[CIMC 2015 Part 2] Journey of the Blue-White Slippers

(Nontopical life update: Current 18.06 homework status: 34% (mildly screwed, probably won’t finish before I leave my cozy home for the U.S. and I usually struggle to get into the mood for homework while traveling, but I guess I’ll have to))

[18.06 status panel: 34%]

(I’ve been spending most of my uptime doing said homework and running errands, and my downtime catching up on Last Week Tonight with John Oliver while farming the Flight Rising Coliseum. And, okay, making the above status panel. Live version here courtesy of Dropbox’s Public folder. No regrets.)

Day 3 (Excursions)

Morning routine snipped. We come to the middle school again to eat breakfast and gather; the contestants will be taking their tests here (accompanied by one bottle of “Buff” energy drink each) while the rest of us will be going on an excursion. Before this happens, though, two Taiwanese contestants ask me and Hsin-Po some math problems. There’s a geometry problem, which I fail to solve:

(paraphrased) In triangle △ABC, ∠A is 40° and ∠B is 60°. The angle bisector of ∠A meets BC at D; E is on AB such that ∠ADE is 30°. Find ∠DEC.

Hsin-Po figures out that, once you guess (ROT13) gur bgure boivbhf privna vf nyfb na natyr ovfrpgbe naq gurl vagrefrpg ng gur vapragre, lbh pna cebir vg ol pbafgehpgvat gur vapragre naq fubjvat sebz gur tvira natyr gung gurl vaqrrq pbvapvqr.1 Then, there’s a combinatorics problem in a book with a solution that they’re not sure about:

(paraphrased) 15 rays starting at the same point are drawn. What is the maximum number of pairs of rays that form obtuse angles?

This happens really close to the test starts and although I have this feeling it’s isomorphic to a notable combinatorial problem, I don’t manage to articulate the isomorphism until it’s too late and they have to go. Indeed, this is more or less equivalent to (ROT13) Ghena: gur tencu unf ab sbhe-pyvdhr naq n pbzcyrgr guerr-cnegvgr tencu vf pbafgehpgvoyr. After thinking though the solution on their book, though, I realize I’ve never seen this proof of said theorem before! (But later I realize it’s actually the just very first proof that Proofs from the BOOK offers. I probably skipped it because it involved induction as well as some algebraic manipulations that looked much less intuitive and natural than they really were, so it didn’t look as cool as the later proofs. Oooooops.)

I suspect I wouldn’t do too well if I had to participate in that contest right then. But anyway, excursion.

After a long bus ride, we arrive at our first destination, Jingyuetan (淨月潭 lit. Clear Moon Lake2), allegedly the sister lake to Taiwan’s own famous[citation needed] Sun Moon Lake. We tour the place on a wall-less car and look at the lake and lots of trees. During a stop, I take some pictures of sunflowers and bees, as well as a stand selling Taiwanese sausages.

[Lake and ferry]

[Bee and sunflower]

[Sausages advertised as from Taiwan!]

The car blares weird music during the tour, such as a version of Für Elise with all the accents on different beats and a disjointed remix of the viral Chinese song 小蘋果 (Little Apple)3 with two other Chinese songs, connected with mumbling English rap segues. We also eat boxed lunches here while sitting on tiny, cramped foam mattresses on the dirt floor.

Our next stop is a museum, where there are lots of ancient historical artifacts I’m not very interested in. I find a collection of certificates involving or quoting Chairman Mao more intriguing:

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(8,9,7)

(streak)

I always tell myself, okay, I will actually just draw something facetiously and get it over with, nobody comes to this blog to admire my GIMP mouse doodles, but then perfectionist tendencies kick in and I get carried away and it ends up taking more than an hour or so.

linear

Okay. I hope I didn’t make any mistakes. 38-hour reprieve. Let it be.

edit: In hindsight, (6,10,6) is likely better. Sorry.

edit 2: None of the words are “multiply” or “multiplies”. Don’t take this too seriously; it sucks.

X + Y (movie)

On Wednesday I got to see a special screening of the film X + Y. You know, the one about the autistic boy who goes to compete in the IMO. You can watch the trailer if you haven’t already.

Disclosure: the ticket was free, courtesy of my math teacher (who appears at 1:06–1:07 in the trailer) having helped the filming process. (I visited once and got to look at some of the cool equipment. Also, far away from everything, one of the director assistants sort of interviewed me. That is the full extent of my contribution, okay?) Except I was also sick with a cold so I might have been kind of miserable. Also I didn’t really have dinner that day, and we got home really late so I had to stay up even later doing homework. So those are the extent of my biases.

I guess this is a review of sorts.

The most important thing I have to say is this: X + Y is not a film primarily about math competitions or the IMO. It is a film about love, about autism, about accepting people who are different, about conquering your own psychological demons, about gender and family and cultural roles. But mostly about love. The film gets big novelty bonus points for a reasonably authentic look at the high school mathematics olympiad scene, but if you go watch this as a former contestant looking to relive some vicarious moments of glory and triumph through hard mathematical work and thinking, I’m pretty sure you’ll be disappointed. None of the main character’s important character development moments are related to becoming better at math. The IMO is largely a well-researched and extensively utilized plot device. (This is one of the acceptable usages of the word “utilize”, okay? Dear classmates: please stop using it as a seven-letter synonym for “use”.)

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Haskell and Primes

“I have been told that any encryption becomes safer if the underlying algorithm is maximally obscured, what is most conveniently done by coding it in Haskell.” — rankk

Functional programming is terribly addicting! Partly I think the completely different way of thinking makes it feel like learning programming, and falling in love with it, all over again. Partly there’s this evil sense of satisfaction from using $s (and later <$>s and =<<s and &&&s) to improve readability for initiated Haskellers and worsen it for everybody else. Partly it’s because Learn You a Haskell for Great Good! is such a fun read — there are too many funny bits to list but my favorite so far is when the author analyzes the first verse of Avril Lavigne’s Girlfriend.

Although I think my code in Haskell tends to be more readable than in other languages, code obfuscation in Haskell is almost natural: all you have to do is refactor the wrong function to be “pointfree”, which means that even though it’s a function that takes arguments, you define it without parameters by manipulating and joining a bunch of other functions. Example (plus a few other tiny obfuscations):

isPrime = liftA2 (&&) (liftA2 ($) (all . ((.) (0 /=)) . rem) (flip
    takeWhile [2..] . (flip (.) $ liftA2 (*) id id) . (>=))) ((<) 1)

QQ wordpress why no Haskell highlighting

Also, for some reason, you can do this in Haskell:

ghci> let 2 + 2 = 5 in 2 + 2
5

(via Haskell for the Evil Genius)


Okay, but seriously now. I wrote this about my journey to learn functional programming in the programming babble post half a year ago:

The main obstacle I have is that it’s hard to optimize or get asymptotics when computation is expensive (a big problem if you’re trying to learn through Project Euler problems, particularly ones with lots of primes).

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Φllotaxy

This is beautiful. Why do they have to make it sound all mysterious and difficult? That’s (the reciprocal of) the golden ratio, by the way.

[Textbook]

Transcript since the resolution is far from awesome: “Most angiosperms have alternate phyllotaxy, with leaves arranged in an ascending spiral around the stem, each successive leaf emerging 137.5° from the site of the previous one. Why 137.5°? Mathematical analyses suggest that this angle minimizes shading of the lower leaves by those above.”

\frac{137.5^\circ}{360^\circ} = 0.3819\ldots = 1 - 0.6180\ldots = 1 - \frac{1}{\phi} = \phi^{-2}

edit: Go figure, the angle itself is called the golden angle.

Matrix Intuition

Stopped by a friend’s house a few days ago to do homework, which somehow devolved into me analyzing what programming language I should try to learn next in a corner, which is completely irrelevant to the rest of this post. Oops.

Anyway, in normal-math-curriculum-land, my classmates are now learning about matrices. How to add them, how to multiply them, how to calculate the determinant and stuff. Being a nice person, and feeling somewhat guilty for my grade stability despite the number of study hours I siphoned off to puzzles and the like, I was eager to help confront the monster. Said classmate basically asked me what they were for.

Well, what a hard question. But of course given the curriculum it’s the only interesting problem I think could be asked.

When I was hurrying through the high-school curriculum I remember having to learn the same thing and not having any idea what the heck was happening. Matrices appeared in that section as a messy, burdensome way to solve equations and never again, at least not in an interesting enough way to make me remember. I don’t have my precalc textbook, but a supplementary precalc book completely confirms my impressions and “matrix” doesn’t even appear in my calculus textbook index. They virtually failed to show up in olympiad training too. I learned that Po-Shen Loh knew how to kill a bunch of combinatorics problems with them (PDF), but not in the slightest how to do that myself.

Somewhere else, during what I’m guessing was random independent exploration, I happened upon the signed-permutation-rule (a.k.a. Leibniz formula) for evaluating determinants, which made a lot more sense for me and looked more beautiful and symmetric

\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n A_{i,\sigma_i}

and I was annoyed when both of my linear algebra textbooks defined it first with cofactor expansion. Even though they quickly proved you could expand along any row or column, and one also followed up with the permutation formula a few sections later, it still felt uglier to me. Yes, it’s impossible to understand that equation without knowledge of permutations and their signs, but I’m very much a permutations kind of guy. Sue me.

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[IMO 2012 Part 6] Mostly Not About Excursions

Yes. I know it’s been more than a month. Blogging motivation decreases, but the responsibility of that stay tuned doesn’t go away.

It’s okay. It’s all worth it because the stuff in the games room is absolutely ridiculous. Warning: huge post.

[IMO 2012 Problem 4 on a cake.

Our old friend, the monster of a functional equation, in edible form. In the games room. Did I mention abso-zarking-lutely ridiculous?


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