On intelligence and intellectualism

Everything that follows is a result of what you see here. (Dr. Lanning’s hologram, in I, Robot (2004))

Intelligence is a subtle and often controversial topic. Because each of us wants to believe that he or she is intelligent, and that his or her nation is more intelligent, and hence superior, to others, the way we think about intelligence is heavily biased and culturally dependent. This makes it very hard to reconcile our own ideas, the ideas of others, anecdotal experiences, and scientific data about intelligence; after all, it’s not even clear whether the word “intelligence” has the same meaning in each context. Most of us are victims of this confusion, but I think that I have been hit particularly hard.

I say this, of course, because my personality and my career up to this point have been heavily influenced by all the cues that suggest that I am a highly, perhaps exceptionally intelligent person, for some definition of the word. In early years, these cues were primarily social; for example, my teachers remarked that I should be moved into a program for gifted students, which eventually happened in grade 4. In later years, as I became better adjusted socially and my classmates more tolerant, I fell naturally into the role of one with extensive knowledge, both trivial and serious. I was also starting to get high test scores in mathematics and science and, eventually, awards at national and international competitions. I gather that some people regarded me as a bit of a role model, and, after opening this blog and examining the stats page, I realized that I had underestimated my fame.

In light of this, what I have recently begun claiming, namely that I am not a highly intelligent person, has generally been met with one of two reactions, namely, that I am either depressed or being modest. It might be true that I am depressed; I definitely have been heartbroken for a few months, although I think that this is drawing to an end. It is not true that I am modest (but I am also sure that I am not as arrogant as some people think I am). The purpose of this post is to elaborate on what I mean when I claim not to be that intelligent, and to defend that statement. But let the reader beware: this does not conform to the standard of a book or a research paper. It contains no scientific data and is backed solely by anecdotal evidence.

My feeling that something was wrong was most likely the result of the convergence of several realizations, the earliest of which began about two years ago.

The first of these arose from my exposure to highly gifted students. I recall in particular the Canadian Computing Competition (CCC), the National Olympiad Finals (NOF) in chemistry and physics, and the International Chemistry Olympiad (IChO); I attended these events and I can say that getting to know the other competitors was a real pleasure and that many interesting conversations were exchanged. What made me uneasy is that I was invariably the least well-rounded and socially adept member of these groups. I had previously believed the common misconception that it is normal for intellectually gifted children to be socially inept and narrowly focused, and therefore I had been satisfied with the way I was. My personal experiences, along with data from Lewis Terman’s longitudinal study of gifted children, directly contradicted this notion. Indeed, there is some evidence for, and a conspicuous lack of evidence against, the hypothesis that all “mainstream” forms of intellectual ability (logical/mathematical, visual/spatial, linguistic/verbal, interpersonal, and so on) are positively correlated. Furthermore, the “general intelligence factor”, g, has been shown to correlate positively with height, longevity, and physical attractiveness, and negatively with divorce rate. There is a lack of evidence suggesting that it correlates negatively with any desirable characteristic. Some studies purport to show a negative correlation between IQ and happiness, but many others fail to find any correlation at all. (Note that “happiness” is hard to define, let alone measure.) Also, it has been shown that high IQ correlates with higher age of first sexual intercourse and first marriage, but I’m not convinced that these are either desirable or undesirable characteristics.

The second also arose from careful observation, and the resulting cognitive dissonance. A widely held belief is that North Americans of East Asian descent are, on average, brighter than their peers of other ethnic origins, and that highly gifted students are overwhelmingly likely to be of East Asian background. These beliefs, at first, seem entirely reasonable. About half of Woburn’s gifted graduating class of ’10 is of East Asian descent, and at least 60% of the contestants in Stage 2 of the Canadian Computing Competition were of East Asian descent in each of the three years I attended. (I have excluded the contestants from Hong Kong and Beijing, for obvious reasons.) In 2009, all four members of the Canadian team in the IOI (International Olympiad in Informatics) were of East Asian descent, and in 2010, all four members of the Canadian team in the IChO and all five members of the Canadian team in the IPhO (International Physics Olympiad) were of East Asian descent. And yet, East Asian Canadians/Americans are rarely the ones to achieve great things, at least based on what I considered great; very rarely do they make breakthroughs in science and technology, at least not of a calibre for which one might be awarded a Nobel Prize or become the CEO of a multi-million dollar corporation, and they are entirely absent from the lofty positions held by such individuals as Albert Einstein, Richard Feynman, Stephen Hawking, John von Neumann, Linus Pauling, and Donald Knuth. One might argue that winning the Nobel Prize is an unrealistic goal, which is true, but isn’t very relevant; it isn’t as though all the East Asians are crammed right below that metaphorical line, and the absence of East Asians is visible when other metrics are considered, too, such as h-index, or impact on education. (For example, textbooks on par with, say, Introduction to Algorithms are rarely written by East Asians, and I can’t think of any popularizers of science, such as Carl Sagan, Martin Gardner, and Bill Nye, of East Asian descent.)

The third is that Bloom’s taxonomy, which I had dismissed in grade 4 because I did not understand it, seems to have some validity. Based on my own observations, I had proposed a multi-stage model of intelligence, which goes roughly as follows:

• The stages from lowest to highest are knowledge, understanding, problem solving, and synthesis.
• One must achieve mastery of one stage in a particular field before moving to the next stage in that field.
• One tends, at any given point, to dismiss lower stages as intellectually juvenile.
• The individuals considered highest-achieving in any field of study are those who reach the final stage.

(The second and third points were inspired by Kohlberg’s model of moral development, as you may have noticed.)
Later, I realized that Bloom had come up with the same ideas a long time ago. Oh well. In any case, whereas I have excelled in certain contests, none of these were heavily design-oriented (such as the University of Toronto Space Design Contest). Furthermore, I am convinced that some olympiad-type contests are more problem solving oriented than others; in particular, the International Mathematical Olympiad (IMO) and its local team selection contests are probably the most heavily loaded, and I was never an IMO contestant, nor have I done particularly well in the Canadian Mathematical Olympiad (CMO), USA Mathematical Olympiad (USAMO), or Asian Pacific Mathematics Olympiad (APMO). Also of note is that I performed poorly (relative to expectation) at IOI 2010, and I think that this has a lot to do with the fact that the problems were of a different style than I anticipated (and prepared for). (The Chinese team also did not do as well as expected, and I conjecture for the same reason. The Americans, on the other hand, did very well.) This puts me at the bottom half of the third stage. (Note that I am considering mathematics, physics, computer science, and chemistry to be a single field, since they all require similar intellectual abilities.)

The fourth is that I have been rejected by several colleges. Last year, I was rejected by MIT, Caltech, Stanford, Harvard, Yale, and Princeton. This year, I was rejected by MIT, Harvard, UChicago, Columbia, and Yale. To be rejected by one college doesn’t mean much; the various rejection letters I have received have all made a note of this. They all sound something like this:

The Admissions Committee has carefully reviewed your application to … and we regret to inform you that we could not offer you admission. Each year thousands of highly talented students apply and thousands must be turned down because of our limited space. Please note that this decision does not reflect any judgement of your character or your ability to succeed. We are sure that you will be offered admission to many other great colleges. Best of luck in your future endeavors.

To be rejected by eight colleges though, some more than once… well, I think that this does mean something. By my final year of high school, I was exceptionally well-versed (for a high school student) in chemistry, computer science, physics, and linguistics; less so, but still highly, in mathematics, biology, and logic. I venture to claim that my personality could have been used as the definition of the word “intellectual”. Why was I rejected? A simple explanation is that I am not very well-rounded. I feel, however, that this alone is inadequate. I think that these colleges can be reasonably expected to consider two important questions when deciding whether to admit a candidate:

1. Can this candidate be expected to fit into the culture of this college and to contribute positively to it?
2. Is this candidate likely to be successful after graduation?

(In other words, when the rejection letter says that the rejection is not a judgement of character or of prospects of success, this is probably a “white lie”.)
How does well-roundedness fit into this framework? It doesn’t really hit on the first point; there are a lot of clubs and societies at these schools and you don’t have to be interested in everything in order to fit in. I think that it probably hits on the second point. Success is never as clear-cut as simply being very good at something, unless you get really lucky; you have to be prepared to handle a wide variety of possible challenges. Furthermore, people who are good at several things are probably more intelligent than people who are good at only one thing, because more intelligent people require less effort to master something, and therefore can handle many things in the time it takes a less intelligent person to handle one or a few. Another interesting observation is that almost everybody from the NOF got into at least one of these schools, and almost everyone who actually got in either has a girlfriend or, at any rate, has had some success with girls. (The exception is one particular girl, who, instead, has a boyfriend. Nobody so far has told me that he is gay, even though intellectuals usually understand on principle that there’s nothing wrong with being gay.) I don’t think it is a coincidence that I’m both uniquely foreveralone and rejected by all those American colleges. Having a one-dimensional, black-and-white personality, as I do, is not attractive, either to girls or to admissions committees.

Based on the last four paragraphs, here is a comprehensive statement about where I stand when it comes to my own ability:

I am of fairly average, perhaps slightly below-average general intelligence. This is backed by the fact that I have no special artistic or musical talent, am inferior athletically and socially, and have a poor reflex response time. (The latter strongly correlates with g.) I have a fair amount of talent in mathematics and the hard sciences, but not the ridiculous amount that most people around me seem to think I have. My success in academic competitions is due to lots of hard work and preparation; in particular, participation in the IOI would never have been possible without the inspiration that the late Maria Plachta provided. By devoting so much time to learning subjects in which I have been held to excel, I was able to surpass the level of many who are more talented than I am. However, I sacrificed other forms of development to attain this level, whereas they did not have to, because they are more talented and required less time to become (almost?) as good as I am. Furthermore, because of my limited intelligence, my mastery of these subjects is shallower than theirs; I am comparable to a student who works every problem in the textbook and, having seen all possible problem types, is prepared to ace any test, without any real understanding of the material. It is for this reason that I have not excelled in mathematical competitions; mathematics is unmatched among human endeavors in depth and breadth, and “working every problem” is simply not feasible, nor is it possible to excel without extremely strong problem-solving skills. I have also shown no particular creative talent; if I have any, it is not visible because my limited intelligence has prevented me from reaching the “synthesis” stage. This pattern of skills seems to be common among students from East Asian backgrounds (although I specifically decline to comment here on “innate” or genetic ability), and probably explains the fact that the success of these students in school often does not carry over into life after school. This does not necessarily imply that these students are worse than their counterparts of other ethnic backgrounds; it merely implies that they are not as much better as they appear to be. (I am not racist, and I will agree just as readily as anyone that an individual, not his or her skin color, determines his or her own success; as a matter of fact, I have utmost confidence in my East Asian friends from the NOF; I am sure that they are no less likely to succeed than their counterparts of other ethnic backgrounds.) The American colleges to which I applied have, without a doubt, received countless applications from students like me, and probably reached the same conclusions I have; this explains why I was rejected, along with scores of other East Asian applicants who appeared to be highly qualified, whereas my highly intelligent friends, who truly are highly qualified, were not. There is no doubt that women find intelligent men attractive, but this statement must be understood in the context of sexual selection, a particular type of natural selection. Men with higher g-factors were once able to better solve problems related to obtaining food and shelter and defeating potential predators; today they tend to be better-equipped to handle the complex and multifaceted intellectual and social challenges posed by modern industrialized society. There is no reason to expect that women should find someone like me, with only mathematical talent, to be attractive, and my average g-factor along with my lack of development in non-intellectual areas makes me uniquely unattractive. My inability to relate to people with different interests and values than my own probably exacerbates the situation by preventing me from being attracted to women who might just happen to be attracted to me. It is not intelligence that made me unpopular and unattractive, but my single-minded devotion to intellectualism. Intellectualism, most often accompanied by deism or secular humanism, is a good life philosophy, regardless of whether you are highly intelligent or not not particularly intelligent, but to be an intellectual and nothing else will probably not end well.

If you met me five years ago, you would probably have been sure that I would one day attend MIT, Harvard, or an institution of comparable prestige, and perhaps go on to be a distinguished professor, as that was my dream at the time. I was living a dream that was never meant to be. My awakening has left me heartbroken (perhaps clinically so), and I have struggled with depression over the course of the last few months. The time has come for me to accept my own mediocrity. Putting this off will only prolong the pain and delay the inevitable.

In many ways this post is like a suicide note; my biological life will continue, but the person I once was will cease to exist. In other ways it is like a letter of resignation from a career that is proving to be altogether too stressful for me to handle. I will no longer identify as an intellectual, as being an intellectual has not served me well; striving for greater conformity will probably make my life easier. I have recently turned eighteen and some would say that my adult life is beginning. I intend to leave behind much of my life up to this point. After the finals of the International Collegiate Programming Contest at the end of this month, I will stop participating in contests altogether, and I will endeavor to forget how well I once performed. I will stop chatting with friends whom I associate primarily through contests, although I mean them no disrespect or ill will and do not intend to prevent them from communicating with me altogether. This will be the last post published on this blog; I may blog again in the future, but it won’t be anytime soon, nor will it be on this site. The old posts, the ones that I have not deleted, will stand forever, in the hope that they will occasionally prove useful to the curious mind who stumbles upon them, but there will be no new posts. In general, I will strive to eliminate my public online identity, which has become well-known only because of my perceived intellectual strength, although I remain committed to maintaining the online judge for Woburn CI’s Programming Enrichment Group, the organization that did more for me than I could ever state in words.

I will stop advocating for civil rights on ideological grounds, and instead argue such issues on practical grounds, if at all. I will no longer seek out intellectual challenges, but whatever it is that life calls on me to do, I will continue to strive to do well. I have always made a point of being a decent human being, and this will remain a priority. But, I repeat, I will no longer be an intellectual. If you think you understand, you probably do not really; if you really understand, you are probably disappointed with me, and rightly so, as I am giving up everything I have stood for, firmly, for what seems like most of my life up to this point.

I have sworn off the study of theoretical computer science, and, following the conclusion of my internship, I will probably buy a laptop with Windows, and stop using Linux, which I had taken up primarily for its friendliness to programmers. I will begin this fall at the University of Toronto studying Physical and Mathematical Sciences. I will not go into academia, and I am unsure of what I intend to do after college, but I will try to avoid going to graduate school if possible. (Right now I am considering being a high school teacher or a private tutor, although, depending on how my internship at Facebook plays out, I might not rule out going into the software industry; I do not expect my competitiveness to be damaged by my decision to no longer actively study algorithms.) Should I happen upon any opportunities to do research, I will almost certainly not take them. I hope I have not discouraged anyone else from being an intellectual or going into academia; rather, I hope that this story of my failure can serve as an example to others. I wish you all the best of luck in whatever paths you pursue.

Until we meet again,
Brian Bi
May 15, 2011

Stay tuned

Today, I turned eighteen. This is not really a big deal.

Legally, it is of little consequence in Canada. You can drive at sixteen — something I have not been particularly eager to do. You can drink and smoke at nineteen — but it’s not as though anyone really cares, and those aren’t exactly my idea of fun either. In the past, you could have sex at fourteen, although the law has become somewhat more complicated in recent years. Of course, eighteen is the age at which you get to vote, but the general election was last week, and there is not likely to be another for another four years.

But so far I haven’t really talked about myself. The significance of the age of eighteen is not primarily procedural, but symbolic and cultural. In North America, eighteen has, traditionally, been the typical age at or around which young people graduate from high school and move out of their parents’ homes. It therefore symbolized a transition from childhood to early adulthood. But not really. I got kicked out of my house around this time last year, and graduated from high school shortly afterward.

Nothing special happens when you turn eighteen. You don’t magically become responsible, mature, self-sufficient, or capable of sound judgement.

So why the post today? Because, despite all that I have written above, a significant change in my life is in the works. If I could, I would tell you more about it today. It is, after all, a date marked off on the calendar, my eighteenth birthday, so it would strengthen the association. However, the phrasing of the next post, or few posts, will depend on the decisions on two more applications for transfer admission that I have submitted, which I expect to receive shortly. Until then, stay tuned. (I promise that the writing will not be as crappy as it is in this post.)

I couldn’t help it

Today

Today, March 4th, 2011, is the day I convinced someone to actually try to do something with his life, to try to achieve something rather than, in his words, “sighing, grieving, sorrows or sadness”.

Does everybody else need to know about this? Nah. But he asked me to make a note of it, to hold him to his promise to himself to change, and this was the most convenient way for me to do so.

(Feel free to disregard this post.)

Why q = CV?

This post will be brief. Why is it that in some textbooks, the definition of capacitance is taken to be the equation ? (Related: )

To me, this is just as bad as stating the equation for density as (or that for velocity as ). It might be correct, but we write firstly to emphasize the importance of the quantity we are actually defining (the density) and secondly because it explains what density really is: a quantity that is higher when you can stuff more mass into a smaller space.

Now, the first time that I read through Fundamentals of Physics by Halliday and Resnick (3 years ago?), I skipped the chapter on capacitance just because I couldn’t understand what the equation was really defining. In retrospect, I was probably just being dim or lazy that day; I should have made more of an effort. But I maintain that what confused me is that the very term capacitance, plus the fact that capacitors store charge, implies that capacitance should express how well a structure can hold charge, and that’s not obvious at all from , because it’s not clear where the is coming from.

If we write the equation as , it becomes much more clear. Not only does this actually define , but it also shows what high capacitance really means: it’s the ability to hold a lot of charge without building up too much of a potential difference (after all, when charging a capacitor, once the potential difference builds up to the level of the EMF, the capacitor stops charging). And we generally tend to think of the as being set up by the , rather than the being a consequence of the capacitance and some potential difference that’s already present, which is what suggests.

Of course, the equation , as presented, has its value too — to cast capacitance as a conversion factor between the stored charge and the established potential. Just as and have their value. Just not as definitions.

Learning Hebrew (לומד עברית)

Surprisingly (or not, depending on how you look at it), I’ve decided to use my Reading Week to read. What am I reading, you may ask?

The first language I learned to speak was Mandarin, but after I got to Canada I started learning English instead. Before long, unfortunately, I had all but forgotten how to read and write Mandarin. I can still understand most of a conversation in spoken Mandarin, and I can say some basic sentences myself, but I was once fluent in the language, and I am no longer.

I started learning French in grade 4, as it is a required subject for all Ontario students; I took French all the way until grade 12. I never had any trouble with grammar, but I always felt weak in actually speaking and listening. It was not until a while after the end of my last French class that I looked back and realized I was more or less fluent in speaking French. However, I still find native French speakers completely incomprehensible.

In grade 10 I took a Spanish course, and in grade 11 I took another. I would have liked to continue with Spanish in grade 12, but I had had to rearrange my schedule slightly. (I can’t remember why this is, but it probably had something to do with the unexpected death of Ms. Plachta and my desire to avoid being a TA for a certain other teacher.) As a result I could not fit it into my schedule, but in any case I went ahead and taught myself the remaining tenses and moods of Spanish verbs. So now I know the grammar of Spanish relatively well, but my vocabulary is severely lacking. I’m not anywhere near fluent.

In grade 11 I also began to study Latin, mostly on my own, since Latin club occurred at the same time as another commitment (I think it was Prefects). Nevertheless, the handouts from the club were my kick-start to Latin grammar — once I had been introduced to the concept of noun declension, the language began to make sense to me, and I pursued its study independently. I never had a comprehensive Latin education, because I never sat down with a textbook and read it cover to cover, but I may look into getting a copy of the Cambridge Latin Course one of these days.

Last summer, as I was selecting my courses for my first term at the University of Waterloo, I thought it would be nice to take a Latin course. As is typical, I could not have my dream schedule because of conflicting lectures and tutorials and so on, and I could not find any section of Latin that didn’t conflict with my other courses. So I decided I’d give Ancient Greek a try instead. Cicero said about the Latin language, Nōn enim tam praeclārum est scīre Latīnē, quam turpe nescīre: Indeed, it is not so much excellent to know Latin as it is a shame not to know [it]. I had begun to feel the same way about Ancient Greek — much of the vocabulary of science comes directly from Latin, yes, but most of the rest comes from Ancient Greek. So I took one course in Ancient Greek last term. Greek was a lot different from Latin and the Romance languages. It was more challenging because the vocabulary is different, but not too challenging because there are still many English words of Greek origin. The grammar was different, but not too different, apparently partially because the Romans often copied the Greeks, but also because it was in the Indo-European tradition. And the script didn’t present much of a challenge, because I had been using the selfsame letters in mathematics for years.

So this Reading Week I’m undertaking a more ambitious project — learning Hebrew. This should be significantly harder for the following reasons: 1) the script is foreign to me; the only letters I have seen in math are א and ב, used to denote transfinite cardinals, and, indeed, the script is not fully alphabetic like those of most (all?) European languages, nor logographic like Chinese characters, but rather an abjad; 2) I am starting with almost no knowledge of the vocabulary, other than loanwords like שבת (shabbat), which I, as a non-Jew, would never use in everyday life anyway; 3) The grammar is expected to be completely different from that of the Indo-European languages.

So far I’ve taught myself to write the Hebrew alphabet with my left hand (after all, the script is written from right to left) and I’m learning some basic vocabulary; apparently, since both identity and possession can be / are expressed without linking words, I can already write some basic sentences. Onward!

(You may be wondering why I would choose Hebrew of all languages. I’m not really sure, to tell the truth. Arabic is more widely spoken, but I’ve heard that it’s extremely difficult. Japanese seems to be popular among my friends, but I’m not sure whether I’m ready to handle all the speech registers. Well, I can’t just sit here and rule out all the other languages in the world anyway.)

Bleargh

I miss the days when the PEG Judge was simple.

The motivation behind complicating it was the implementation of Analysis Mode: that is, after a contest has completed, its problems should be moved to the main problem set. But I made a really bad design decision here — storing main Judge archive problems and contest problems in the same place, that is, a MySQL table called ‘problems’. Effectively this means that, in order to determine whether a problem belongs to the main Judge, it’s necessary to carry out a join between the problems table and the contests table (which contains the start and end times for each contest). And certain queries on the submissions then require first joining the problems and contests tables and then joining this with the submissions table. Because these tables have been steadily growing, load times have become, in some cases, a few seconds, which I find unacceptably slow, since the introduction of Analysis Mode. It’s funny because I’m going to do an internship with Facebook this summer, and I’m sure they would never have hired me if they had seen the way I’ve been using SQL queries recently. (I’m kidding, but the SQL really is that bad.)

The trouble is that the Judge was never designed with this kind of growth in mind, or all the features that have been implemented since its inception. It looks like I might have to overhaul the backend at some point. Given that Hanson overhauled the backend when he took over from Guru, I should’ve seen this coming.

The short-term solution would be to separate the main judge from the contest judge, and one of the easiest ways to do this would be to set up a subdomain and a separate database. In this way all those joins for checking whether a problem is in an active contest or a contest that has ended or … would be eliminated. I originally didn’t do this because it would break the SSL (not that SSL is very important for a site like this, of course), as multiple name-based virtual SSL hosts from the same IP and port are impossible (see this). But then I remembered that my host actually gave me two IP addresses, so I can have wcipeg.com point to one and contest.wcipeg.com point to the other. I plan to implement this at some point during this year (I can’t make any guarantees, because it would take a lot of work, much of which would simply be reversing the damage I’ve done).

In the long term, if we are to continue supporting all the features we currently have without unacceptably compromising the speed, it will become necessary to write a custom database engine, as there seems to be quite a significant discrepancy between what we need and what SQL is intended to deliver. For example, it makes sense to store submissions in a flat file sorted by date, because then, almost all the time new submissions would be added to the end of the file (which is fast), deletions would be infrequent, and changing the dates on submissions would probably never be necessary. (By “submissions” I mean the metadata; the actual files themselves are not a big deal.) In other places, where deletion may occur more frequently, other data structures may be more appropriate. I’m not sure whether this will ever get done. Maybe the Judge won’t even last that long; after all, it’s not clear who’ll be left to motivate Woburn students three years from now. Or maybe it’ll last, but I’ll become too busy to administer it, and, like Hanson before me, have to pass it on to someone younger (probably one of the PEG members who are currently in grade 10).

An “impossible integral”

On Friday, Bosco asked me how to evaluate . It turns out it is not necessary to carry out the integration, as the question only requires knowing the derivative, which is given by one of the two parts of the Fundamental Theorem of Calculus (“the integral as an antiderivative”). But he went on Wolfram|Alpha to find the antiderivative. You can view the output here. He asked me how one would obtain this result.

Wolfram|Alpha, of course, uses Mathematica as its backend for solving integrals and other math problems, and although it can now show steps for some integration problems, one should keep in mind that the algorithms that Mathematica uses to evaluate integrals are not at all like those used by humans. I’m not sure how Wolfram|Alpha shows human-like derivations, but I am led to suspect that it uses some sort of heuristic search, where the vertices of the graph are expressions and the edges are simple transformations. I say this because it occasionally gives very poor but correct results (example). In any case, steps are not shown for nonelementary antiderivatives.

Nevertheless, just by examining the output I was able to draw some conclusions about how to derive the solution using “human” algorithms. In particular, the fact that appears in the antiderivative suggests the substitution , that is, . Making a substitution similar to this results in a transformation of the integrand into the form . We follow up with the substitution , which converts the hyperbolic functions into circular ones, and obtain an integrand of .

I was stumped about how to proceed, so once again I turned to Wolfram|Alpha. The part puzzled me, but I realized that this could be re-written as . Differentiating the output did not help — of course the result was , but it looked like magic. It was the factor of 12 that ultimately tipped me off — this had to have resulted from the integration by parts trick (where a multiple of the original integral appears on the right side, so it can be moved to the left side and solved for) and the part looked like the part of the integration by parts formula. After that, it all fell into place, although somewhat painfully.

You can view the full derivation here [source], if you are bored enough.

Two points to note:
1. I would never have been able to solve this if I had not already known the answer.
2. Nevertheless, I would still like to say the following: In many mathematical texts and papers all you will ever see is a “forward” derivation similar to what I have shown in the PDF I linked to (and I’m not talking specifically about this problem). What I’ve described in this post should give you some idea of the actual thought process that goes into creating these apparently magic forward derivations, which initially seem as though they could only have arisen through divine inspiration.

Damn

The 2011 ACM ICPC World Finals, an annual programming competition between teams from post-secondary institutions consisting of three members each, which was slated to be held in Egypt at the end of the month, has been postponed for at least three months.

Here is the email received from the ICPC organizers:

Dear Participant,

The 2011 World Finals is postponed.

Contact your travel agent or airline for a refund or travel voucher.
Consular Travel Warnings should make it easier for you to avoid penalties.

The earliest date will be the last week of May.
Please block the last week of May on your calendar.

Please block the last week of June on your calendar.
The month of July and the first two weeks of August are also under consideration.

We hope to announce the date by February 10th.
We plan to announce both the place and date by February 28th.

I look forward to seeing every one at a spectacular World Finals later this year.

Bill

P.S.
If you need accommodations in Sharm El Sheikh from February 20 – March 5,
please contact <worldfinalsoffice@acmicpc.org>

Looks like I’m going to have to ask for a week off, since I’m (hopefully) going to be on an internship at that time (with whom, though, is not yet certain).

Regardless of whether or not the World Finals can be held in Egypt at all this year, I can only hope for a quick resolution to the civil unrest. There’s not too much more I can say, because such issues are complex and it’s hard to get enough information to form a balanced view.

Handwaving geometric interpretation

Last night a friend asked me how to derive the moment of rotational inertia for a disk (). I confirmed that one treats the disk as a collection of infinitely thin nested hoops, but then wondered whether there is a geometric interpretation that sidesteps the need to break out the full power of calculus.

I was pretty tired, so I spent some time fruitlessly hoping an idea would pop into my head. It was not until after I had gotten into bed that I realized that any geometric realization of the problem would have to end up “looking like” the integral. So much, in fact, that this post is pretty worthless and not at all cool.

The proposal is to “flatten” the disk out into a triangle with base length and height . Imagining the base to be horizontal, each horizontal cross-section of the triangle, taken at distance from the top, represents the radial cross-section of the disk at radius . We see that if we take a horizontal slice of the triangle with height near , we get an area of approximately , which is the same as the mass of an infinitely thin hoop with radius .

So, this is not a particularly exciting setup; right now, taking the area of the triangle: : only gives back the mass (but at least that proves we haven’t made a mistake yet). We don’t want but rather . So we take each horizontal cross section (length ) and convert it into a square prism with length and base of infinitesimal 4-thickness. We then stack all these square prisms in 4-space to get a simplex with base and height .

In a slightly handwavy argument, I go on to claim that since the area of a triangle is , and the volume of a pyramid is , the 4-volume of a 4-simplex is . (This is handwavy because the formula is derived using the integration power rule anyway, which is also exactly what you need to solve this problem by integration, so the approach really is equivalent, as I said before.) So we obtain , as desired.

We see that, in general, the integral can be recast as the hypervolume of a -dimensional simplex with height and base ; this comes out to , which is… exactly the power rule.