Watching Love Like Galaxy episode 6 (around 33 minutes in), I was super intrigued by the riddle Yuan Shanjian presents about his well’s water level, and then I proceeded to waste an hour understanding Niaoniao's solution and two hours creating this post. Without further ado, read on to find out how you too can win a jar of 千里醉 wine and catch Ling Buyi’s eye the next time you time travel to ancient China!

THE PROBLEM

Using only a 3-foot ruler, measure the distance between the well opening and water surface. (Notice how Niaoniao cheats by also using a tree branch and her brain.)

NIAONIAO’S SOLUTION

Niaoniao’s script: 

井径二尺半，

立三尺木于井上，

从木末望水岸入径一尺。

My translation: 

The well diameter is 2.5 feet.

Prop the 3-foot ruler on top of the well.*

Looking at the water edge from the top of the stick**, the stick enters*** the diameter by 1 foot****. 

“Foot”notes: 

*Refer to fig. 1.

**Refer to fig. 2; water edge refers to "where the water surface touches the well side on the opposite side of the well from where the 3-ft ruler is”. Refer to fig. 3 for where exactly she’s looking. 

***This is a little confusing, but it's essentially "the ruler juts into the well by 1 foot.” Refer to fig. 4 for my interpretation of Niaoniao’s pov. 

****Refer to fig. 5 for how Niaoniao measures the “1 foot”.

Figures:

Figure 1: Step 2, Niaoniao propping ruler on well

Figure 2: Step 3, Niaoniao looking at water edge from stick top

Figure 3: Step 3, Niaoniao's line of sight, side view

Figure 4: Step 3, Niaoniao's perspective

Figure 5: Step 3, how Niaoniao measures “1 foot”

THE MATH

THE ANSWER

Hence, the distance between the well opening and water surface is 4.5 feet!! Go time travel with confidence now.

SOME MORE NOTES FROM ME

The 3-foot ruler has to be perpendicular to the water surface. 

This solution assumes the well diameter is constant from the top to bottom. 

This solution ignores water refraction, which could be accounted for by Snell’s Law as a dear discord friend pointed out. 

(Refering to step 3) This solution ignores that the angle that Niaoniao uses to view the water edge has to be highly accurate because being just a few degrees off could create a big difference in her measurement of how much the ruler juts into the well. 

I translated 尺to foot. It’s not technically the same thing as the American foot of 12 inches, but people call it the Chinese foot. 

I've been informed that her name is Niaoniao, not Niuniu, and that Niaoniao isn't even her name, it's her nickname. Please excuse me if I forgot to change Niuniu to Niaoniao somewhere. 😅

“Pure math,” he replied. “How is that different from”—she laughed—“regular math?" Gillian asked.

“Well, regular math, or applied math, is what I suppose you could call practical math,” he said. “It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will—the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.”

“Do you mean imaginary geometries, stuff like that?” Laurence asked.

"It can be, sure. But it's not just that. Often, it's merely proof of—of the impossible yet consistent internal logic of math itself. There's all kinds of specialties within pure math: geometric pure math, like you said, but also algebraic math, algorithmic math, cryptography, information theory, and pure logic, which is what I study."

“Which is what?” Laurence asked.

He thought. “Mathematical logic, or pure logic, is essentially a conversation between truths and falsehoods. So for example, I might say to you ‘All positive numbers are real. Two is a positive number. Therefore, two must be real.’ But this isn't actually true, right? It's a derivation, a supposition of truth. I haven't actually proven that two is a real number, but it must logically be true. So you'd write a proof to, in essence, prove that the logic of those two statements is in fact real, and infinitely applicable.” He stopped. “Does that make sense?”

“Video, ergo est,” said Laurence, suddenly. I see it, therefore it is. He smiled. “And that's exactly what applied math is. But pure math is more”—he thought again—“Imaginor, ergo est.”

[...]

“[The] law isn't so unlike pure math, really—I mean, it too in theory can offer an answer to every question, can't it? Laws of anything are meant to be pressed against, and stretched, and if they can't provide solutions to every matter they claim to cover, then they aren't really laws at all, are they?" He stopped to consider what he'd just said. “I suppose the difference is that in law, there are many paths to many answers, and in math, there are many paths to a single answer. And also, I guess, that law isn't actually about the truth: it's about governance. But math doesn't have to be convenient, or practical, or managerial—it only has to be true.

“But I suppose the other way in which they're alike is that in mathematics, as well as in law, what matters more—or, more accurately, what's more memorable—is not that the case, or proof, is won or solved, but the beauty, the economy, with which it's done."

“What do you mean?” asked Harold.

“Well,” he said, “in law, we talk about a beautiful summation, or a beautiful judgment: and what we mean by that, of course, is the loveliness of not only its logic but its expression. And similarly, in math, when we talk about a beautiful proof, what we're recognizing is the simplicity of the proof, its ... elementalness, I suppose: its inevitability."

“What about something like Fermat's last theorem?" asked Julia.

“That's a perfect example of a non-beautiful proof. Because while it was important that it was solved, it was, for a lot of people—like my adviser—a disappointment. The proof went on for hundreds of pages, and drew from so many disparate fields of mathematics, and was so—tortured, jigsawed, really, in its execution, that there are still many people at work trying to prove it in more elegant terms, even though it’s already been proven. A beautiful proof is succinct, like a beautiful ruling. It combines just a handful of different concepts, albeit from across the mathematical universe, and in a relatively brief series of steps, leads to a grand and new generalized truth in mathematics: that is, a wholly provable, unshakable absolute in a constructed world with very few unshakable absolutes.” He stopped to take a breath, aware, suddenly, that he had been talking and talking, and that the others were silent, watching him. He could feel himself flushing, could feel the old hatred fill him like dirtied water once more. “I'm sorry," he apologized. "I'm sorry. I didn't mean to ramble on.”

“Are you joking?” said Laurence. “Jude, I think that was the first truly revelatory conversation I’ve had in Harold’s house in probably the last decade or more: thank you.”

A Little Life by Hanya Yanagihara

Fair warning, I have basically no formal exposure to that kind of thing. 

My math lover’s heart tells me that if you do it right, it should tell you all that you need to know. But my math lover’s mind tells me that in many cases it would be virtually impossible to calculate everything with total accuracy (unsatisfying, when you’re used to absolute proof in pure math).