## 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:

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.

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.

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.”
“Apoptosis.”
“That is correct.”

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.

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.

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.

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.”
“Correct.”

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?”
“Zero?”
“Correct!”

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.

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.

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