title

The unexpectedly hard windmill question (2011 IMO, Q2)

description

The famous (infamous?) "windmill" problem on the 2011 IMO
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/windmillthanks
Home page: https://www.3blue1brown.com
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:
https://www.imo-official.org/problems/IMO2011SL.pdf
You can find data for past IMO results here:
https://www.imo-official.org/
Viewer-created interactive about this problem:
https://www.reddit.com/r/3Blue1Brown/comments/d0b0qw/interactive_windmill_visual_program_download_link/
And another:
https://aalluri7.github.io/windmill/
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.
https://youtu.be/WFAEHKAR5hU
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These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim
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.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
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.
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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: http://3b1b.co/subscribe
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detail

{'title': 'The unexpectedly hard windmill question (2011 IMO, Q2)', 'heatmap': [], 'summary': 'The 2011 international math olympiad featured 563 participants from 101 countries, with lisa sauermann achieving a perfect score, and the chapter analyzes the windmill process problem, highlighting that problem 6 had the lowest number of perfect scores (6 students) and problem 2 had the fewest perfect scores (22 students) out of 563 participants, illustrating the challenge of problem 6 despite being introduced differently. the chapter also explores windmill puzzle problem solving and emphasizes the importance of identifying invariants in solving complex systems.', 'chapters': [{'end': 77.541, 'segs': [{'end': 33.755, 'src': 'embed', 'start': 2.944, 'weight': 1, 'content': [{'end': 9.085, 'text': 'Every year, more than 100 countries send six of their brightest teenagers, or the occasional prepubescent prodigy,', 'start': 2.944, 'duration': 6.141}, {'end': 13.546, 'text': 'to represent them at the International Math Olympiad, commonly known as the IMO.', 'start': 9.085, 'duration': 4.461}, {'end': 20.488, 'text': 'Considering that each country has its own elaborate system of contests leading to their choice of six representatives,', 'start': 14.306, 'duration': 6.182}, {'end': 26.049, 'text': 'the IMO stands as the culminating symbol for the surprisingly expansive and wonderful world that is contest math.', 'start': 20.488, 'duration': 5.561}, {'end': 33.755, 'text': 'The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day.', 'start': 26.869, 'duration': 6.886}], 'summary': '100+ countries send top teens to imo, a math test with 6 reps, 2 days, 3 qs', 'duration': 30.811, 'max_score': 2.944, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM2944.jpg'}, {'end': 92.367, 'src': 'embed', 'start': 61.874, 'weight': 0, 'content': [{'end': 63.355, 'text': "But that's not why this test was interesting.", 'start': 61.874, 'duration': 1.481}, {'end': 69.137, 'text': 'Out of all these prime problem solvers, only one of them, Lisa Sauermann from Germany, got a perfect score.', 'start': 63.895, 'duration': 5.242}, {'end': 75.24, 'text': 'And the only thing standing between the next two runners up that year and a perfect score was problem number two.', 'start': 69.677, 'duration': 5.563}, {'end': 77.541, 'text': 'And this problem is beautiful.', 'start': 75.92, 'duration': 1.621}, {'end': 85.284, 'text': "And despite evading many of the world's best mathematicians of their age, the solution is something that anyone watching this video can understand.", 'start': 78.001, 'duration': 7.283}, {'end': 88.605, 'text': "So let's begin by reading through it carefully.", 'start': 86.284, 'duration': 2.321}, {'end': 92.367, 'text': 'Let S be a finite set of at least two points on the plane.', 'start': 89.386, 'duration': 2.981}], 'summary': 'Only one out of many prime problem solvers, lisa sauermann from germany, got a perfect score in an interesting test, with the solution to a challenging problem being accessible to all.', 'duration': 30.493, 'max_score': 61.874, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM61874.jpg'}], 'start': 2.944, 'title': 'International math olympiad', 'summary': 'Provides a glimpse into the 2011 international math olympiad, featuring 563 participants from 101 countries, with lisa sauermann achieving a perfect score and the runners-up being just one problem away from perfection.', 'chapters': [{'end': 77.541, 'start': 2.944, 'title': 'International math olympiad: a glimpse inside', 'summary': 'Delves into the international math olympiad, highlighting the 2011 event with 563 participants from 101 countries, where only one, lisa sauermann from germany, achieved a perfect score, while the next two runners-up were just one problem away from perfection.', 'duration': 74.597, 'highlights': ['Lisa Sauermann from Germany achieved a perfect score in the 2011 International Math Olympiad.', 'The 2011 International Math Olympiad had 563 total participants representing 101 countries.', 'The next two runners up in the 2011 Olympiad were just one problem away from a perfect score.']}], 'duration': 74.597, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM2944.jpg', 'highlights': ['Lisa Sauermann from Germany achieved a perfect score in the 2011 International Math Olympiad.', 'The 2011 International Math Olympiad had 563 total participants representing 101 countries.', 'The next two runners up in the 2011 Olympiad were just one problem away from a perfect score.']}, {'end': 301.167, 'segs': [{'end': 183.287, 'src': 'embed', 'start': 141.383, 'weight': 4, 'content': [{'end': 143.245, 'text': 'This process continues indefinitely.', 'start': 141.383, 'duration': 1.862}, {'end': 145.327, 'text': "Alright, that's kind of fun.", 'start': 144.246, 'duration': 1.081}, {'end': 147.729, 'text': 'We keep rotating and changing the pivot.', 'start': 145.907, 'duration': 1.822}, {'end': 150.231, 'text': 'And you can see why they call it a windmill process.', 'start': 148.289, 'duration': 1.942}, {'end': 154.295, 'text': 'And you can also see why they specified that no three points lie on one line.', 'start': 150.511, 'duration': 3.784}, {'end': 158.618, 'text': "You wouldn't want to run into the ambiguity where you don't know which pivot to switch to.", 'start': 154.855, 'duration': 3.763}, {'end': 161.881, 'text': "Okay, so, with all this setup, what's the question??", 'start': 159.539, 'duration': 2.342}, {'end': 167.723, 'text': 'Show that we can choose a point P in S and a line L going through P,', 'start': 162.842, 'duration': 4.881}, {'end': 173.145, 'text': 'such that the resulting windmill uses each point of S as a pivot infinitely many times.', 'start': 167.723, 'duration': 5.422}, {'end': 174.945, 'text': 'All right.', 'start': 174.585, 'duration': 0.36}, {'end': 177.486, 'text': 'Depending on your tolerance of puzzles.', 'start': 175.525, 'duration': 1.961}, {'end': 181.187, 'text': "for puzzles' sake, you might wonder why would anyone care about such a question?", 'start': 177.486, 'duration': 3.701}, {'end': 183.287, 'text': "There's a very good reason, in fact.", 'start': 181.807, 'duration': 1.48}], 'summary': 'Demonstrate windmill process using each point infinitely many times.', 'duration': 41.904, 'max_score': 141.383, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM141383.jpg'}, {'end': 240.404, 'src': 'embed', 'start': 201.463, 'weight': 6, 'content': [{'end': 204.587, 'text': 'maybe a difficult counting setup or an elaborate geometric construction.', 'start': 201.463, 'duration': 3.124}, {'end': 208.312, 'text': "But problem two, it's an unusually pure puzzle.", 'start': 205.188, 'duration': 3.124}, {'end': 210.533, 'text': "And in some ways, that's its charm.", 'start': 209.053, 'duration': 1.48}, {'end': 216.675, 'text': 'Proving that some initial condition will result in this windmill hitting all the points infinitely many times.', 'start': 211.074, 'duration': 5.601}, {'end': 219.076, 'text': "well, that doesn't test your knowledge of a particular theorem.", 'start': 216.675, 'duration': 2.401}, {'end': 222.117, 'text': 'It tests if you can find a clever perspective.', 'start': 219.696, 'duration': 2.421}, {'end': 224.398, 'text': 'But that blade cuts both ways.', 'start': 222.937, 'duration': 1.461}, {'end': 230.62, 'text': 'Without resting on an existing result from math, what could possibly prepare someone to study for something like this?', 'start': 224.958, 'duration': 5.662}, {'end': 234.701, 'text': 'And in fact, that brings us to the second unusual thing about this problem.', 'start': 231.3, 'duration': 3.401}, {'end': 240.404, 'text': "Based on the results, I'm guessing that it turned out to be much harder than the contest organizers expected.", 'start': 235.521, 'duration': 4.883}], 'summary': 'Problem two is an unusually pure puzzle, testing clever perspective and turned out to be much harder than expected.', 'duration': 38.941, 'max_score': 201.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM201463.jpg'}, {'end': 301.167, 'src': 'embed', 'start': 265.023, 'weight': 0, 'content': [{'end': 268.806, 'text': 'only 22 of them got full marks for question number 2.', 'start': 265.023, 'duration': 3.783}, {'end': 275.532, 'text': 'By contrast, 170 got a perfect score on problem 5, which is supposed to be about the same difficulty,', 'start': 268.806, 'duration': 6.726}, {'end': 279.815, 'text': 'and more than twice as many got a perfect score for problem 3,, which is supposed to be harder.', 'start': 275.532, 'duration': 4.283}, {'end': 287.542, 'text': 'Some of you might notice that only 6 students got full points for problem 6 that year, so by some measure that was the hardest problem on the test.', 'start': 280.856, 'duration': 6.686}, {'end': 294.841, 'text': 'In fact, the way I introduced things earlier was a little disingenuous, the full data would suggest that problem 6 was the real clencher.', 'start': 288.314, 'duration': 6.527}, {'end': 301.167, 'text': "But what's strange is that if you look at the results of those 6 students who solved this hardest problem,", 'start': 295.421, 'duration': 5.746}], 'summary': '22 students achieved full marks on question 2, 170 on problem 5, and only 6 on problem 6, making it the hardest.', 'duration': 36.144, 'max_score': 265.023, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM265023.jpg'}], 'start': 78.001, 'title': 'Math olympiad problem analysis', 'summary': 'Analyzes math problems from an olympiad, including the windmill process, an unusually pure puzzle, and the difficulty of international mathematical olympiad problems. it reveals that problem 6 had the lowest number of perfect scores (6 students) and problem 2 had the fewest perfect scores (22 students) out of 563 participants, indicating problem 6 was the most challenging despite being introduced differently.', 'chapters': [{'end': 183.287, 'start': 78.001, 'title': 'Windmill process math problem', 'summary': 'Discusses the windmill process math problem, outlining the process and requirements to show that a point and line can be chosen to utilize each point of a set infinitely, with a focus on the importance of the question.', 'duration': 105.286, 'highlights': ['The problem involves showing that a point and line can be chosen to utilize each point of a set infinitely.', 'The windmill process starts with a line going through a single point and rotates clockwise around the pivot until it meets another point, then the new point becomes the pivot.', 'The problem is significant due to its mathematical challenge and the practical implications it holds.']}, {'end': 240.404, 'start': 183.727, 'title': 'Unusually pure puzzle', 'summary': "Discusses problem two of an olympiad, an unusually pure puzzle that tests one's ability to find a clever perspective, and it's revealed that the problem turned out to be much harder than expected.", 'duration': 56.677, 'highlights': ["Proving that some initial condition will result in the windmill hitting all the points infinitely many times doesn't test your knowledge of a particular theorem but rather if you can find a clever perspective.", 'The problem turned out to be much harder than the contest organizers expected, based on the results.']}, {'end': 301.167, 'start': 241.104, 'title': 'Imo problem difficulty analysis', 'summary': 'Discusses the difficulty of international mathematical olympiad problems, revealing that problem 6 had the lowest number of perfect scores (6 students) and problem 2 had the fewest perfect scores (22 students) out of 563 participants, indicating problem 6 was the most challenging despite being introduced differently.', 'duration': 60.063, 'highlights': ['Problem 6 had the lowest number of perfect scores (6 students) out of 563 participants, suggesting it was the hardest problem on the test.', 'Only 22 students got a perfect score on problem 2, indicating its difficulty compared to other problems.', '170 students got a perfect score on problem 5, which is supposed to be about the same difficulty as problem 2, highlighting the varying difficulty levels among problems.', 'More than twice as many students got a perfect score for problem 3 compared to problem 2, indicating its higher difficulty level.']}], 'duration': 223.166, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM78001.jpg', 'highlights': ['Problem 6 had the lowest number of perfect scores (6 students) out of 563 participants, suggesting it was the hardest problem on the test.', 'Only 22 students got a perfect score on problem 2, indicating its difficulty compared to other problems.', 'More than twice as many students got a perfect score for problem 3 compared to problem 2, indicating its higher difficulty level.', '170 students got a perfect score on problem 5, which is supposed to be about the same difficulty as problem 2, highlighting the varying difficulty levels among problems.', 'The windmill process starts with a line going through a single point and rotates clockwise around the pivot until it meets another point, then the new point becomes the pivot.', 'The problem involves showing that a point and line can be chosen to utilize each point of a set infinitely.', "Proving that some initial condition will result in the windmill hitting all the points infinitely many times doesn't test your knowledge of a particular theorem but rather if you can find a clever perspective.", 'The problem is significant due to its mathematical challenge and the practical implications it holds.', 'The problem turned out to be much harder than the contest organizers expected, based on the results.']}, {'end': 959.292, 'segs': [{'end': 326.848, 'src': 'embed', 'start': 301.167, 'weight': 0, 'content': [{'end': 307.513, 'text': 'all of whom are clearly phenomenal world-class problem solvers, this windmill puzzle evaded 5 out of 6 of them.', 'start': 301.167, 'duration': 6.346}, {'end': 313.583, 'text': 'But again, this problem is not hard because of the background knowledge it demands.', 'start': 309.841, 'duration': 3.742}, {'end': 315.383, 'text': 'It asks only for insight.', 'start': 313.983, 'duration': 1.4}, {'end': 322.726, 'text': 'So how do you approach something like this? The first step with any puzzle is to simply play around with it and get a feel for it.', 'start': 316.284, 'duration': 6.442}, {'end': 326.848, 'text': "And it's always good to start simple and slowly get more complicated from there.", 'start': 323.087, 'duration': 3.761}], 'summary': '5 out of 6 world-class problem solvers failed to solve the windmill puzzle, emphasizing the need for insight and experimentation.', 'duration': 25.681, 'max_score': 301.167, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM301167.jpg'}, {'end': 384.595, 'src': 'embed', 'start': 359.29, 'weight': 3, 'content': [{'end': 365.195, 'text': "Looking back at the problem, it's asking you to show that for some starting position of the line, not any position,", 'start': 359.29, 'duration': 5.905}, {'end': 367.917, 'text': 'the process will hit all the points infinitely many times.', 'start': 365.195, 'duration': 2.722}, {'end': 375.205, 'text': 'So, for an example like this if you start with a line going through that troublesome middle point, what happens?', 'start': 370.12, 'duration': 5.085}, {'end': 381.071, 'text': "And again, at this point, we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", 'start': 375.766, 'duration': 5.305}, {'end': 384.595, 'text': 'You want to believe a result before you try too hard to prove it.', 'start': 381.552, 'duration': 3.043}], 'summary': 'Problem requires showing infinite hits for specific line position', 'duration': 25.305, 'max_score': 359.29, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM359290.jpg'}, {'end': 772.64, 'src': 'embed', 'start': 743.038, 'weight': 1, 'content': [{'end': 748.603, 'text': 'So once more, we have a cycle which hits all of the points and which ends in the same position where it started.', 'start': 743.038, 'duration': 5.565}, {'end': 752.086, 'text': "This time it takes 360 degrees, but that doesn't matter.", 'start': 749.183, 'duration': 2.903}, {'end': 755.929, 'text': "As the cycle continues, it'll hit all the points infinitely many times.", 'start': 752.526, 'duration': 3.403}, {'end': 763.675, 'text': 'Stepping back, there are two important lessons to take away from this puzzle.', 'start': 760.293, 'duration': 3.382}, {'end': 766.396, 'text': 'The first one social, and the second one mathematical.', 'start': 764.095, 'duration': 2.301}, {'end': 772.64, 'text': 'Once you know the solution and if you kind of ponder it a bit and turn it around in your head a couple times,', 'start': 767.397, 'duration': 5.243}], 'summary': 'Cycle hits all points infinitely, with 360 degrees. two lessons: social and mathematical.', 'duration': 29.602, 'max_score': 743.038, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM743038.jpg'}, {'end': 898.304, 'src': 'embed', 'start': 864.496, 'weight': 2, 'content': [{'end': 866.437, 'text': "We're finding what's called an invariant.", 'start': 864.496, 'duration': 1.941}, {'end': 870.139, 'text': 'Topologists do this when they count the number of holes in a surface.', 'start': 867.118, 'duration': 3.021}, {'end': 873.861, 'text': 'Physicists do this when they define the ideas of energy and momentum.', 'start': 870.679, 'duration': 3.182}, {'end': 878.164, 'text': 'or in special relativity when they define more abstract ideas like proper time.', 'start': 874.321, 'duration': 3.843}, {'end': 882.387, 'text': "As a student, it's easy to take for granted the definitions handed down to you.", 'start': 878.925, 'duration': 3.462}, {'end': 886.791, 'text': 'But the more puzzles you solve where the insight involves an invariant,', 'start': 882.888, 'duration': 3.903}, {'end': 891.194, 'text': 'the more you come to appreciate that each one of these definitions was once a clever discovery.', 'start': 886.791, 'duration': 4.403}, {'end': 898.304, 'text': "Terence Tao, one of the greatest modern mathematicians and the world's youngest IMO medalist,", 'start': 893.803, 'duration': 4.501}], 'summary': 'Discovering invariants leads to appreciating clever mathematical definitions.', 'duration': 33.808, 'max_score': 864.496, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM864496.jpg'}], 'start': 301.167, 'title': 'Windmill puzzle problem solving and invariant', 'summary': 'Explores windmill puzzle problem solving, challenging the audience to find a starting position of the line hitting all points infinitely, and emphasizes the importance of playing around before a rigorous proof. it also illustrates a windmill puzzle where a line passing through points changes colors, hitting all points infinitely due to the invariant of the number of points on each side staying constant during rotation, stressing the importance of identifying invariants in solving complex systems.', 'chapters': [{'end': 467.741, 'start': 301.167, 'title': 'Windmill puzzle problem solving', 'summary': 'Explores the windmill puzzle problem, challenging the audience to find a starting position of the line that will hit all the points infinitely many times, and emphasizes the importance of playing around and getting a feel for the puzzle before finding a more rigorous proof.', 'duration': 166.574, 'highlights': ['The windmill puzzle evaded 5 out of 6 world-class problem solvers, highlighting its difficulty level.', 'Emphasizing the importance of playing around with the puzzle and starting simple before getting more complicated.', 'Exploring the behavior of the windmill around different numbers of points and its cyclic nature.', 'Challenging the audience to find a starting position of the line that will hit all the points infinitely many times, and the importance of finding a rigorous proof for productive ideas.']}, {'end': 959.292, 'start': 467.741, 'title': 'Windmill puzzle: invariant and persistence', 'summary': 'Illustrates a windmill puzzle where a line passing through points changes colors, hitting all points infinitely many times due to the invariant of the number of points on each side staying constant during rotation, emphasizing the importance of identifying invariants in solving complex systems.', 'duration': 491.551, 'highlights': ['The windmill puzzle demonstrates the invariant concept in mathematics, where the number of blue and brown points remains constant during the rotation process, hitting all points infinitely many times, illustrating the importance of identifying invariants in solving complex systems.', 'The chapter highlights the significance of recognizing invariants in mathematics and physics, comparing it to topologists counting holes in a surface or physicists defining concepts like energy and momentum, emphasizing the value of discovering and leveraging invariants in problem-solving and understanding real-world applications.', "The puzzle serves as a valuable lesson in understanding invariants, as emphasized by Terence Tao, one of the world's youngest IMO medalists, who likened mathematical puzzles to fables, stories, and anecdotes that are important in understanding real-life problems, indicating the relevance of such contrived puzzles in preparing for solving real-world problems by identifying invariants."]}], 'duration': 658.125, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/M64HUIJFTZM/pics/M64HUIJFTZM301167.jpg', 'highlights': ['The windmill puzzle evaded 5 out of 6 world-class problem solvers, highlighting its difficulty level.', 'The windmill puzzle demonstrates the invariant concept in mathematics, hitting all points infinitely many times.', 'The chapter highlights the significance of recognizing invariants in mathematics and physics.', 'Challenging the audience to find a starting position of the line that will hit all the points infinitely many times.']}], 'highlights': ['Lisa Sauermann from Germany achieved a perfect score in the 2011 International Math Olympiad.', 'The 2011 International Math Olympiad had 563 total participants representing 101 countries.', 'Problem 6 had the lowest number of perfect scores (6 students) out of 563 participants, suggesting it was the hardest problem on the test.', 'Only 22 students got a perfect score on problem 2, indicating its difficulty compared to other problems.', 'More than twice as many students got a perfect score for problem 3 compared to problem 2, indicating its higher difficulty level.', '170 students got a perfect score on problem 5, which is supposed to be about the same difficulty as problem 2, highlighting the varying difficulty levels among problems.', 'The windmill process starts with a line going through a single point and rotates clockwise around the pivot until it meets another point, then the new point becomes the pivot.', 'The windmill puzzle evaded 5 out of 6 world-class problem solvers, highlighting its difficulty level.', 'The windmill puzzle demonstrates the invariant concept in mathematics, hitting all points infinitely many times.', 'The chapter highlights the significance of recognizing invariants in mathematics and physics.']}