#trigonometry proof

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. 😅