This problem seems hard, then it doesn't, but it really is

The famous (infamous?) “windmill” problem on the 2011 IMO
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The author of this problem was Geoff Smith. You can find the full list of problems considered for the IMO that year, together with their solutions, here:
You can find data for past IMO results here:
Viewer-created interactive about this problem:
And another:
I made a quick reference to “proper time” as an example of an invariant. Take a look at this minutephysics video if you want to learn more.
These animations are largely made using manim, a scrappy open-source python library:
If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
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Stream the music on Spotify:
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe:
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50 Replies to “This problem seems hard, then it doesn't, but it really is”

  1. I'm not sure that the handling of the even case is quite correct. As you correctly note, after the line has rotated 180 is it parallel to but displaced from the original line. You then seem to be relying on rotating another 180 brings it back to the original position and you conclude from that that it has hit every point, as in the proof for the odd case. But wait! After 360, the orientation is the same, so you cannot apply the color swap reasoning because the two lines now have the same orientation.

    You don't need to go 360 to handle the even case. Go 180, just like in the odd case. Even though the 180 rotated line is parallel to the original line rather than coincident with it, you can still use the color swap argument at that stage because it is easy to see that there are no points between the two parallel lines. (If any of the points were between the parallel lines, it would throw off the counts).

  2. Woah, thinking I have one of those geniuses who did IMO in my class, and like 4 in the entire school make me think how I even managed to get there…

  3. I've often said that the hardest thing about learning something new, is remembering what it was like to not know.

    When we learn something new our minds are inclined to disregard any conflicting old way of thinking, but we need to be careful about what we let our minds forget. If we're going to be able teach others what we've discovered, we must be able to remember what it was like to not know so that we can empathize with those who've yet to learn it and meet them on their level.

  4. @7:00 omg I solved the concept st this point. I knew that if half the points were on one side or the other it’d never go outside. I don’t know enough about proofs to do that part though.

  5. Does the starting position make a difference? In the explanation, 3B1B showed a system of 4 points with one in the middle, and he showed that the line never accesses the middle pivot if it begins pivoting on one of the outer 3. So – even with the same field of pivot points, doesn't the position of the line affect the outcome?

  6. Three minutes into the video "oh so you made a clock but with varying distance, neat". End of video "wtf is going on here, what did I just watch, I don't understand anything he just said"

  7. I sent this video to a friend and told him not to watch it until he decides to see the solution, but the solution happens to be on the preview frame.

  8. Well, I got it – one just need to draw a line outside of all the dots, and "tie" that. Then remove all the points that is on that outer curve. Then try to repeat it layer by layer until you have no inner dots left. The last point, or the last points on line – is a valid starting point(s).
    The whole pattern of switching rotatring points is actually a directed graph. And somehow it feels cool just not to get any deeper withing the whole set than the layer from which we started. Thats why you obviously get infinite loop if you start on very outter point – the cycle in blind to the inner dots.

    Really a beautiful one. Yet, without some mindset and proper visualisation it could be unexpectedly harder than it, I guess, was meant to be.

  9. "it is clear that each point is touched an infinite number of times" … huh? I was thinking about convex hulls, some measurement for being internal, how internal?, do some points touch most but not all? are there special points? Brain spinning. Very clear analysis as always, thank you. Now got to watch it again because I'm sure I missed something.

  10. I have an other problem related : what does happen in 3D ? More precisely, I preserve the rotating line, but I replace the 2D plane by the 3D space, the non-aligned points by non coplanar lines, and replace the "rotation around the intesected point" by a "rotation around the intersected line" ? For the sense of the rotation, choose an arbitrary sense for the first one, and at each step, when the rotating line L meet a new line L2 at M2, consider the plane P generated by the preceeding line L1 and L2, and choose for the rotation around L2 the only sense such that the two parts of L outside the segment [M1M2] goes each one to the corresponding other side of P (M1 is the intersection of L and L1). For instance, if all the lines are parallel, by projection parallel to this common diirection, the problem is equivalent to the 2D problem.

  11. Like, after this explanation, I'm pretty sure I understand the solution, and could probably explain it to someone else. however, There is not a chance in hell I could ever figure it out on my own.

  12. Just about as soon as he finished reading the proof question. The way that I figured it out is by watching the diagram at the bottom play out while he was narrating the problem, I realized that the problem came from S being in the middle, and it seemed to start sorting itself out as soon as it touched Q, so there was a point to having the first point be on the outside. After that, I realized that if you just start at Q and remove P entirely, then you would have replacement P’s, or in my thinking, other points in the middle, so the problem was having points in the middle. What I thought by middle was that it wasn’t out the outside of the shape, and my brain automatically thought of it like a shape with points in the middle because my brain likes organization. If you remove all points besides for the outline of a shape and start the windmill, you still could get get a problem due to the angle the windmill is at, and my brain didn’t like that chaos, so it assumed that it should be on the outside, rotating around the shape. This leaves the best solution by having it be tangent-like in a circle-like shape.

    This solution is correct but not the solution they were looking for so I would probably get points off unfortunately.

  13. How here is a meta-puzzle: which real mathematical problem this toy problem came from? I do know the possible research area and can imagine where a similar reasoning is applicable, but I am very curious, where exactly it happened.

  14. 4:54 and 4:59, is this an editing mistake? I'm so confused with the chart, almost everyone had above 5 on P2 at 4:54, and few had 0 on P6

    But now at 4:59 it's revered??

    Fuck i didn't even get to the problem solving and I'm already confused lol

  15. Why couldn’t you just use 3 points, or even 2 to prove this true? It will hit them forever and go around infinitely, so it follows the rules of the question… Or am I just wrong?

  16. I'm glad I didn't skip any of this (recommended) video to be more time efficient. I learned something about myself. I was pretty quick to pick parts of this up and 2 out of 3 times before you began to explained it on the video.

  17. Pictoral/Factoral; some facts are also the same name as answers to Math functions, probable sabotage of English and other languages.

  18. Casual proof: the line keeps rotating clockwise. A line rotating in a plane will touch every point in that plane, infinitely many times. QED

  19. I know I'm late, but the ending to this video left me amazed at how poetic it was to compare this windmill problem to Don Quixote's imaginary windmills

  20. Sooo your channel is HUGE and this is an older video but just want to state that this is the 2nd one I've watched and I'm VERY glad to have this channel as I go from programmer to Data Scientist (basically Statistics meets programming) (:

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