Mei arrives a little after Lan Wangji. She is also excited at the prospect of Combat Math. She perhaps did not read the course description very carefully.

Wei Wuxian is sitting half out of his chair and spinning his sliderule. "Everyone ready to punch math?"

Masozi followed Wei Wuxian from their previous class together, which was pretty interesting. 

He's SO excited about more math! Though he's a bit stressed about it all being in Mandarin. He makes sure to sit right next to Wei Wuxian, as soon as he's finished checking the entire room very carefully for mals. 

On every one of their desks is a piece of paper which says:

1. Wetting your feet
1.1 Estimate how many liters are in a barrel of oil and how many barrels of oil China imports every year.
1.2 The mass of the sun is 2 · 1030 kg. Estimate the mass of earth and the ratio mass of sun / mass of earth.

This isn't about combat at all.

This is the best math class ever.

This is the worst math class ever.

Masozi puzzles over the Mandarin characters for a bit, and even recognizes a few, but eventually has to admit defeat on understanding the question and ask Wei Wuxian to translate. 

….Those questions are SO UNFAIR how is he supposed to know literally anything that could help him out here???

“Wei Wuxian?” he whispers. “Is the sun bigger or smaller than the earth - am I supposed to know the answer to that already…?”

Wei Wuxian immediately begins singing:

The sun is a mass of incandescent gas
A gigantic nuclear furnace
Where hydrogen is built into helium
At a temperature of millions of degrees

The sun is hot

It is so hot that everything on it is a gas
Iron, copper, aluminum, and many others

The sun is large

If the sun were hollow, a million Earths could fit inside
And yet, the sun is only a middle-sized star

The sun is far away

About ninety-three million miles away!
And that's why it looks so small

And even when it's out of sight, the sun shines night and day!

"Except," he says at the end, "it's actually not gas, it's plasma, which is what happens when you heat things up so much that they stop being gas."

"Do you guys think we're allowed to say the mass of the earth is one earth mass? That's a real unit of measure they use it for planets." 

"I think it's worth a try to see if the school will accept it."

"- So if a million earths fit inside then it's a millionth of - no wait sometimes things take up the same space but are heavier. I......would the sun sink or float on water?"

This class is extremely weird! Not in a bad way but Masozi feels like he's mostly going to end up learning things totally unrelated to math. 

"I think the sun is denser than water?"

"The sun is way less dense than water. But it's still a lot bigger than the earth in mass, you can tell because the earth revolves around it but I don't actually remember how much one earth-mass is in kilograms so I don't know how much bigger?" 

"Well, a million earths could fit inside, so probably it's a million times larger?"

"That's volume, though, and the sun is less dense than water so it has to be less dense than rock..."

He trails off. Confidently disagreeing with people is hard.

"Oh, that's a good point. How much less dense is water than rock?" He starts scribbling on a piece of paper.

"Yeah, rocks definitely sink–" Masozi stops. "Oh! I know how many miles it is from Malawi to Johannesburg. If you know how big that would be on a map of the whole world, then we can guess how many miles around the whole earth is! And then we could figure out its volume from there and then its weight - does anyone have a rock and a scale so we could actually check how dense rock -" 

He doesn't understand this class but it involves a lot of Wei Wuxian smiling so he's not very upset by it. 

"It depends on the kind of rock I think?" 

While they were distracted a second paper appeared on their desks:

Simple Methods for Order-of-Magnitude Approximation:

Divide and conquer: Split a complicated problem into manageable chunks, especially when you must deal with tiny or huge numbers, or when a formula naturally factors into parts.

Guess: Make a guess before solving a problem. The guess may suggest a method of attack. For example, if the guess results in a tiny or huge number, consider using divide and conquer. The guess may provide a rough estimate; then you can remember the final estimate as a correction to the guess. Furthermore, guessing—and checking and modifying your guess—improves your intuition and guesses for future problems.

Talk to your gut: When you make a guess, ask your gut how it feels. Is it too high? Too low? If the guess is both, then it’s probably reliable.

Lie skillfully: Simplify a complicated situation by assuming what you need to know to solve it. For example, when you do not know what shape an object has, assume that it is a sphere or a cube.

Cross-check: Solve a problem in more than one way, to check whether your answers correspond.

Use guerrilla warfare: Dredge up related facts to help you make an estimate.

Punt: If you’re worried about a physical effect, do not worry about it in your first attempt at a solution. The productive strategy is to start estimating, to explore the problem, and then to handle the exceptions once you understand the domain.

Be an optimist: This method is related to punt. If an assumption allows a solution, make it, and worry about the damage afterward.

Lower your standards: If you cannot solve the entire problem as asked, solve those parts of it that you can, because the subproblem might still be interesting. Solving the subproblem also clarifies what you need to know to solve the original problem.

Use symbols: Even if you do not know a certain value—for example, the energy density stored in muscle—define a symbol for it. It may cancel later. If it does not, and the problem is still too complex, then lower your standards.

Masozi is pretty sure that contains MANY Mandarin characters he's never seen before! He goes through it and circles everything that he does recognize and spends a bit of time breaking down bits that he thinks are 'radicals' like Lan Xichen explained. 

And then turns to Wei Wuxian. "It says - small, here? And this says something about parts and something about - cutting? I think I might be remembering that wrong though. And this says something about being tall? ....Sorry I can't actually figure out what this paper says can you help please." 

Wei Wuxian reads it out loud to him in English.

Why is this math class about trusting your gut! The whole point of math is that there are numbers and you do things with the numbers and you know what the rules are and at absolutely no point do you have to trust your gut! 

Masozi thinks this math class is great! It's - actually kind of the way he does math all the time in his head anyway? The main problem is that it expects him to know things like 'the sun is bigger than the Earth.' 

"Wei Wuxian?" he asks. "Why does the sun being bigger mean that the earth goes around it?" 

"It's, um. About gravity? Bigger things have more of it, and if something smaller is close enough it gets... caught... I'm not explaining this right, am I." 

(It does not seem to have occurred to Kinsei to wonder why Masozi is in a Mandarin-language math class with him.) 

Math is supposed to have One True Answer! The lesson says she should lie?? Why does math involve lying!

Also this math class is in Mandarin why does everyone keep talking in English. Mei doesn't understand anything and she is going to die and Ayako will have to make it to graduation alone because Mei is too stupid to understand how to lie to math.

Well, Masozi is going to slightly increase the total amount of Mandarin being spoken by asking Wei Wuxian to read the instructions to him out loud in Mandarin so that he can copy out phonetically in English letters the sound that goes with each character. 

Is this time for a lesson in the International Phonetic Alphabet? Wei Wuxian thinks it is absolutely time for a lesson in the International Phonetic Alphabet.

Amazing! 

Why is this class in English. Lan Wangji thought it was in Mandarin. He understands absolutely nothing that is happening here. Math is supposed to have rules. Math is one of his favorite academic subjects because there is always a right answer and a wrong answer.

He is going to sit over here next to Mei who also doesn't speak non-spellcasting English.

Kinsei looks over at the other people who seem like they're having a bad time and haven't been speaking exclusively English.

He is not quite brave enough to actually get up and introduce himself to two people with power-sharers, though. 

Mei had not quite realized before that "some kids take classes in languages they don't know so they can also get language credit for it" would mean that her math class might be someone else's Mandarin class, or how distracting that would be.

She looks at Lan Wangji as he sits down by her. Well, upside, he won't be talking a language she barely understands. Downside, she does not think he is going to produce more explanation of their assignments than "mn".

Maybe that Japanese kid over there has some idea what they're doing. "Hey, I'm Mei Masane. Are you good at math?"

"I'm Kinsei Saito. I'm. Good at math usually?" 

And now that someone else has made the first move it's probably fine to move so he's next to Mei now? 

The rules say that he is supposed to do his homework and follow class instructions. The class instructions say he's not supposed to follow the rules of math. He is very quietly having a meltdown.

Mei suggests to the person currently sitting next to her that perhaps they could trade seats with Saito-han and thanks them as they vacate their chair.

Right okay thank you. The fact that that is a thing people can just do is kinda terrifying but it's fine probably? 

"Thank you. Uh. I can try to explain the assignment I think? But I'm good at... trigonometry, and things, this isn't really..." 

"--oh, so you had a question about the earth orbiting the sun-- so you know how when you drop an apple it falls to the ground? That's because big things tend to pull small things towards themselves. We call that 'gravity.' And the sun is much bigger than the earth so it keeps pulling the earth towards it."

“….Then why doesn’t the earth just fall into the sun?”

"Great question! Because the earth was going very fast this way"-- he draws a line-- "and then the sun was here" --dot-- "and so they averaged out into a curve like this"-- he draws an ellipse-- "that's also how spaceship orbits work."

"Did you know that people went to the moon?"

Masozi goes very still. "- What? How?" 

"Okay, so, when you shoot something out of the back of something, it pushes it forward. That's called thrust. And if you build a machine that makes a really big explosion for long enough it can push the machine high enough that you can reach the moon. --I really need a visual aid for this, maybe I can whip something up in shop."

Masozi's eyes go out of focus for a few seconds as he tries to - process all of that, fit it into anything he understands -

- oh right he's not safe he has to watch for mals and he has to remember that no one else is careful enough and his survival here depends on himself alone– 

He takes a deep breath. "I - that'd be really interesting but I think I should try to learn what's in this class first and not get distracted?" 

"Oh, right," he says, as if this just occurred to him. "So, uh, barrels of oil in China-- a barrel is like a meter tall, right--"

Lan Wangji is having a very small emotional crisis. 

First of all, the math is telling him to break the rules of math and he doesn't understand how to do this. You're not supposed to just guess on math problems. There's a right answer.

Second, Wei Wuxian keeps smiling, which is normally good, except now he's smiling at Masozi and being interested in Masozi and Lan Wangji has never gotten him to smile that much. They're saying something in English and Lan Wangji doesn't speak English and they could be saying anything. Maybe Wei Wuxian is confessing his love to him. Maybe they're going to get married and Lan Wangji has accustomed himself to a life of celibate devotion but it is one thing to be accustomed to a life of celibate devotion and another to live it while your love is getting married to someone else and smiling at him. Lan Wangji wants those smiles to be reserved for himself alone.

Maybe he could stab Masozi with Bichen.  

"So, uh," Kinsei tries again, "I-- I think it's asking us to try and answer the questions without really doing... math about it?"

Masozi is still raptly listening to Wei Ying's explanation of how to answer the barrels-of-oil question. 

"How can you answer a math question without doing math?"

"I think they're not really math questions? Like-- it wants to know the size of the earth and how much bigger the sun is, but it hasn't told us the numbers we'd need to actually do math to get a real answer, and instead we got an instruction sheet that says to just go ahead and make a guess that feels right?" 

Mei puzzles over this. How are they going to be graded? Are they going to fail if they don't somehow guess the right number with no information? Are they supposed to go to the library and do research on the size of the sun? How could it possibly grade them on whether their guess felt right? Do they lose any points if they just make up random numbers and write them down? It feels pointless to ask any of this out loud because it's not like anyone else knows how they're being graded. 

Mei looks down at her assignment sheet angrily.

"Fine. 'Question one: Estimate how many liters are in a barrel of oil and how many barrels of oil China imports every year.' A barrel of oil is like, a meter tall and half a meter across. So that's, what, a quarter of a cubic meter? So if there's a thousand liters of oil in a cubic meter then there's a quarter of that in a barrel. And there's, I don't know, a billion people in China? So if everyone gets a barrel a year then that's 250 billion liters of oil. I have no idea how right that's supposed to feel! I'm pretty sure it's wrong! But it's probably less wrong than, like, '5'??"

Mei has no idea if she's doing too much math or not enough and this is not the sort of thing that one is normally uncertain about in a math class!

"...that does sound a lot less wrong than five but that's not how I'd do it, the way I'd do it is that China is the biggest importer of oil in the world so it has to be a lot, like, on the order of millions of barrels a lot, except I think oil is usually measured in barrels per day so actually it's a couple orders of magnitude more than that?" 

"I guess probably people use more than a barrel of oil a year. Oil is used for lots of things."

"Yeah. So if China has a billion people, I think it has a more than that but I don't know how much more and that sounds right, and they each use five barrels of oil a year on average, then that would be about five billion barrels a year? Which sounds at least not obviously wrong for a big country with a lot of factories?" 

"If somebody who knew anything about the topic told me that number I would totally believe them. And I don't see how we could get a better guess without more information, so..."

Mei writes down "250 liters" and "5 billion barrels" next to question one and grumpily contemplates the vastness of the sun.

Kinsei does the same. 

"For the Earth I'm tempted to put that it's one earth-mass, that's a real unit of measurement, they use it for planets? It doesn't say you have to give the answer in kilograms. But that doesn't tell us how much smaller it is than the sun since I don't remember how much an earth-mass is." 

"I really don't see how we're supposed to guess this at all. There's nothing to go on! The Earth is big and the sun is even bigger by some totally mysterious amount. It's not like you can picture them side by side and go 'ah yes the sun is so many Earths across'!"

"...well, it says the sun is 2 with thirty zeroes kilograms, so we know we're working with the kind of scale where it's easier to say that things have thirty zeroes than to write out all the zeroes they have?" 

"Great, so the Earth has fewer than 30 zeroes and more than 3 zeroes. Uggh I think I remember hearing once how many Earths you can fit in the sun but I can't remember how many it was! A hundred sounds too small and a million sounds too big?"

"The Chinese boy who was speaking English said it was a million earlier, but that's volume not mass so it doesn't help us very much...

...wait actually yes it does, it means we know the ratio has to be less than a million?" 

"Can we not just count the number of zeroes in a million and take them away from thirty and go with that?"

"...yeah, we can probably do that? A million is one times ten to the seventh so that'd be... two times ten to the twenty-third? If it had exactly one millionth the mass of the sun? Except we know it has more because the earth is denser so I'm going to call it two times ten to the twenty-fourth." 

Mei writes down two times ten to the twenty-fourth. "Great, that's two questions done."

She looks at the rest of the worksheet and groans.