent math visualization videos -- some of the best I&#x27;ve ever seen.

This is hella cool.<p>&quot;Knowing when the math is hard is way harder than the math itself&quot;<p>But then maybe the math is hard only because it&#x27;s not being explained well? (I hope it&#x27;s uncontroversial to suggest that our current methods of teaching math are not the best of all possible worlds.)<p>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.<p>I kinda think that we should teach math as fast as we can so that we can concentrate on the stuff that&#x27;s really hard, not just <i>apparently</i> hard because someone is being coy with the easy routes.

Since this is a puzzle that will certainly nerd-snipe a number of us, could someone who already watched the video tell us if there is a &quot;spoiler&quot; moment in it or if we could watch it bit by bit in case that we get stuck?

This is great, the visualization is so helpful. Even better I think would be if the point field was counter rotating and scrolling such that the windmill was constantly falling forwards and backwards either side of being vertical, keeping the two sets of points bisected and on either side of the screen.

Mmh intuitively, I would have thought that the proof would involve the enveloppe of points. If the line starts with a section that crosses inside the enveloppe of the set of points then it remains so, and hits all points, while a line that starts outside remains outside and so can avoid some points.<p>Any formal proof along those lines?

I enjoyed watching this similarly insightful video [1] on &quot;The hardest problem on the hardest test&quot; of the Putnam Competition.<p>[1] <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=OkmNXy7er84" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=OkmNXy7er84</a>

The video gives excellent intuitions. But do try writing down a rigorous argument after watching it!

From the same channel : <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=jsYwFizhncE" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=jsYwFizhncE</a> overview of very elegant connection between blocks collision and PI.

Great presentation. I wonder whether all correct solutions submitted on the contest day had the same solution.

Now I really want to know what question 6 was and an equally informative explanation what made it so hard!

I&#x27;m going to have to watch this later, because the cat I&#x27;ve got on my lap appears to be deeply alarmed by the blinking eyes of the &quot;pi&quot; avatar.