
The “Windmill” Problem on the 2011 International Mathematical Olympiad [video] - bobbykrk
https://www.youtube.com/watch?v=M64HUIJFTZM
======
rvz
The approach to solving this problem looks very elegant to viewers
with/without a mathematical background and the author's use of visual
explanations towards solving it step-by-step helps untangle the ambiguities in
this puzzle.

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.

~~~
gbjw
The author is Grant Sanderson
([https://en.wikipedia.org/wiki/3Blue1Brown](https://en.wikipedia.org/wiki/3Blue1Brown))
who has an undergrad degree in math from Stanford and worked at Khan Academy
before starting his YouTube channel 3Blue1Brown. Also, the student who is
mentioned in the video (Lisa Sauermann) as having solved this problem at the
2011 IMO (and attaining the only perfect score) just recently started as a
Prof. at Stanford
([http://web.stanford.edu/~lsauerma/](http://web.stanford.edu/~lsauerma/)) as
a 27 year old.

~~~
jedberg
Wow. She went straight from getting her PhD at Stanford to teaching there.
That's almost unheard of. She must not only be a brilliant mathematician but
an amazing teacher too.

~~~
rsj_hn
Not only do teaching skills play no role in getting tenure at a place like
Stanford, teaching is actually a threat to research productivity and research
universities will hire professors who do as little teaching as possible,
leaving most of it to older who professors who are no longer productive
researchers or to post docs, teaching assistants or other staff.

To understand how this works, the researcher will apply for some grant, say
$300,000 to study some question in geometry. Now, why does a mathematician
need grant money when their only tools are a paper and pencil (maybe a laptop
with Tex installed)? First, the university gets 1/3 of that money as
"overhead", so the researcher is left with $200,000. Then, the researcher will
pay to "buy out" his teaching load which is more money paid to the university,
say $150,000 to not teach 2 classes for a year. With the remaining $50,000, he
may spend money to fund a post doc to come and assist him for a semester.
Again, that money goes to the university. So the researcher may get $300,000
but it all ends up in the pocket of the University, which in turn pays him a
good salary with the assumption that he keeps the grants coming. A place like
Stanford gets about 1/3 of its funding from these research grants, 1/3 from
its endowment, and 1/3 from tuition. It hires researches to get the grants,
grad students and adjuncts to teach, and the sports teams and other events
help with endowment.

Thus research professors are hired on the basis of their ability to avoid
teaching loads, not on their teaching skills.

~~~
jedberg
> say $150,000 to not teach 2 classes for a year

Is that why she is teaching two classes in her first semester, including one
which is lower division? Usually you don't put the crappy teachers in the
lower division classes, you give them graduate seminars.

I appreciate your cynicism, but based on her teaching load, I'm going to guess
that she is _also_ a good teacher.

~~~
jimmyvalmer
> Usually you don't put the crappy teachers in the lower division classes

Since when? The hard-and-fast rule is junior faculty are assigned intro
classes. We often give youthful teachers higher marks than crusty, doddering
emeriti, perhaps for good reason, perhaps not.

> I appreciate your cynicism, but based on her teaching load, I'm going to >
> guess that she is also a good teacher.

GP did not question her teaching ability, but your inference of said ability
from her impressive ascent at Stanford. It's a bit like inferring LeBron James
must really be mature since he entered the NBA straight from high school.

------
chongli
This phenomenon, whereby a person who knows the "trick" 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!

~~~
OscarCunningham
Tanya Khovanova has a paper
[https://arxiv.org/abs/1110.1556](https://arxiv.org/abs/1110.1556) with a list
of some more of these problems with easy solutions that are hard to find. She
calls them "Jewish Problems" because they were used by USSR universities to
discriminate against Jewish applicants. The applicants would fail to solve
them but the university would justifiably be able to claim they were easy.

~~~
dxbydt
Hey, thank you so much for posting this! I picked a random problem ( the one
on logs) in that paper and solved it under 1 minute. I feel like a million
bucks now!

------
eruci
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.

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.

~~~
boyobo
Just pick a point and rotate a line around that point continuously. Keep track
of the number of points on the left. Since this count is essentially
continuous (the jumps are of size 1), at some point during this rotation you
will have an almost-balanced configuration (same number of points on left and
right, up to parity error).

~~~
eruci
Yes, but it must be a point with certain properties, such that the same number
of points are on either side of the line initially, otherwise, it would not
work.

~~~
boyobo
I am giving an alternate proof of your implicit statement "There exists a line
which has the same number of points on each side". You did this by computing
duals of cell arrangements. I am arguing that you don't need to do that.

The proof I outlined will work for any point. Initially it might have the
wrong number of points on both sides but for some rotation it will have the
correct number of points on both sides.

------
x3n0ph3n3
This guy makes excellent math visualization videos -- some of the best I've
ever seen.

~~~
ranie93
Yes! I really enjoy them as well-- and he has his animation software on
github: [https://github.com/3b1b/manim](https://github.com/3b1b/manim)

~~~
Chris2048
Would be interested in how this stacks against mathbox2*

* [https://acko.net/blog/mathbox2/](https://acko.net/blog/mathbox2/)

------
carapace
This is hella cool.

"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.

~~~
boyobo
Are you trying to say that math is hard because the teachers are withholding
all the tricks?

> 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?

~~~
carapace
> Are you trying to say that math is hard because the teachers are withholding
> all the tricks?

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/](http://iconicmath.com/) or "Proofs without Words"
[https://en.wikipedia.org/wiki/Proof_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.

~~~
boyobo
> Check out William Bricken's "Iconic Math"
> [http://iconicmath.com/](http://iconicmath.com/) or "Proofs without Words"
> [https://en.wikipedia.org/wiki/Proof_without_words](https://en.wikipedia.org/wiki/Proof_without_words)
> or the other 3Blue1Brown videos for that matter.

Yes, those references demonstrate that _some_ mathematical facts can be
demonstrated easily, given the right presentation.

I agree that there is large gap between current mathematical presentations and
the optimal presentation.

I am curious how effective the optimal presentation is. How easy can we make
math? We have to be careful of the trap that the 3b1b video warns us against -
when we understand something it is very difficult to put yourself in the shoes
of a beginner. Something that looks like an elegant and clear presentation may
seem like gibberish to the beginner (look at the YouTube comments on the 3b1b
video).

I tried to google "Alan kay teaching calculus smalltalk" and didn't find
anything. I am curious to see how much he actually taught the kids. It's clear
that the typical college student, after taking a typical calculus class,
doesn't really get the point of calculus. They may be able to follow some
algorithms for differentiating and integrating but I don't think they
understand when they can apply calculus.

~~~
carapace
I went and looked the Alan Kay thing up. It's not calculus, just acceleration.
Sorry! It's described in "The Real Computer Revolution Hasn’t Happened Yet"
[http://www.vpri.org/pdf/m2007007a_revolution.pdf](http://www.vpri.org/pdf/m2007007a_revolution.pdf)
in the section titled "Children Discover, Measure, and Mathematically Model
Galilean Gravity"

I think an important piece of the puzzle (no pun intended) is customized
presentations with feedback for individually-tailored education. But then it
occurs to me that group activity is also crucial for learning, eh? I'm not an
expert.

------
_Microft
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 "spoiler" moment
in it or if we could watch it bit by bit in case that we get stuck?

~~~
lelf
The problem:

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.

~~~
mlevental
if not all points in P are coplanar then what does it mean to rotate
clockwise? perpendicular to which axis?

~~~
bcyn
They are all coplanar, but no three points are colinear. The first sentence
just means that there are at least 2 points in the set.

------
gorgoiler
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.

------
d--b
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.

Any formal proof along those lines?

~~~
ghusbands
By envelope, you likely mean the convex hull. By "crosses inside the
envelope", you probably mean it intersects the hull. That does not work. If
you have a triangle inside a triangle and a central point inside that, that
gives you seven points. If you start on the inner triangle, intersecting the
outer triangle (convex hull), you can repeatedly visit the points of the
triangles without touching the central point.

------
tromp
I enjoyed watching this similarly insightful video [1] on "The hardest problem
on the hardest test" of the Putnam Competition.

[1]
[https://www.youtube.com/watch?v=OkmNXy7er84](https://www.youtube.com/watch?v=OkmNXy7er84)

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

------
sAbakumoff
From the same channel :
[https://www.youtube.com/watch?v=jsYwFizhncE](https://www.youtube.com/watch?v=jsYwFizhncE)
overview of very elegant connection between blocks collision and PI.

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

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

~~~
authoritarian
Numberphile has a couple videos about it

[https://www.youtube.com/watch?v=Y30VF3cSIYQ](https://www.youtube.com/watch?v=Y30VF3cSIYQ)

[https://www.youtube.com/watch?v=L0Vj_7Y2-xY](https://www.youtube.com/watch?v=L0Vj_7Y2-xY)

~~~
AlexCoventry
The actual question is at
[https://www.youtube.com/watch?v=Y30VF3cSIYQ#t=5m31s](https://www.youtube.com/watch?v=Y30VF3cSIYQ#t=5m31s)
, if you don't want to hear a 5 minute rant about how hard it is.

------
AlexCoventry
I'm going to have to watch this later, because the cat I've got on my lap
appears to be deeply alarmed by the blinking eyes of the "pi" avatar.

