There’s hope for you yet

Tatami Galaxy is tied with Ping Pong: The Animation for my favorite anime of all time, and this “simulator” that my good friend jonman over at Fine Whining and Breathing came up with captures a lot of what the story is trying to tell. If you’ve ever had a time of your life that you spent in constant regret, agonizing over what you could have done differently to make things better, check this out and brace for impact.

Tatami Galaxy Simulator 2016

1. Listen to this song in the background.

2. Read this extract:

MC: “I feel like something went awry with my life.”

Master: “Your life has not yet begun. You are spending overtime inside of your mother’s belly.”

MC: “It’s not that bad. I successfully obtained a life that people would be jealous of. But something is missing. Is this really it? There’s got to be some more meaningful life out there…More rose-colored, more sparkling! There might have been some college life with not a single dark cloud that would have satisfied me!”

Master: “What’s the matter? Are you half-asleep?”

MC: “I got to where I am by believing in my own potential! I’m not sure I’m saying it right…But why does my heart feel so cold? Maybe there’s a choice I should have made that would have led to some other possibility? Maybe the choices I made in my first year were wrong!”

Master: “You cannot use the word ‘possibility’ without limitations. Can you become a bunny girl? Can you become a pilot? Can you become a famous singer, or a superhero who saves the world with his powers?”

MC: “No, I can’t.”

Master: “Perhaps you could. But if you keep focusing your gaze on that which is unrealistic, you never will. The root of all your evil is in always relying on one of your other possibilities to get your wish. You must accept that you are the person here, now, and that you cannot become anymore else other than that person. There is no way that you can lead some worthwhile college life and feel satisfied. I guarantee it, so have confidence! There is no such thing as a rose-colored campus life, because there is nothing rose-colored in this world. Everything is all a bunch of colors mixed up, you see.”

King Disaster’s Math Manifesto

King Disaster’s Math Manifesto

This summer, rather than selling my soul to some corporate overlord and grinding myself into the dirt to make some big bucks working on Wall Street, I chose to stay at my college and do some academic research. I was assigned a mentor and we worked together to simulate the kind of project that one would take on in grad school or as a post doc. Eventually, we settled on the topic of random polytopes. Essentially, that means that you pick a bunch of points in \mathbb{R}^d, take their convex hull, and see what happens. The results are the 11+ page pdf I linked at the top of this post.

I can’t say I was thrilled with the project. Due to a couple of unfortunate lapses in communication, I got stuck with a topic which I did not find particularly interesting. (From what I gathered, the program administrators did not tell my mentor about my research preferences, nor did they inform him that he was supposed to do most of the legwork getting background material and a well-defined question prepared.) We also spent a large chunk of our time just doing background reading, finding what seemed like a feasible question, and then realizing that someone had already solved it in greater generality. When we finally did reach questions which had not been exhaustively answered, they were, simply put, way too hard to answer, especially within the span of five weeks. As a result, the paper I produced is purely expository. I am of the opinion that it brings some clarity and intuitive understanding to the results which we survey, but I suppose I’m biased.

Despite all this, I certainly enjoyed myself more than I have at any other job/internship. I think that research may be a good fit for me, if I can just find a field which intrigues me enough. I think perhaps a *slightly* more applicable area of math–for instance, theoretical statistics or random matrix theory–is more my speed. Even though I find answering abstract questions rewarding, if I’m going to devote as much time and energy to a project as I did to this one, I’d like a compelling reason to plod forward when the going gets tough. As I’ve heard on many occasions, perseverance is far more critical to research than sudden miraculous inspiration.

Overall, I think it was a positive experience. I got some more practice writing a technical paper, at the very worst, and I did learn about some pretty interesting stuff. (See the part in the paper about affine perimeter, and the part about making a convex chain in a triangle.) I think I’d like to give it another try moving forward, and next time I’ll do a better job of picking a damn project.

Putnam 2009-A1

I was about to go to sleep, but I haven’t posted in a while and this one is quick. I don’t think I have anything very insightful to say about how I came across this solution, since it came to me without much fiddling around. So, rather than trying and failing to give a good explanation, I’ll just leave a hint. The full solution will be included with my next post.

2009-A1. Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane?

Good luck!

Good luck!

Putnam 2013-A1

2013-A1. Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them.

 Obvious first step: what the hell does an icosahedron look like? Well, it looks like this:

Icosahedron

Also known as a d20. Nerd.

For any of you nerds that have played DnD before, this is a d20. Hope that helps.We’re looking for things that share a vertex, so we should probably look for things which share a vertex. We can see from the picture that about each vertex, there are five equilateral triangles, so we want to show that two of these have the same number. 

How can we do that? It seems hard to pin down one specific number that would have to occur twice. For instance, we can make 19 faces 2 and 1 face 1, or we could make 13 faces 3. In the first case, some 2’s will be touching, but in the second case, some 3’s will be touching. This is probably not going to work.

If we can’t prove what we want directly, we’ll have to try and prove it indirectly. That is, we’ll assume that the statement is false and find a contradiction. When is the statement false? Well, each of the 5 faces about each vertex must be different. Again, this statement by itself doesn’t help us a whole lot, because there a lot of nonnegative integers to choose from. So we look back at the problem statement for guidance. The fact that the sum of the faces must be 39 narrows our choices down a lot.

39 is not a particularly large number, not when we have 20 faces to consider AND the smallest number we can put on a face is 0. The average number on a face must be less than 2. This doesn’t give us a lot of wiggle room, so let’s see if we can find our contradiction here. We want to make our numbers as small as possible. If all the numbers about each vertex are different, that means that the smallest they can be is the integers 0, 1, 2, 3, and 4. Those add up to 10, so at best we can have a sum of 10 about each vertex. Great! If we can decompose the icosahedron into 4 set of these bad boys, we’ll be done, because then our sum is 40 which is too big.

…unfortunately, a quick glance at the picture above and a little experimentation will quickly show that this can’t be done. But we shouldn’t scrap this idea yet, because it seems like it’s going somewhere. The problem is that we can’t look at a bunch of these other things without having them overlap, in which case we’re overcounting some faces and not counting others at all. Instead of trying to pick some subset of these 5 faces around 1 vertex, let’s consider all of them instead. By the symmetry of the icosahedron, we’ll be overcounting each face the same amount of times.

We get one set of 5 faces around a vertex for each vertex, so there are 12 of them. If each has 0, 1, …,4 around it, that means the sum around each of these is 10, so the sum of all of them is 120. Note that all we’re doing is adding the value of a face each time it occurs in one of these sets of 5. That happens every time a vertex of the face is the center of a set of 5, which will happen once for each of its vertices. Since each face has three vertices, that means we’re adding each face to the total 3 times, so in total we end up with 3 times the sum of all the faces. Thus 3 * (sum of the faces) = 120, so the sum of the faces is 40. Last I checked, that’s greater than 39.

What does this all mean? Remember, what we just did was create the absolute smallest sum of all the faces when no two faces with a shared vertex have the same number on them. This lower bound ends up being greater than the sum which we know that the faces have, so it can’t be done! This means that our assumption must have been invalid, i.e. it’s not possible that there are 5 different numbers for each set of 5 faces around a vertex. So there exists some set of 5 about a vertex with the same number. These share a vertex, so we’re done.

I like this problem because a) it’s cool, and b) it shows that you shouldn’t always give up on an approach that seems promising at the first sign of trouble. Sometimes you just need make a few tiny adjustments to find yourself at the answer.

Some clarification

Some clarification

A friend and I were talking about my B Gata H Kei review, and he pointed out that I didn’t address a large part of the story, specifically Yamada and Kosuda’s interaction with the Kanejou siblings. To be honest, I thought that the post was dragging on, so I decided to not address it and hope that none of my 0 readers would notice. Alas, my prayers were left unanswered, my hopes dashed by jonman‘s sharp eye.

Anyway: I was wrong to suggest that there is nothing more than raging hormones between Yamada and Kosuda. There is plenty of evidence that suggests otherwise, primarily in the form of the Kanejou family. Kanejou Kyoka is a smokin’ hot transfer student who vies with Yamada for the position of “biggest bitch alive” “queen of the school.” In order to assert her dominance, Kyoka decides to try and steal Yamada’s boyfriend. If Kosuda were really just looking for a wet hole to fill, he probably would have fallen for her seduction and tried to bone her while the two alone in her room.  Similarly, Kyoka’s brother Keiichi—an extremely attractive Harvard Business School student—takes a liking to Yamada and asks her to be his girlfriend. Even after she learns that he is a virgin (one of the most important qualifications for her boyfriend), she still turns him down in favor of Kosuda. It’s plausible that Kosuda rejected Kyoka on a purely physical basis. Maybe he just finds Yamada more attractive. It seems unlikely; after all, a boob in hand is worth two in the bush. (xDDD) Regardless, there isn’t a chance in hell that Yamada would have turned down Keiichi if the only thing she had on her mind was getting her freak on. Clearly there’s something more going on here.

What the hell is my gripe, then, you ask? Sure, it’s out of wedlock, but my centuries-old puritanical fundamentalism and I can eat shit, right? They’re in love, nothing wrong with some dirty dirty sex, yeah?

This brings us to my main issue with the show. If I were just watching some horny, pea-brained teenagers trying to get it in, that wouldn’t have been so bad. That’s pure degeneracy, and it’s here to stay. I’ve been mostly desensitized. What bothers me is that Yamada and Kosuda—and society as a whole—have managed to defile love itself. The situation at hand is a double edged sword. On one hand, I’m glad to see that it’s still hard for us to divorce love and lust entirely. The downside of that, of course, is that as sex becomes more cheap and meaningless, one of two things will necessarily happen. One possibility is that we sever the connection between love and sex. If we do this, we lose an incredibly powerful means of expressing a critical—perhaps the most critical—emotion.  If we can’t separate the two, then the corruption will continue to spread, eventually engulfing love itself. This seems to be where Yamada and Kosuda are taking things. Their love was born of degeneracy (their desire to fuck each other without an emotional attachment) and breeds even more degeneracy (their even stronger desire to fuck each other overshadowing their emotional connection). What’s worse, these days, their story passes as “sweet” and “cute.” We’ve grown so accustomed to the filth that we dwell in that B Gata H Kei seems downright adorable in comparison. And that makes me sad.