Elementary school math, again

Remember when you were five years old and you learned that one plus one is two? Remember when you were just about to enter middle school or sometime around there and you learned about finding x? And how you had to add some stupid constant on both sides or divide through by a variable and you were like, “What’s so goddamn important about this x anyway?”

And then you went to high school and finding x through simple addition and subtraction gradually got easier, but then you learned harder ways to find x for seemingly no reason. And if you got far enough in high school math you learned about some seemingly black magic where x was zero but apparently not really zero because of something called a limit?? What the heck?? And what were these fancy elongated S symbols?

And then you went even further than that, you went so far that you stumbled onto abstract algebra and realized that almost everything that you learned when you were little could be thought of as operations equipped on the set of integers \mathbb{Z} with the algebraic structure of a group or ring.

Let me explain.

By algebraic structure I just mean that the integers (which math people like to call \mathbb{Z}) have some super obvious but also really nice properties that we’ve taken for granted in elementary school, because not all sets are created equal and \mathbb{Z} just happens to be super privileged.

For example, there’s this property that for any two integers a and b, adding them up will give you another integer. Really obvious, right? In fancy speak we say that the integers are closed under addition, or: \forall a,b\in\mathbb{Z}, a+b\in\mathbb{Z}.

Then, there’s the property of associativity which should be obvious to us since we’ve seen it jammed down our throats so many times in grade school. Formally, \forall a,b,c\in\mathbb{Z}, (a+b)+c=a+(b+c).

Next, there’s the property that there exists an additive identity 0 such that every integer plus 0 equals itself. Again, really obvious. In fancytalk, \forall x\in\mathbb{Z}, \exists 0\in\mathbb{Z} such that 0+x=x and x+0=x.

Finally, there is the property that there exists an additive inverse -a such that every integer plus its additive inverse equals the identity, 0. This is obvious: -a+a=0, or a+(-a)=0, any way you slice it.

So these four properties being satisfied, as obvious as that is, defines a special object in mathematics called a group. A group is a set G equipped with some operation such as addition, multiplication, composition, rotation, or what have you; as long as that set with that operation satisfies these four group axioms. So when we try and solve for x in one-step equations like in seventh grade, what we were really doing was working in the additive group of integers. I won’t get into it now, but if we can also throw multiplication and the distributive property in the recipe of a group, we get an even spicier object called a ring.

So why define this horribly formal thing? Why do we need to call this the “additive group of integers?” Why can’t we just do our damn addition problems? Well, for starters, it’s freaking cool. The first time I saw this stuff my mind was blown. We just grow up assuming that this is just the way it is, and we add 5 on both sides to cancel the -5 in x-5 just because that’s how it’s done. We just follow a strict recipe for solving equations without knowing why. Well, this is the why.

Second, group theory can be applied to pretty much anything and everything that might have such a structure. The next basic additive group that we could talk about are the additive group of integers modulo n, otherwise known as cyclic groups because the integers just keep recycling around and around like numbers on a clock. The circle of fifths, an analog clock, the days of the week, anything that has a cycle can be isomorphic to the cyclic group of integers.

What the heck does isomorphic mean? The formal definition is kind of long to explain from scratch, but intuitively it means that two groups are isomorphic to each other if one group seems like another group in disguise. For example, the days of the week {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} equipped with the operation of “progressing to the following day” (yes, operations can be defined in an informal way like this!) is isomorphic to the more abstract representation of this group, \mathbb{Z}/7\mathbb{Z}, defined as {0, 1, 2, 3, 4, 5, 6} with the operation of addition and the added caveat that 7=0. (Formally speaking, this definition is horribly incorrect, but I’m just trying to put it in a way that makes sense.) For example, 8 is equal to 7+1, which is equal to 0+1, which is equal to 1. So 8 is 1 modulo 7. In the same vein, 25 is 7+7+7+4=0+0+0+4=4, so 25 is 4 modulo 7.

Why are they isomorphic? Because the former group (the days of the week) looks like it could be the same thing as \mathbb{Z}/7\mathbb{Z}. We can see this if we assign the element “Sunday” to 0, “Monday” to 1, “Tuesday” to 2, and so on all the way to “Saturday” being 6. With this one-to-one assignment (keyword being one-to-one! or, formally, bijection) we can see that “progressing to the following day” is the same thing as “adding by 1” in \mathbb{Z}/7\mathbb{Z}.

If you’re beginning to realize it (and start getting your mind blown), you’ll see that “If today is Friday, what day of the week will it be 4 days from now?” and “What is 5+4 in \mathbb{Z}/7\mathbb{Z}?” are the same question. Instead of counting “Saturday, Sunday, Monday, Tuesday!” we can say “5+4 is 9, which is 2 mod 7” and if we wanted we could see that 2 corresponds to Tuesday, as we had assigned earlier. Obviously this is a really easy example, but what if I asked what the day of the week would be 1000 days from now? You can’t count those days one by one ad infinitum.

You might be saying, “Well, only a stupid person would do that. A realistic person would break the 1000 up into multiples of 7, with the knowledge that every 7 days it’ll be the same day of the week. Then we get 7(142)+6=1000, so we’d just need to count 6 days instead of 1000.”

Dude. You just did the same exact thing as computing 1000 modulo 7.

Of course, cyclic groups are still pretty basic. There are a huge variety of groups that are insane in their own right when explored. But I’ll leave that up to you to explore, if this blog post sparked your curiosity even a little bit.



I am a double agent (kind of)

Subtitle: I have so many things I want to do with my life and there’s no time to do any of them in time because I’m leaving for college in less than a month.

Why the title? I’ve been looking for a good proofwriting course that is pretty easy to get into so that I could have an easier time learning some higher-level math courses if the time comes. That search is finally over thanks to many of my friends that decided to go to Caltech instead of MIT (I will miss them so much and it makes me a little sad every time I think about it). Caltech’s curriculum is theoretical from the start, and it shows since they have an into math class that freshmen can take over the summer called Math 0.

The full class apparently comes equipped with video lectures and the ability to send assignments to graders that give super helpful, detailed replies and annotations. But I was only able to get my hands on the PDF notes and problems, which honestly is more than enough seeing as I’m having a lot of fun figuring out the problems on my own and I’m halfway done. If anyone’s interested in the PDFs, Caltech or some administration would probably shoot me, so just give me a private message or email me (found in my About page).

So another thing that I’ve really put off but I really want to do is review 18.02, something I touched on last blog post. Denis Auroux is honestly such a wonderful human being and I love listening to his lectures and his amazing accent and his perfect handwriting and speed-erasing. I’m really sad he’s not at MIT anymore (he teaches at Berkeley now). ASEing out of 18.02, a.k.a. getting full credit for it after passing an equivalent final exam, would mean I could pursue math a semester early like differential equations or linear algebra instead of reviewing what I already know (or maybe even take an introductory engineering course or other elective). I’ve gone through most of 18.02’s material once, and I took a course on the differential half of multivariable calculus last summer, so I’m hoping I can just pass by winging it at this point.

Other weird things that I committed myself to:  a real estate salesperson license. It shouldn’t be that hard, just a lot of time to learn the material so I can pass the exam. And I have to slog through a 75 hour timed course online before I take that. The good thing is I technically have until December until the online course times out, so I’ll be doing that when I have truly reached the pinnacle of boredom.

Speaking of boredom, some other goals and activities I’ve been trying to do when I have free time this summer: get a 99 on Oldschool. I have no idea why I started playing again (I checked in-game that I first made my account over ten years ago) but one thing led to another and me and three other school friends memed our way back into Gielinor. I’m talking we would Skype almost all day for a period of several weeks just grinding out the early game, fueled by our desire to simply keep up with each other. Now there is much less unity, but me and my very close friend still play from time to time. Since I’m at 87 fletching right now and it’s profitable (if only by a small amount) to fletch magic longbows until 99, I’m just going to string those anytime I have a few minutes of free time. Those minutes add up.

Another videogame-related activity that I seem to have lost all desire for: League. The only time I play is usually very late at night if friends are on. When I get to college, I’m assuming that I will get my ass handed to me with all the coursework, so that coupled with the fact that the game doesn’t really appeal to me much anymore means that I’ve weaned myself off pretty effectively. It’s still fun to watch LCS, though, don’t get me wrong. (Congrats to Doublelift for getting 1000 season kills!)

And finally, I got a DSLR. My first one, actually; it’s the Canon Rebel T6i. I literally just got it and haven’t even read the manual yet, and I’ve just taken a couple practice shots, but I feel like I’m already starting to fall in love. That coupled with two new whiteboards that I bought (a smaller one for my house and a huge one for my dorm room) would probably make for a revitalization in Youtube videos.

And I also want to learn Java further, but at this point when I finally get to doing that I’ll probably be learning it for my major anyway.


Okay, I’m going to end this blog post here because I really want to tackle this Math 0 problem: Let a,b\in\textbf{R}. Show that if a+b is rational, then a is irrational or b is rational.



Integration by parts for multiple functions (n>2)

Let me preface this by saying that the things I’ll get into aren’t even remotely useful from a practical standpoint. This is basically for shits and giggles. They might be useful if you don’t have a calculator/computer and want to rip your hair out, however.

If you take/took BC Calculus or Calc II you’ve probably heard of the integration by parts technique. You can look at it as an inverse product rule that lets you take the integral of a product of two functions f(x)=u and g(x)=v. An example would be

\displaystyle \int x^2\sin{x}\ dx.

In the above case you would apply IBP twice to decrement the power of x^2 to a constant. If you continue on I’m just going to assume that you’ve seen this already and know how to do these kinds of problems.

What if we wanted to find the integral of a product of three functions, or four, or n? Sometimes such an integral might not be expressed in terms of elementary functions, but there’s probably an approximation out there using a computer. Sometimes the solution exists, however.

For two functions, the solution to integration by parts is

\displaystyle \int u\ dv = uv-\int v\ du.

This is derived by taking the integral on both sides of the product rule. So to find the n-function integration by parts formula, it makes sense to first derive the general n-function product rule.

Let’s start small with the product of three functions, uvw. Taking (uvw)' is equivalent to (uv)'w+(uv)w' by treating uv like a single function. Now we use the product rule again on uv to get

\displaystyle (uvw)'=u'vw+uv'w+uvw'.

This seems to make sense; for a product rule of three functions, there are three terms with each function having one and only one derivative for each term. Let’s see if this trend continues for four functions f_1 f_2 f_3 f_4. We can consider f_1 f_2 f_3 as an entire function and apply the product rule to get (f_1 f_2 f_3 f_4)' = (f_1 f_2 f_3)'f_4+(f_1 f_2 f_3)f_4'. But we already found what (f_1 f_2 f_3)' is! We just used different letters. So we now know that

\displaystyle (f_1 f_2 f_3 f_4)'=f_1'f_2f_3f_4+f_1f_2'f_3f_4+f_1f_2f_3'f_4+f_1f_2f_3f_4'.

Our pattern is now confirmed! We can continue like this on towards infinity, since if we do it this way we have a formula for an (n-1)-function product rule that we can apply towards the n-function product rule. The only small challenge is formulating this pattern concisely. Using a combination of pi and sigma notation, we can state the following proposition.

Proposition 1 (General product rule). Let f_1, f_2, \cdots, f_n be n different differentiable functions of x. Then

\displaystyle \frac{d}{dx}\prod_{i=1}^{n}f_i = \sum_{i=1}^{n}\frac{f_i'}{f_i}\prod_{j=1}^{n}f_j.

Now that we have the general product rule, we just have to take the integral of both sides with respect to x to get the general integration by parts formula. Doing so, we get

\displaystyle \int\prod_{i=1}^{n}f_i = \int\sum_{i=1}^{n}\frac{f_i'}{f_i}\prod_{j=1}^{n}f_j\ dx.

Although this is technically a valid formulation of the general IBP formula, I claim that integral signs are interchangeable, that is,

\displaystyle \int\sum_{i=1}^{n}f_i\ dx = \sum_{i=1}^{n}\int f_i\ dx.

This is due to the sum rule of integration: \int(f+g)\ dx=\int f\ dx + \int g\ dx.

Proposition 2 (General integration by parts). Let f_1, f_2, \cdots, f_n be n different continuous and differentiable functions of x. Then integration by parts is given by

\displaystyle \prod_{i=1}^{n}f_i = \sum_{i=1}^{n}\int\frac{f_i'}{f_i}\prod_{j=1}^{n}f_j\ dx.

This isn’t in a ‘useable’ form per se yet since we need to expand that sum and product notation to actually apply it in an integration, but this is as clean as the theorem is going to get. To see IBP in its familiar form, we make the substitutions u_1=f_1(x), u_2=f_2(x),\cdots,u_n=f_n(x) to get

\displaystyle \prod_{i=1}^{n}u_i = \sum_{i=1}^{n}\int\frac{du_i}{u_i}\prod_{j=1}^{n}u_j.

Let’s see what this form looks like when n=3.

\displaystyle \prod_{i=1}^{3}u_i = \sum_{i=1}^{3}\int\frac{du_i}{u_i}\prod_{j=1}^{3}u_j

\displaystyle u_1 u_2 u_3 = \int\frac{du_1}{u_1}(u_1u_2u_3)+\int\frac{du_2}{u_2}(u_1u_2u_3)+\int\frac{du_3}{u_3}(u_1u_2u_3)

\displaystyle \int u_1 u_2\ du_3 = u_1u_2u_3 -\int u_2u_3\ du_1 - \int u_1u_3\ du_2.

This last form is finally something we can work with when trying to find, say, the fiendish integral

\displaystyle \int xe^x\cos{x}\ dx.

I’ll leave the work up to you if you so choose, but using the n=3 IBP formula I got a solution of

\displaystyle \int xe^x\cos{x}\ dx = \frac{e^x(x\cos{x}+(x-1)\sin{x})}{2}.

That’s all for now.

An Optimal Buildpath

Fair warning: if you don’t play League of Legends this post may not be interesting to you. Or it still might. Who knows. Oh, and for people who actually want to use relevant versions of these statistics, this post was made in the middle of Patch 6.2.

Armor is a statistic that reduces incoming physical damage taken. It’s similar to magic resist, which is another defensive stat that reduces incoming magic damage taken. The Wikia article I linked does a good job of explaining how armor is calculated, but i’ll restate it here: all incoming physical damage to your champion is multiplied by a factor of \frac{100}{100+A} where A stands for armor. For example, if you have 100 armor, you’ll receive \frac{100}{100+100} or 0.5 times all incoming physical damage.

It sounds like you can theoretically stack armor forever and reduce your incoming physical damage to a multiplier so low that you’d take close to zero damage at all times. This graph of the multiplier function shows you why that’s not possible:


Screen Shot 2016-01-31 at 10.46.29 AM

The horizontal axis is armor, while the vertical axis is the multiplier. Even at 400 armor, which is an absurdly high amount, you would still receive 20% of incoming physical damage which is relatively high compared to what you invested.

Another reason why stacking solely on armor is a bad idea is because of the notion of effective health. Effective health is the amount of damage required to kill you, taking resistances into account. If you build resistances, your EH will always be higher than your HP. If you have a lot of armor but not that much base HP, it won’t matter how much armor you have since it doesn’t take a lot to kill you. In addition, physical damage isn’t the only source of damage: magic damage and true damage (damage that is dealt directly to your HP without exception) is also prevalent. Here’s a simple example: if you have 3000 HP and 100 armor, you have an EH of 6000, which is equivalent to having 1500 HP and 300 armor. The downside to having 300 armor is when the opposing team has a good source of magic damage as well, making your armor useless.

Since every point of armor requires you to take 1% more of your maximum health in physical damage to be killed, armor doesn’t have diminishing returns per se, but it’s a much better idea to find an optimal balance between HP and armor to have the optimal EH from a frontline tank’s perspective.

It’s apparent that investing in straight health is useful because it raises EH regardless of magic or physical damage. In contrast, armor will only raise EH if physical damage is concerned, and magic resist will only raise EH is magic damage is concerned. Health is a universal defense for physical, magic, and true damage. Of course, there is also a point where simply buying HP gives diminishing returns with respect to EH. Simply put, the more well-rounded the enemy team composition is in terms of physical/magic damage, the more highly you should invest in health. If the enemy composition is almost entirely AD (physical) or AP (magical), then they made a giant mistake as long as you can abuse it: stacking the appropriate resistances here are much more helpful than raw health due to the percentage nature of how armor and magic resist works.

There’s another advantage to armor: it makes healing EH easier. Most healing abilities on champions heal a flat amount of HP. If you only buy health, it makes heals weaker in terms of effective health (it’s just a 1:1 healing to effective health increase ratio), but investing in resistances forces enemies to go through your HP bar more slowly, making the healing:EH ratio greater than one.

So the question that remains is: how do you know what to invest in at a given stage of the game? Should you prioritize flat health or resistances? To answer that, we need to optimize our EH function given a set amount of money c. How do we do that?

We first need some unit of account to gauge how much armor, magic resist, or HP is really worth. Fortunately, there are three “basic” items that grant solely armor, MR, or health. These items are Cloth Armor (300g, +15 armor), Null-Magic Mantle (450g, +25 magic resist), and Ruby Crystal (400g, +150 HP) respectively. Dividing through, we get unit costs of 20g/+1 armor, 18g/+1 MR, and 2.67g/+1 HP. Note that armor actually costs a bit higher than MR.

The tricky thing is that given constant resistance r, investing in +1 hp raises effective health by \frac{100+r}{100}. For example, if I have 200 armor, each point of health I invest in raises EH by 3. Taking it the other way around, every point of armor I invest in given constant health h raises EH by \frac{h}{100}. For example, investing in +1 magic resist with 2000 health gives 20 additional EH. (This is all assuming single-type damage, and not split.)

Since 2.67g gives +1 HP and 20g gives +1 armor, 1g of HP will give you \frac{100+r}{2.67(100)}=\frac{100+r}{267} EH, and 1g of armor will give you \frac{h}{20(100)}=\frac{h}{2000} EH. We now know the value (in terms of effective health) of both armor and health: the key here is to recognize that setting these values equal to each other means that our buildpath is optimized. Doing so and solving for h, we get roughly h=750+7.5r. This equation is what we’ve been looking for.


The horizontal axis is armor, the vertical axis is HP. If you find yourself above the line (aka to the left), invest in armor to shift right and get yourself back on the equilibrium line. If you’re below the line, invest in health to shift up.

Magic resist is pretty much the same thing. Repeating the calculations done above gives a graph of h=670+6.7r.

I won’t get into specifics, but the equilibrium line will shift upwards if the incoming damage is hybrid instead of just one type. This is because mixed damage favors building flat health. You should still build resistances, but not as much as if the enemy composition is mostly one type of damage.

The exact defensive items that you should buy I won’t get into, as this post is already getting a bit long. But if you can multiply your resistances by seven and add 700 to that total in your head, you should be in good shape to make some better item choices on the Rift.



National Science Bowl 2016: Long Island Regionals

The National Science Bowl is a science and math trivia competition run by the U.S. Department of Energy. More than 9,000 high school students and thousands of teams from all over the nation compete first in a regional competition to clinch an all-expenses paid trip to Washington, D.C. for the national finals.

The competition format is very similar to University Challenge, a British trivia show in which many universities and colleges in England compete. Teams are comprised of four players, one of whom serves as the captain. Starter or “toss-up” questions are asked by moderators in subject areas including mathematics, chemistry, biology, physics, earth science, astronomy, and energy. Questions can be either multiple choice or short answer. Buzzing in and getting a starter right earns you points. You can interrupt the moderator as he asks the question to give an answer quicker, but if you’re incorrect, the other team gets points and gets a chance to answer the starter for themselves for another 4 points. If you get the starter question correct, you get an additional bonus question in the same category as the starter for 10 points. Unlike the starter, you can confer with your teammates, but only the captain is allowed to deliver the final answer.

Personally, I think this scoring system is slightly absurd. The value of a bonus is 2.5 times greater than a starter, which means that even if a team is thirty points ahead, the other team only has to string together two lucky starters and bonus questions to put themselves back in the game. (A team can potentially earn 18 points from a single question!) Some very interesting comebacks happened today because of this exact scoring system. The truth is that the name of the game is to secure starter questions for your team, as starter questions are a gateway to the high-scoring bonus questions. For exceptionally competitive teams, most of the meta revolves around interrupting the toss-ups in an optimal way so that you buzz before the other team and have a high probability of getting the toss-up correct. Here’s an example: in the final, an energy multiple-choice question was read. (There are four possible choices; w, x, y, or z.) The Farmingdale captain buzzed as soon as the second choice was beginning to be read, knowing well that the reader would finish reading the second choice due to suboptimal reflexes. Knowing that the first two choices were incorrect, the captain guessed choice z correctly without even hearing choice y. For him, the 50/50 chance was well worth the chance to answer the bonus.

But enough of the minute details in strategy. I want to talk about how the actual event played out for my team, Walt Whitman High School, as well as the other teams from Long Island.

This year’s regional competition was held in Brookhaven National Labs in Upton, New York. It’s quite an impressive campus. Thousands of employees work there annually, while thousands more scientists come here as guests for their research. The most well-known building is probably the National Synchotron Light Source II (NSLS-II), a $912 million project that allows scientists to image fundamental particles and materials with extreme intensity, among other things. Our competition was held in Berkner Hall, a more modest convention center.


My team’s day started around 8 AM in the auditorium where we were welcomed to the NSB and briefed on the competition rules.



After that, everyone split for their first round-robin match. There were twenty high school teams that were divided into groups of five: Group A, B, C, and D. We found ourselves in Group B with Farmingdale, Stony Brook, Jericho, and Huntington. It turns out our first match of the day was in the auditorium itself against Stony Brook.

setting up 1

Here’s a perspective with me at the table. “Seat three” was a position I would assume for the rest of the competition. I like edge seats.

table 1

Round 1: (0-1) Stony Brook 28 – 66 Whitman (1-0)

Another note on how matches are scored: matches are broken up into eight-minute halves with a two-minute break in between. During this break, coaches are allowed to talk with their team and substitute players are allowed to replace teammates. We had a relatively small lead at the half due to some first-time nerves and interrupting too many starters. We calmed down in the second half and extended our lead enough to finish with a comfortable win.

“The separation of fingers and toes in a human embryo is due to what highly controlled-interrupt-B three.”
“That is correct.”

room d

Our second match was held in a less dramatic venue called “Room D.” It was quite small and required no microphones. This was Huntington’s first match, which was unfortunate since they forfeited their first match to Farmingdale since they came late.

setup 2

Round 2: (0-2) Huntington 28 – 54 Whitman (2-0)

To be honest, this match was pretty much a repeat of the first round, except we were able to keep our composure a bit more and gave away fewer free points due to interrupts.

“The citric acid cycle is a series of chemical reactions that takes place in which biological process? w. aerobic respiration, y. lactic acid fermentation-interrupt-B captain.”
“Aerobic respiration.”
“That is correct.”

We have a fairly well-rounded team. Mark specializes in biology, Zenab is an all-rounder, Alex is very talented in math, physics, and some esoteric general science, Shelbi does chemistry, and I specialize in math, physics, and some biology.


Mark Theodore Meneses, a junior, in the flesh.


Our coach Scardapane gives me a thumbs up.

hunt vs jeri

Round 3 was a bye for us, which meant we got to sit out and watch the other four teams in our group duke it out. Farmingdale, the favorite to win the whole thing, was sitting rather comfortably at two wins and no losses, and they would improve it to three this round. We decided to watch Jericho (left) vs. Huntington (right) in the auditorium. Surprisingly, Huntington destroyed Jericho by about 50 points, something I surely didn’t expect as Jericho is a pretty well-respected high school.

The back of the Jericho squad’s matching Science Bowl shirts say, “National Science Bowl: when your personal best isn’t good enough.”

Back to Room D for the fourth round.

setup 3

Round 4: (1-2) Jericho 7276 Whitman (3-0)

This was the closest match we had by far. We were trailing 22-44 at the half. Scardapane told us to calm down and play smart. And play smart we did: we gave away no free points in the second half and strung three bonuses together in a row to surge to an 18 point lead. With thirty seconds remaining, Jericho got a starter and took the remaining time to answer the bonus correctly. Thankfully, the time ran out before the moderator could start another question; if Jericho had secured one more starter they could have triumphed in a buzzer beater. Lots of handshakes and “good games” were exchanged afterwards.

“In a three-dimensional coordinate axis system, the z axis is oriented with respect to the x and y axis by-interrupt-A three.”
“The right hand rule.”

But now we faced the real challenge. Only the top team in the group from the round robin would advance to the double elimination playoff stage. Both Whitman and Farmingdale were undefeated. Whoever won the next round would move on. Could we prevail?

Round 5: (3-1) Whitman 18 – 122 Farmingdale (4-0)

I managed to squeak out 18 points for the team by answering a physics starter and a math toss-up/bonus. Otherwise, all we could really do was throw our hands up and watch in awe as the Farmingdale Harvard-bound captain made us bend over backwards ass up as he destroyed our rectums singlehandedly. That dude was on another level. The physics starter I did answer, however, I’m pretty proud of. It went something like this:

“Consider an infinite parallel circuit in which the first branch has one resistor, the second branch has two resistors, the third has three, and so on to infinity. Given that the resistors all have a resistance of 1 ohm, find the total resistance of the circuit… B three?”

In case you needed an explanation: \frac{1}{R_{t}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n} given n branches. The sum turns out to be the harmonic series 1+\frac{1}{2}+\frac{1}{3}+\cdots to infinity, which by definition approaches infinity. The reciprocal of infinity is (informally) zero.

And so ended our National Science Bowl run. We consoled ourselves with some prepackaged lunch and cookies.

lunch 2

Mark and I decided to stay and watch the rest of the Bowl. Mt. Sinai, the winner of Group D, gave a surprising decisive 84-38 victory over Farmingdale in the first double elimination round. Sinai’s captain, Alex, was an absolute beast at chemistry. It was almost unnerving how good he was. He’d answer chemistry multiple choice questions as if they were short answer. In fact, he answered 100 percent of every. single. chemistry question leading up to the finals. That’s insane. It felt like he was some shady hidden-OP anime character.

Farmingdale’s loss meant they had to win three matches in a row to win the whole thing, therefore sweeping the losers’ bracket. In the meantime Mt. Sinai edged out the regional winners two years ago, Great Neck, in the semifinals.

gn vs sinai

Great Neck lost to Farmingdale 20-126 in the loser’s finals, so it was time for a rematch: Farmingdale v. Sinai on the main stage.



This time, the questions just didn’t fall in Alex’s favor. The few chemistry questions that got asked were gamed perfectly by Farmingdale’s captain Suraj. Farmingdale took the whole thing with subsequent 118-30 and 102-50 wins.


GGWP to Farmingdale! We lost to the eventual winners, which makes me feel a little better. This was my first and last time participating in NSB, and I had a lot of fun. Hopefully Whitman can go all the way next year. If you’re in high school and not a senior, I highly recommend making a team and participating!


From left to right: Shelbi, Mark, Zenab, me, Alex