The teacher taught mathematics at a private junior high school. That year, he was in charge of six classes of first-year students, from Class A to Class F. With forty students in each class, he was responsible for a total of 240 students. During testing season, his arms felt as though they would get tendonitis from all the grading. Once the first semester ended, differences in the students' academic abilities began to stand out. It was particularly noticeable between the children who went to cram schools and those who did not. The students who couldn't keep up with the lessons had careless or apathetic attitudes during class. He couldn't exactly tell them to go to cram school. Every day was a struggle of trial and error to prevent anyone from falling behind. The first class of the second semester was Class A. Once the textbook had progressed to where it was scheduled to be, he decided to use the remaining time for a diversion. "Everyone, do you know how many days there are in a year?" "Three hundred and sixty-five!" Voices rose. Since the teacher had closed his textbook, a relieved atmosphere washed over the classroom—thankfully, annoying math was done for the day. "Well, that means there are also 365 possible birthdays. There are 40 people in this class. Now, what do you suppose the probability is that two people in this class share the same birthday?" The students began buzzing. One person raised their hand. "Isn't it 40 out of 365?" A few others nodded in agreement. "Wrong," the teacher said. Another child raised their hand and called out loudly, "8 out of 73!" The teacher gave a wry smile. "Thanks for simplifying the fraction. But you're wrong. It's much higher." This was a problem dealing with complementary events. It was actually material learned in high school, but he posed this problem—also known as the birthday paradox—to first-year junior high students because he wanted them to experience the wonder of mathematics: how something your intuition tells you must be one way leads to a completely different answer. "The probability that two people share the same birthday in a class of 40 is about 89%." "Eh? Why does it turn out like that?" Everyone threw out questions all at once. Wondering about things like that. Wanting to know why it happens. This ought to be the necessary and sufficient condition for coming to like mathematics—or at least, for not giving up on it. "Listen. You're all getting mixed up, thinking about the probability of someone sharing *your* specific birthday. The chance of someone having the exact same birthday as you among the remaining 39 people is about 10%. But it doesn't have to be someone sharing *your* birthday. As long as any two people in the class, including yourself, share a birthday, even just one pair, that's fine. Doesn't it feel like the probability should be much higher then?" With that, the teacher wrote the formula on the blackboard. He didn't explain it. The formula for complementary events was still too difficult for first-year junior high students. He just wanted to show them that it could be solved using mathematics. Instead, he said, "Let's actually check," and handed out cards he had prepared in advance. "Write your name and birthday on these." He collected the completed cards from everyone and, splitting the task with that day's student on duty, posted them on the blackboard in order of earliest to latest birthdays. The teacher finished lining up all the birthdays, stepped back a bit from the blackboard, and folded his arms. "Well, how about that." It seemed he had drawn the 11% chance. All forty birthdays were completely different. "In reality, if you have 23 people in a class, the probability that two people share a birthday exceeds 50%." Concluding with that, seeing the disappointed looks on the students' faces, he ended up feeling like he'd done something bad. ◆ During the lunch break, the teacher thought about what happened in Class A's lesson. If even a single pair had shared a birthday there, he could have really captured everyone's hearts. Bad luck. Oh well. The probability would be even higher in the next class. In his Class B lesson that afternoon, the teacher repeated the same thing. "In a class of 40, the probability of two people sharing a birthday is about 89%." In front of the students teasingly saying, "No way," he wrote out the formula and handed out the cards. "Write your name and birthday." He collected the cards just like before and posted them in a line on the blackboard. *Ah*, he thought. It was the 11% chance again. Cold sweat broke out. The birthdays were neatly scattered. Good thing he had prepared just in case. "I had everyone in Class A write their birthdays on cards, too. Across 365 possible birthdays, we have 80 students combined from Class A and Class B. If the probability of two people sharing a birthday were 80 out of 365, it would only be about 22%. But if you use this formula, the probability of two people sharing a birthday among 80 people works out to 99.9914%. In other words, practically 100%. I'll show you the proof." The teacher posted the Class A cards he had kept lined up in advance, interleaving them among the Class B cards on the blackboard. There, this should create at least one pair with the same birthday. Thinking that, while minding the closing chime that should be ringing soon, he performed the task briskly. And yet. And yet... The teacher said, "Huh?", his smile stiffening. The students stared intently at the cards on the blackboard. Among the 80 cards, not a single one shared a birthday. That's absurd. He had calculated it at home the day before—could he have made a mistake? That figure of 99.9914%. Dumbfounded, the teacher let the chalk slip from his hand, and white powder shattered on the floor. ◆ In the empty staff room after school, the teacher swallowed hard for the nth time. His throat was terribly dry. The teachers at the surrounding desks were starting to pack up to go home. Some were rallying people to go out for drinks. Yet that math teacher stared intently at the top of his desk, as if completely deaf to the surrounding voices. On top of the desk were 240 cards. They were the cards with the birthdays of every single student in the six first-year classes he taught. He had them written during these past two days of class. In not a single one of those classes had he been able to make them experience the wonder of the birthday paradox. On the contrary, even the students' incorrect formula of 240 out of 365—a phenomenon that ought to exceed a 65% chance—had failed to occur. 240 cards, all with different dates. He calculated it over and over with a confused head. Over and over again. The probability of two people sharing a birthday among 40 people is 89.12%. The probability of two people sharing a birthday among 80 people is 99.9914%. And the probability of two people sharing a birthday among 240 people is... 99.999999999999999999999... When he counted, there were 33 nines after the decimal point, and an 8 finally appeared in the 34th place. By this point, he couldn't calculate it accurately himself, so he had asked a university peer remaining in the lab to figure it out for him. It was an unmistakable number. Mathematically aside, statistically and in daily life, a probability to be ignored. In other words, an event that simply could not be overturned. And that impossible thing was right in front of him. All 240 birthdays were different. What is this? A chill ran through him, and goosebumps broke out all over his body. Was mathematics the thing that was wrong? Had he stepped outside the laws of this world and wandered into some unfathomable realm? "Aren't you going home yet?" Another teacher called out to him, snapping him back to reality. "Ah, yes." "What's that?" The other teacher peered over with a look of intense curiosity at the cards on his desk, causing him to hurriedly sweep them into his desk drawer to hide them. "I thought they might make good teaching materials. But, you know." He stood up briskly, muttered something about heading to the restroom, and left the staff room. As the teacher relieved himself in the restroom, a phrase he had learned back in university suddenly came to mind: Occam's razor. It was supposed to be a maxim stating that if two theories exist to explain the same phenomenon, the simpler one is the better theory. Rather than assuming that the impossible—a 0.00000000000000000000...1% chance—had occurred, wasn't there a much simpler explanation? Washing his hands, leaving the restroom, and walking down the hallway, he passed the principal's office. At that moment, a thought crossed his mind. A much simpler explanation for the 240 different cards. But the moment he tried to grab onto it, another terrifying darkness opened its jaws. An eerie darkness he couldn't possibly touch. A shudder wracked his body. What am I thinking? Overcome with vertigo, the teacher hurried down the hallway and collapsed back into his seat at his desk in the staff room. As if the school had intentionally gathered only students with different birthdays... That's ridiculous. Is he saying the private junior high school organization he belonged to was operating with some hidden agenda? That's absurd. He struck his head to shake off the impossible thought. It made a surprisingly loud *thud*, and the remaining teachers cast creepy, uninhibited glances his way. ◆
That night, the teacher brought the cards home and noticed something. On a single card, he saw the traces of Occam's razor: the simplest answer to explain an incomprehensible phenomenon. Looking at the date on that card and the student's name, the teacher tried to picture their face. A hazy, dark face wavered vaguely in his mind, its contours undefined. Strange. To be sure, they hadn't been a prominent student. Yet, the teacher was convinced. This wasn't some bizarre occurrence; it was artificial and premeditated. The next day, he asked a general affairs staff member to let him check the student roster. Just as he thought. The teacher's conjecture was correct. It wasn't that an impossible thing had happened by defying probability, nor were students with different birthdays chosen for enrollment for some unimaginable reason. No way. ...Could it really be that the students had all been lying together? It was a complete blind spot. Even after checking the students' dates of birth on the roster, he still couldn't believe it. In every single class, there were students who had faked their birthdays. In Classes B and D, there were two pairs, and in Classes C, E, and F, there was one pair each of students who shared a birthday, but in every case, one of them had lied and written a false date on the card. Furthermore, students who shared a birthday with someone in another class—the ones whose math lessons came later—had reported false birthdays. The only class where no coincidental shared birthdays existed was the very first one, Class A. The 40 people in Class A just happened not to fall into that 89%. Learning about what happened in Class A's lesson, the children in the subsequent classes colluded to put on an act. He had been brilliantly deceived. First-year junior high school students, pulling something like this. Should he tell their respective homeroom teachers? To the students, it would probably be a funny story about how they beat the teacher, but he felt he couldn't let it go like this. During class, he hadn't sensed that attitude at all. The way they coordinated it felt almost bizarre. First, however, he called a few approachable students over during the break and asked them about the cards. They all hesitated at first, but eventually answered timidly: they had been asked to do it by a student in Class A. Apparently, every class had been asked by a single student from Class A to lie about their birthdays. Extracting that student's name, the teacher felt a sickening sensation that was neither satisfaction nor its opposite. That student from Class A had written "November 31st" as their birthday on the card. Of course, November only has up to 30 days. The teacher had noticed it the day he brought the cards home, looking at that student's card date: the students were lying. However, upon checking the students' real birthdays on the roster and conducting interviews, he found that only that one student from Class A had lied in such a peculiar way. That student didn't share a birthday with anyone in Class A. Yet, for some reason, they had faked their birthday—with an impossible date like November 31st, no less. Class A was the very class where he had first talked about the birthday paradox. And he had only thought of it the day before the lesson. Why would that student try to lie at that exact moment? No matter how much he thought about it, he couldn't understand. After school, the teacher called that student into the counseling room. When they walked into the room, his very first thought was, *Ah, was that what their face looked like?* He dropped the cards in front of the student sitting in the seat opposite him. "Why did you ask everyone in the other classes to lie?" He lined up the cards of the students who had written false birthdays on the desk. "And you—what is this ridiculous date of November 31st? If you find my class boring, fine, be bored. But don't do things that disrupt the students who are trying to listen seriously." When the teacher spoke in a harsh tone, the student raised their eyes from the cards on the desk with a lifeless expression. "Why did I lie, you ask?" Their words were slow and flat. "It wasn't a lie. But I had people in the other classes write lies because I knew I would be left out." "Left out?" "The idea that someone is supposed to share your birthday—that's a terrible kind of leaving out." Not quite understanding what the student was talking about, the teacher asked for confirmation: "You lied because you'd be left out?" The student nodded. "Teacher. Have you ever considered that your own birthday might actually be different?" "What on earth are you talking about?" "What if February 30th is actually your birthday, Teacher? When would you hold your birthday party?" "Stop saying ridiculous things. I'm going to get angry." The teacher raised his voice. Yet maybe that was just him resisting the unidentified sense of pressure he felt from the student sitting before him. Did a student like this really exist in this school? For some reason, he felt like he was about to lose his grip on that fact. "Teacher. Do you happen to know what birth sign November 31st has?" Suddenly, he felt as though there was a void right in front of him. Though they had the shape of a student, a delusion crossed his mind that what lay on the other side was a bottomless pit. "It's Cetus, the Sea Monster." The deep abyss peeked out from beyond pitch-black holes for eyes and a mouth. There was no expression. "Sunami, you..." Were they really a student at this school? No, were they even human? Pressing both hands against his trembling knees, the teacher instinctively leaned his body backward.
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