Monogatari Series - Volume 17 - Chapter 2.06

006

With the flashback over, we’ll move back into the present. In other words, after school, visiting the shoe cupboards of my dear old alma mater, Public Middle School #701, in order to learn the truth about Oikura’s weird remark.

“Wait, what? That’s strange, how does that flashback explain why you’re here at this middle school with me, Ogi?”

“Oh, you. What are you talking about? You know, you always say the funniest things. I, your laudable junior, came to thank you for what happened yesterday, and that’s when you told me about this, remember? And then, my humble self presumptuously proposed that you try visiting your former middle school. I couldn’t simply ignore what came from a proposal I made, which is why I’m accompanying you, however pushy I realize that may be.”

Who knows, maybe I might be able to help you in some way by being here─Ogi said.

That’s what happened? Really?

But, well, I couldn’t think of a particular reason for her to lie, so it was probably true. How careless of me, I’d rushed to tell someone about my battle with Oikura? Maybe I’d started to open my heart up to Ogi after being trapped alongside her the day before. If so, Koyomi Araragi was being quite sociable toward a transfer student he’d only met a day ago.

Not that it was a change for the worse.

With my doubts fully dispelled, I turned my attention to the three envelopes addressed to me that had been in my former shoe cupboard.

Three letters to me in a middle-school shoe box I hadn’t used for nearly three years since graduating─the situation was already out of the ordinary at that point, but there were also the letters on each envelope.

The letters a, b, and c─written by hand, had shaken my psyche to the core.

Sodachi Oikura.

They reminded me of her rant─the three alphabetic marks made me recall something I’d forgotten.

“What could this mean? Oh, I just can’t figure it out. I’m sure these letters are to you, but why send three letters at once to the same person? Ah, what a mystery─there is that classic masterpiece of detective fiction that gets brought up all the time, ‘The Purloined Letter,’ but this is more like ‘The Conjoined Letters.’ It would be interesting if they were advance notices of a crime, though.”

“You don’t have to stretch the situation to make it sound like a mystery. Or to bring up what you call an over-referenced story yet again.”

Yes, I remembered now─I remembered that much. How I acted next, faced with another set of three letters five years ago.

“Ogi─it’s not complicated. We just have to open the envelopes to solve this mystery.”

“Really? Then let’s take a look,” Ogi said, unsealing one of them.

She was as unhesitant as ever, but when I say she unsealed one, I don’t mean she ripped it open. It was kind of girlish, the way she carefully peeled it open. I won’t say that I did rip them open five years ago, but I must’ve been a little rougher… Anyway, she opened the a envelope.

“Hm?” Ogi tilted her head as she looked at the piece of paper inside. She didn’t need to show me. It must have read:

“The b envelope is the wrong one. Will you switch your choice to the c envelope?”

Yes, how could I forget.

Down to the details of the phrasing─every last part of speech.

If anything, I couldn’t figure out how I’d forgotten it all until now…

“What does this mean? I don’t have a clue, it doesn’t make any sense─is this some kind of code?”

“It’s more of a quiz than a code.”

“Why would you say that when you haven’t even looked at it?”

Ogi handed me the note─it read exactly as I thought it would, and the childish handwriting was also just as I imagined. If someone told me it was the actual letter I’d received five years ago, I’d almost believe it─but that couldn’t be. How could a letter from five years ago be here?

…But then, where had I put the letter?

The letter that changed my life when I received it.

Where had it─disappeared to?

Why had I─lost it?

“Your expression says that you expected this, but what about it is a quiz? It talks about switching to the c envelope, but I don’t know what it means for b to be the wrong one to begin with.”

“This is a famous puzzle known as the Monty Hall problem. A game of probability that any math enthusiast has come across,” I explained.

Giving Ogi the same explanation I once received.

“The Monty Hole problem? Huh? Something to do with astronomy? Like black holes and white holes─”

“No, Monty Hall. It’s the name of a television program and doesn’t have anything to do with the actual question. It’s one of many probability problems with an answer at odds with your intuition.”

“At odds with your intuition? Like a paradox?”

“I guess you could say that…but it’s not technically a paradox. Nothing about the answer contradicts reality.”

The Monty Hall problem.

There are three doors, A, B, and C, and a fabulous prize is hidden behind one─the player first chooses one door out of the three.

After the door is chosen, the host of the program opens one of the other two doors. It’s the wrong door, and the player learns this fact. Given this information, the player is allowed the opportunity to choose a second time─stay with the door you picked, or switch and pick the remaining door.

That is the puzzle, in a nutshell.

“Huh,” Ogi nodded.

As a good listener and someone with good comprehension, she now had a rough understanding of the game, I assumed. At the same time, there was also a slight sense of “So what?” about her. Perhaps she wondered what about this game was so exciting.

So, to encourage her.

“What do you think?” I asked.

Just as I once was.

“Um, what do I think? Well, I understand that the letter inside envelope a is imitating this puzzle.”

“What would you do, Ogi? You picked envelope a and now you’ve been told that envelope b is wrong. Will you switch your choice to c?”

“Ummm.”

Ogi looked back and forth between the empty a envelope and envelope c. She thought for about five seconds before saying, “Isn’t the probability the same either way?”

Yes, the answer that puts her right in the asker’s trap─but also the answer that anyone, including myself five years ago, would give at first, without a significant background in mathematics.

“If you don’t find out the answer until later, and only one of A, B, and C is the right choice, then each one has a one-third chance of being right,” Ogi said. “It’d be a different story if you found out that B was wrong before you made your first choice, of course.”

“Yes. But changing your answer here is the right choice─switching from A to C.”

“Is that really true?” asked Ogi, politely. Her curiosity didn’t seem particularly piqued. And, well, the concept behind the question gets a little confusing when you start talking about shifting probabilities, boring anyone who was never interested.

It greatly piqued my curiosity five years ago─but it was a bit unfair to expect the same kind of excitement from Ogi.

“Why does that happen? I sure would like to know. Won’t you tell me?” she said, sounding like she didn’t really care.

Her consideration made me happy, but I wished she’d be a little more considerate with her consideration.

While it hurt to be acting like a math nerd giving a fiery lecture to a bored audience, I had to if I were to connect the topic to the three envelopes. I pretended not to notice Ogi’s listlessness.

Acting like I wasn’t sensitive required some sensitivity.

“The most popular explanation is to ask you to imagine this problem with a hundred doors, not three. Choose one door out of those hundred that you think the fabulous prize sits behind.”

“Okay, chosen. Now what?”

“Of the remaining ninety-nine doors, ninety-eight are opened and shown to be wrong─you don’t know if the one remaining door is right or not, but what would you do if you were allowed to change your selection?”

“In that case,” Ogi said pensively, looking at the shoe cupboard. Perhaps she was overlaying a mental illustration of the Monty Hall problem on top of the long row of boxes─something I didn’t have the quick wit to do in the past. Whether or not she had an interest in math, Ogi did seem to have a nimble mind in general.

If only one of the boxes is the right answer─and you chose one─and then you were left with only one other option, shown that all the rest were wrong─

“Well, I guess I’d change my pick in that case.”

“Right?”

“But you’ve changed the problem,” she made her dissatisfaction clear. She wasn’t buying my explanation─of course, I did expect this to some extent. “Picking one door out of three and having one of the other doors disappear doesn’t seem like the same problem as picking one door out of a hundred and having ninety-eight of the others disappear.”

“Well, yeah…”

It’s obvious in this case that the one final option, the survivor of 1/99 odds, seems more correct than the 1/100 choice you first made. But it’s hard to go from that and successfully appeal to someone’s impression that the same goes with three doors for the same reason─naturally, because the problem has to do with math, not impressions.

“Then let’s go with the solution that I heard.”

I decided to back down and try another approach─sometimes a detour can prove to be a shortcut.

It seemed to be the quickest way. The shortest path isn’t always the most expedient one.

“First, let’s think about if A is the right answer. In this case, switch choices and you’ll always be wrong. It doesn’t matter whether the game show host opens door B or C, the player is guaranteed to lose by changing doors. Therefore, not switching is the right move─therefore, it’s better not to switch if A is the correct answer. Right?”

“Yes. I get that.”

“Then let’s think about it when B is the right answer. The host has no choice but to open door C if the player has chosen A, one of the two incorrect doors. In other words, the player only has two choices, A or B. Switch and you’re right, don’t switch and you’re wrong─so it’s better to switch if the answer is B.”

“I see. Well, I get that too.”

“Finally, when C is the right answer─this follows the same pattern as when the answer is B. Given that the player has chosen A and the right answer is C, the host can only open door B. This gives you the two options of A or C, where not switching is wrong and switching is right, making it better for you to switch.”

“It─does?”

“Imagine the paths toward getting the answer right for all three cases, A, B, and C. There are two cases where switching is better, and one where switching leaves you worse off. In other words, not switching gives you a one-third chance of getting it right, while there’s a two-thirds chance that switching is beneficial.”

And of course, the calculations are the same if the player picks door B or door C─which is why the optimal action for the player to take in the Monty Hall problem is to change their selection.

This proof left my first-year middle schooler self in shock─but while I wouldn’t call Ogi’s reaction apathetic, it was still on the level of, Ah. Okay, I understand.

So it didn’t leave a high schooler stunned… Yes, maybe these kinds of math problems hit hardest when you’re in late elementary to middle school. In that case, I’d encountered it at the right time.

Well, maybe not encountered.

I was introduced to it─taught it.

By the individual who had left three envelopes in my shoe box.

“This is kind of an aside,” said Ogi, “but did this TV show know this when they ran the game? Was it a program meant to amuse viewers by letting them watch players be fooled by their human instincts into being unable to pick the optimal solution?”

“No, apparently not─it seems like no one thought a player could double their chances until it was pointed out in a magazine, not the staff working on the show or its viewers. I guess you could say that’s weird…”

It really was weird.

Why else would someone come up with a game involving such odd mechanics─if they thought that your chances stayed the same, how would the game be any different from just choosing one door out of the three? Even if it was meant as a kind of countdown, it seemed so meaningless.

It had become a famous question, the Monty Hall problem, precisely because someone had shown its solution to be so counterintuitive─but the problem existing in the first place felt like some sort of nauseating inversion of ends and means, almost like if aberrations existed prior to aberrational phenomena.

As if children existed before their parents, and that’s what was truly weird about it─how had the creator of the problem come up with the game?

“Heh. I see─well, I guess it’s suggestive.”

“Hm? What’s suggestive of what?”

“Oh, nothing. I’m just talking to myself─for now. No need to worry, we’re not going to get to that for a while. So, to summarize and apply this to the envelopes, you’re saying that it’s the right answer to change our selection from a, the first envelope I opened, to c.”

But I already did open the a envelope, didn’t I, Ogi pointed out mercilessly. I wished she’d overlook the fact. These three envelopes weren’t some kind of project put together by a TV show─that wasn’t who sent them.

The person who did, through my shoe locker, was a first-year middle schooler at the time, like me.

“In that case, let’s open the c envelope─let’s play right into expectations. And what’s this? A map? There seems to be some kind of marking on the map, too,” Ogi said in an overly explanatory style. She didn’t delay opening the c envelope for a single moment once she knew it was the right answer. Though I wasn’t fully on board with it, I could learn something from her drive.

Had I possessed half of her drive during that morning’s commotion, it never would have ended in that awful way. I’d have been able to stop Oikura, or if not her, then Senjogahara─

“In other words, we should go to the place shown on the map? Huh… It doesn’t seem awfully far. This isn’t─a treasure map, is it? And by the way, what was in the b envelope? Let’s take a look.”

Ogi briskly opened the b envelope as well.

What drive…

She had no intention of playing by the rules─or rather, she was playing by an entirely different set of them. Firm rules, rules that made any others just about meaningless.

“Oh? This envelope was empty from the start. Does that mean it’s the wrong choice? Hm─the Monty Hall problem. But this entire chain of events only worked out because we opened the a envelope first. Wouldn’t it not make any sense if I’d started by opening the b or c envelope?”

“Well, yeah─but it was unlikely that you would. If you have three envelopes marked a, b, and c, most people are going to start by opening a.”

“Ah, is that so. Yes, yes, I see. Hmm─what a clever grasp of human psychology. And look at me, I went and sided right with the majority. It seems that whoever placed these letters in your shoe box had a lot of confidence in their own intellect. Though I don’t see the sender’s name on either side of the envelopes.”

So, naturally, Ogi continued.

“We’re going to the place shown on the map next─a journey tracing your memories. We’re on a tour, tracking a young Araragi’s footsteps.”

“Yeah… That’s right,” I said, reminiscing.

Actually, we could just cancel the tour then and there, now that I remembered most of everything that had happened. In other words, I could tell Ogi that our journey had come to an end, and perhaps that would have been the right thing to do as her senior─I’d made her tag along for my sake. But I couldn’t stand not going, not after coming this far.

Going there─going to the place that a young Araragi frequented every single day one summer.

I had to go.

“Let’s, Ogi. To the coordinates shown on that map─wait, what?” I found myself saying again. Because at some point, she’d disappeared from in front of the shoe cupboard─no doubt acting before waiting to hear my reply.

Come on, give me a break.

Cutting short my attempt to look cool…

Just how aggressive could one person be? And why bother checking with me if she was just going ahead with it? What was she doing ditching her travel companion, even if we were at my alma mater? All this ran through my mind as I chased after Ogi. She might already be past the school gates with that drive of hers, I thought. But I didn’t have to exhaust myself going after her, she’d stopped only a little ahead of me.

Had she decided to wait on me and my slow decision-making abilities?

She stood at the shoe-cupboard corner for the second-years.

Vacantly staring at one of the labels.

“Sorry to make you wait,” I apologized. She’d been the one to go ahead without asking, but I wasn’t about to criticize her for that.

“Oh, no, it’s fine, you fool. Don’t you worry,” answered Ogi, before starting to walk again. I’d gotten pretty used to the way she called me a fool, but it did surprise me when it came out of nowhere.

“Hm…”

I passed my eyes over the nearby boxes and found Sengoku’s name on one. Well, of course, she’d have one since she was a student here but─hmmm? It kind of seemed like Ogi had been staring at this particular one─was I imagining things?