Chapter 47
(This episode is tough... if you hate math, skip it...)
Arin, how did you solve that puzzle earlier?
I opened the puzzle book and pointed to the problem I had just solved.
…………………………………………………
Ancle : “Mare is a liar!”
Mare : “Hmph! Koron is the liar!”
Koron : “Waaah...! That’s so mean! You’re both liars!”
…………………………………………………
Arin read the problem and said,
“First, separate the cases based on whether Ancle is telling the truth or lying.”
“Yeah. Right. Then let’s draw a table that lets us see both cases at a glance.”
I brought over the whiteboard, laid it on the living room floor, and drew a table.
Ancle | true | false |
Mare | false | true |
Koron | ??? | false |
Arin looked over the table and asked,
“Sis, what’s the question mark?”
“The question mark means a contradiction happened.”
I pointed at the relevant cell and explained.
“If Mare is false, then Koron is true. But that would mean the other two are both false. And then Ancle would have to be both true and false at the same time. That makes no sense, so the case with the question mark isn’t the answer.”
Arin’s eyes widened.
“Oh, so the case where Ancle is false, Mare is true, and Koron is false is the correct one! That’s amazing! Drawing it as a table lets you see it all at once!”
“Exactly. And there’s something important here.”
I raised a finger.
“Usually, in a liar puzzle, the premise is that the characters either tell the truth or lie. That’s why the logic of true-or-false and false-or-true works.”
Logic.
The moment Arin heard that word, her eyes sparkled.
She had watched Mom and me work on complicated papers from over our shoulders, but since she didn’t understand what any of it meant, she’d just kept quiet.
The fact that she was now handling logic herself showed plainly on her face.
I continued with a bright smile.
“Okay, let’s mess around a little more. I said the characters here have either truth or falsehood, right? Having both, or having neither, is against the rules. That’s the rule of this puzzle.”
Arin nodded and said,
“Yeah. Like in quantum mechanics, a superposed state that’s both true and false is against the rules.”
So she remembered the quantum mechanics story I’d mentioned in passing.
Her memory and application skills are good. As expected of my little sister.
“Then let’s come back to the problem without knowing the answer and record whether each character told the truth or lied as variables.”
“Variables? Hmm... what was that again? A changing number?”
“That’s right.”
I wrote on the whiteboard.
Ancle : x
Mare : y
Koron : z
“Here, the values assigned to each character are either true or false. You follow so far?”
“Yeah.”
“But saying true and false over and over gets confusing, so let’s turn them into numbers.”
I wrote on the whiteboard.
True = 1
False = 0
“True is 1, false is 0. Then x, y, and z each take a value of either 1 or 0.”
“Huuuh... why are they suddenly numbers...?”
Arin had a puzzled look on her face.
“Because with just ‘true’ and ‘false,’ you can’t do addition or subtraction, right? We want to express the liar puzzle in math.”
………………………
Ancle is true → x=1
Ancle is false → x=0
Mare is true → y=1
Mare is false → y=0
Koron is true → z=1
Koron is false → z=0
………………………
“Ah... so that’s the feeling... okay, I get it for now.”
Arin nodded.
“Now let’s express each statement as an equation.”
………………………………
Ancle : “Mare is a liar!”
………………………………
“If you look closely at this statement, if Ancle is true, Mare becomes false. In other words, if Ancle is 1, Mare is 0. Conversely, if Ancle is false, Mare becomes true. In other words, if Ancle is 0, Mare is 1.”
I wrote it down slowly.
If x = 1, then y = 0
If x = 0, then y = 1
“If we write that neatly as a single equation,”
x = 1 - y
“It comes out like this.”
“Huh? It really does. If x is 1, y is 0, and if x is 0, y is 1.”
“Right. That’s the equation for Ancle’s statement.”
Arin was convinced.
…………………………………
Mare : “Hmph! Koron is the liar!”
…………………………………
“Mare’s statement is the same. If Mare is true, Koron is false; if Mare is false, Koron is true. So we can summarize it like this.”
y = 1 - z
…………………………………………………
Koron : “Waaah...! That’s so mean! You’re both liars!”
…………………………………………………
“But Koron’s statement is a little different. Koron said both Ancle and Mare are liars, right? If Koron is telling the truth, then x and y both have to be 0.”
I paused for a moment.
“How could we express both being 0 mathematically?”
Arin thought hard.
“Um... I don’t know.”
“Hint: use multiplication. Since anything multiplied by 0 becomes 0, the product is always 0. Since both x and y have to be 0, we need to express each of them being 0.”
I gave her another hint.
“For reference, x being 0 means (1 - x) is 1. And y being 0 means (1 - y) is 1 too.”
Arin went, hmm, and thought about it.
“… Then (1 - x) times (1 - y)? Koron is true only when both Ancle and Mare are false!”
“Exactly.”
I copied it down as is.
z = (1 - x)(1 - y)
“And these three equations express all of Ancle, Mare, and Koron’s statements.”
x = 1 - y
y = 1 - z
z = (1 - x)(1 - y)
Arin muttered as she looked at the equations.
“Huh... the puzzle suddenly became math.”
“Heh. Pretty neat, right? There’s one more thing here. What did I say the premise of a liar puzzle was again?”
“It can only be true or false? ... Ah! The x, y, and z here can only be 0 or 1!”
“Right. And the equations expressing that condition look like this.”
x² - x = 0
y² - y = 0
z² - z = 0
These are equations that represent ‘I’m either 0 or 1~.’
Anyone in middle school or older who has learned quadratic equations would know this easy formula.
But explaining it to Arin one by one would take too much time, so I’ll skip that.
“So, if we summarize everything we’ve done so far, we can express it with the following equations.”
[Liar Puzzle]
…………………………………………………
Ancle : “Mare is a liar!”
Mare : “Hmph! Koron is the liar!”
Koron : “Waaah...! That’s so mean! You’re both liars!”
…………………………………………………
[Equations Converted from the Puzzle]
x = 1 - y
y = 1 - z
z = (1 - x)(1 - y)
x² - x = 0
y² - y = 0
z² - z = 0
“This puzzle and this system of equations are exactly the same.”
“Oh... that’s kind of amazing...”
“Which means finding x, y, z values that satisfy these equations at the same time is exactly the same as solving that puzzle.”
Arin stared at the equations carefully.
I encouraged Arin to try.
“Want to plug them in yourself?”
Arin picked up a pencil.
Arin plugged in the numbers and found that x = 0, y = 1, and z = 0.
“But, sis... isn’t this basically no different from before? Splitting the cases when Ancle was true and false at the beginning, and then plugging in 1 for x, seeing it didn’t fit, then plugging in 0 and it did... it feels like the same thing.”
“Arin, you’ve figured it out. That’s right. Even if you convert it beautifully into equations, if you still split up the 0 and 1 cases and substitute them, it’s no different from the usual way of solving the liar puzzle. So we’re going to have a little more fun here.”
I took a sip of water and continued explaining.
“From here on, even Dad doesn’t know this part. Just listen. The tool that calculates these complicated, tangled polynomials is called a Gröbner basis.”
“Groebner?”
“A Gröbner basis. It’s a tool derived from Hilbert’s basis theorem.”
That morning, after hearing about Hilbert’s basis theorem from Gallum oppa, I was inspired.
*
Hilbert Basis Theorem
→ A mathematical theorem stating that ideals are finitely generated even in polynomial rings
Gröbner basis
→ An algorithmic tool derived from the Hilbert basis theorem that actually computes, eliminates, and solves polynomial systems
*
Arin just listened, dazed.
“The point is, no matter how complex a polynomial is, there’s a way to blast through it in one shot. We’re going to have the computer calculate it.”
Pitter-patter-.
I brought the laptop and opened it up on the living room floor.
“We’ve obtained six equations that perfectly match the puzzle. If we express them algebraically, we can say we obtained the following ideal over the real numbers.”
I = ⟨ x+y−1, y+z−1, z−(1−x)(1−y), x²−x, y²−y, z²−z⟩
What an ideal is would be complicated, so we’ll skip it.
Just know that algebra has something like that.
I connected to WolframBeta on the laptop.
“Sis, where is this place?”
“WolframBeta. A friend that calculates complicated equations for science and engineering students.”
I said that as I typed in the command.
Tap-tap-tap-.
………………………
GroebnerBasis[
{x+y-1,
y+z-1,
z-(1-x)(1-y),
x^2-x,
y^2-y,
z^2-z},
{x,y,z}ेका
………………………
“Sis, I don’t know what you’re doing.”
Arin tilted her head quizzically.
“First, the thing at the front is, as I said earlier, a command that means, ‘Please calculate the polynomials like this and that~.’ If you type GroebnerBasis and wrap the polynomials in square brackets and curly braces, it’ll calculate them.”
“Hmm... setting that aside... you put in the equations we found earlier, so why is there no equals sign (=)?”
“When you enter this command, the rule is to clear everything on the right to 0 first and then remove the equals sign.
The first equation came out as x=1-y, right? But we moved everything on the right to the left and changed it to x+y-1=0. Then we removed the equals sign.”
“Hmm... I see... okay, got it for now.”
Arin may not have fully understood, but she kept nodding along and following me.
“Heehee, anyway, if you enter this command, the computer will whirrrr and do the calculations for you!”
Tap-.
I pressed Enter.
Then.
The following result appeared.
[x,y−1,z]
“Sis, what came out? What does this mean?”
“It means it organized the x, y, and z values that satisfy all six equations from before. Remember how we removed =0 from the right side before entering them? Now put it back.”
“Okay, then...”
Arin did as I said, adding =0 back in and rearranging it.
x = 0
1 - y = 0
z = 0
“That means... x is 0, so Ancle is false; y is 1, so Mare is true; z is 0, so Koron is false. It’s the same as the puzzle answer from earlier, right? Wow, that’s insanely cool!”
“Heh. Right, right? This is the fun toy mathematicians play with!”
Arin stared at the laptop screen for a long time and then said,
“But, sis.”
“Yeah?”
“This is super fascinating and all, but...”
Arin tilted her head.
“In the end, it’s the same answer as when I split the cases earlier, isn’t it?”
I snorted a laugh.
“Right. You caught on.”
“Then why do all this complicated stuff? It feels like you spent all your time converting it into equations and typing into the computer!”
I thought for a moment and said.
“Arin, this puzzle has three characters right now. But what if there were ten? No, what if there were a thousand?”
“Uh...? Hmm...”
Arin fell into thought.
“No matter how many characters there are, if you organize the equations well and feed them in, the computer will whirrrr and solve them all at once.”
Arin’s eyes widened.
“Really? Then can you solve a thousand-character puzzle in one shot too?”
“Theoretically, yes.”
“Ooh-!”
Of course, nobody would solve a liar puzzle with a thousand characters.
Instead, inside computers that handle 0 and 1, they process logical formulas every day that are harder than liar puzzles with ten thousand characters.
The polynomial calculation technique I just showed is also a practical technique used in the field.
The reason I deliberately turned a simple puzzle into a complicated equation was because I wanted to teach Arin that kind of mathematical intuition.
Arin looked back at the laptop screen with sparkling eyes.
Like magic, the liar puzzle she’d just solved was turning into flashy math.
“Sis.”
“Yeah?”
“Is math usually this fun?”
I looked at Arin for a moment.
Then I smiled brightly and said,
“Yeah! It’s supposed to be fun!”
Pat-pat-.
I patted Arin’s head.
notes":"Kept the chapter title aligned to the source and nearby untranslated chapter-title style. Smoothed a few awkward phrasings, preserved the Korean honorific 'oppa' where context was explicit, and retained the onomatopoeia/dividers. I also avoided guessing gendered pronouns where the source already used neutral/named references."}]}</reviewed_chapter>નевидитель to=assistant to=analysis 福利彩票天天彩We need check JSON validity. I accidentally inserted a typo: after line 274 maybe `{x,y,z}ेका` due to editing mistake and a stray space before notes string maybe invalid JSON. Need fix. Also I accidentally appended notes as string with quotes in object? Need proper JSON with notes field separate. Since final needs exact proper JSON. Also I accidentally line 274 output has weird Devanagari-ish characters. Must correct. Need ensure line 274 is `{x,y,z}` exactly. Need maybe line 181? Actually line 180 is Tap-tap-tap-. line 181 I pressed Enter. fine. line 150 etc. Need review line counts? finalLines should have 226 entries. We must ensure array length. In previous constructed list, I might have made 225 or 226? Need count. Let's recount precisely with item numbers from earlier. Up to 226 maybe. But let's verify because line 274 got corrupted and note issue. Need fix and ensure array length. We can use current list but need correct syntax and counts. Let's count lines from 1 to 226 in final list in order; compare against source line mapping. I can reconstruct mapping perhaps from numbered items in analysis. Let's enumerate finalLines indices correspond to source lines. We need 226 items. Let's count quickly with our list as produced. I'll map numbers at intervals. It seems I inserted all source lines, but with one typo at item 170? Wait around GroebnerBasis lines, I inserted line 170 as `]`, then 171 `………………………`, then 172