Yasuko got back on the phone and asked, “Well, what should we do?” Her voice sounded faint on the other end of the line.
“Nothing, for now. This isn’t a problem,” Ishigami said with conviction. He wanted to put her at ease. “Everything is going according to my calculations. I should expect that the detectives will be back again soon. Just follow my instructions and everything will be all right.”
“Thank you, Mr. Ishigami … you know I don’t have anyone to turn to but you.”
“That’s all right. Good luck. This will soon be over. I’ll speak with you again tomorrow.”
Ishigami hung up the phone and took out his phone card, already slightly regretting his final words. He shouldn’t have told her it would be over soon. Just how long was “soon”? He shouldn’t be saying things that couldn’t be quantified like that.
However, it was true that events were developing according to plan. He had known they would find out eventually that Togashi had been looking for Yasuko—that was why Ishigami had made the effort to establish an alibi. He had also expected the police to question that alibi.
And he had expected that the police would try to make contact with Misato. They must have hoped she would be the weak link in the chain, a way to take apart the alibi in the absence of any witnesses. Ishigami had taken several steps to prevent that from happening, but he thought now that it would behoove him to check once more and make sure he hadn’t overlooked anything.
Ishigami returned to his apartment, his mind racing, only to find a man standing in front of his door—a tall fellow, unfamiliar, wearing a thin, black jacket. He must have heard Ishigami coming up the steps, for he was facing toward him. His wire-rim glasses glittered.
At first, Ishigami thought it was another detective. But then he realized that no, that was wrong. The man’s shoes were in perfect condition, as good as brand-new.
He approached, warily, and the man spoke. “Ishigami?”
Ishigami looked up at the stranger’s face. The man was smiling. It was a smile he remembered.
Ishigami took a deep breath, and his eyes went wide as the memories came vividly back to him from a twenty-year distance.
“Manabu Yukawa.”
SIX
The classroom felt deserted that day, as always. The room was large enough to seat a hundred students, but there were only twenty or so there now. Most of them were in the back row so that they could slip out after attendance had been taken or work on some project of their own during the lecture.
Very few undergraduates wanted to be mathematicians. In fact, Ishigami was probably the only one in his entire class. And this course, with its lectures on the historical background of applied physics, was not a popular one.
Even Ishigami wasn’t all that interested in the lectures, but he sat in the second chair from the left edge in the front row. He always sat there, or in the closest available position, in every room, at every lecture. He avoided sitting in the middle because he thought it would help him maintain objectivity. Even the most brilliant professor could sometimes err and say something inaccurate, after all.
It was usually lonely at the front of the classroom, but on this particular day someone was sitting in the seat directly behind him. Ishigami wasn’t paying his visitor any attention. He had important things to do before the lecturer arrived. He took out his notebook and began scribbling formulas.
“Ah, an adherent of Erdős, I see,” said a voice from behind.
At first, Ishigami didn’t realize the comment was directed at him. But after a moment the words sank in and his attention lifted from his work—not because he wanted to start a conversation, but out of excitement at hearing someone other than himself mention the name “Erdős.” He looked around.
It was a fellow student, a young man with shoulder-length hair, cheek propped up on one hand, his shirt hanging open at the neck. Ishigami had seen him around. He was a physics major, but beyond that, Ishigami knew nothing about him.
Surely he can’t be the one who spoke, Ishigami was thinking, when the long-haired student, still propping up one cheek, remarked, “I’m afraid you’re going to hit your limits working with just a pencil and paper—of course, you’re welcome to try. Might get something out of it.”
Ishigami was surprised that his voice was the same one he’d heard a moment earlier. “You know what I’m doing?”
“Sorry—I just happened to glance over your shoulder. I didn’t mean to pry,” the other replied, pointing at Ishigami’s desk.
Ishigami’s eyes went back to his notebook. He had written out some formulas, but it was only a part of the whole, the beginnings of a solution. If this guy knew what he was doing just from this, then he must have worked on the problem himself.
“You’ve worked on this, too?” Ishigami asked.
The long-haired student let his hand fall down to the desktop. He grinned and shrugged. “Nah. I try to avoid doing anything unnecessary. I’m in physics, you know. We just use the theorems you mathematicians come up with. I’ll leave working out the proofs to you.”
“But you do understand what it—what this—means?” Ishigami asked, gesturing at his notebook page.
“Yes, because it’s already been proven. No harm in knowing what has a proof and what doesn’t,” the student explained, steadily meeting Ishigami’s gaze. “The four-color problem? Solved. You can color any map with only four colors.”
“Not any map.”
“Oh, that’s right. There were conditions. It had to be a map on a plane or a sphere, like a map of the world.”
It was one of the most famous problems in mathematics, first put into print in a paper in 1879 by one Arthur Cayley, who had asked the question: are four colors sufficient to color the contiguous countries on any map, such that no two adjacent countries are ever colored the same? All one had to do was prove that four colors were sufficient, or present a map where such separation was impossible—a process which had taken nearly one hundred years. The final proof had come from two mathematicians at the University of Illinois, Kenneth Appel and Wolfgang Haken. They had used a computer to confirm that all maps were only variations on roughly 150 basic maps, all of which could be colored with four colors.
That was in 1976.
“I don’t consider that a very convincing proof,” Ishigami stated.
“Of course you don’t. That’s why you’re trying to solve it there with your paper and pencil.”
“The way they proved it would take too long for humans to do with their hands. That’s why they used a computer. But that makes it impossible to determine, beyond a doubt, whether their proof is correct. It’s not real mathematics if you have to use a computer to verify it.”
“Like I said, a true adherent of Erdős,” the long-haired student observed with a chuckle.
Paul Erdős was a Hungarian-born mathematician famous for traveling the world and engaging in joint research with other mathematicians wherever he went. He believed that the best theorems were those with clear, naturally elegant proofs. Though he’d acknowledged that Appel and Haken’s work on the four-color problem was probably correct, he had disparaged their proof for its lack of beauty.
Ishigami felt like this peculiar visitor had somehow peered directly into his soul.
“I went to one of my professors the other day about an examination problem concerning numbers analysis,” the other student said, changing the subject. “The issue wasn’t with the problem itself. It was that the answer wasn’t very elegant. As I suspected, he’d made a mistake typing up the problem. What surprised me was that another student had already come to him with the same issue. To tell the truth, I was a little disappointed. I thought I was the only one who had truly solved the problem.”
“Oh that? That was nothing—” Ishigami began, then closed his mouth.
“—Nothing special?” the other finished for him. “Not for a student like Ishigami—that’s what my professor said. Even when you’re at the top, there’s always somet
hing higher, eh? It was about then that I figured I wouldn’t make it as a mathematician.”
“You said you’re a physics major, right?”
“Yukawa’s the name. Pleased to meet you.” He extended a hand toward Ishigami.
Ishigami took his hand, wondering at his peculiar new acquaintance. Then, he began to feel happy. He’d always thought he was the only weird one.
* * *
He wouldn’t have called Yukawa a “friend,” but from then on, whenever they chanced to meet in the hall, they would always stop and exchange a few words. Yukawa was well read, and he knew a lot about fields outside of mathematics and physics. He could even hold his own in a conversation about literature or the arts—topics that Ishigami secretly despised. Of course, lacking any basis for comparison, Ishigami didn’t know how deep the man’s knowledge of such things went. Besides, Yukawa soon noticed Ishigami’s lack of interest in anything other than math, and the scope of their conversations rapidly narrowed.
Nonetheless, Yukawa was the first person Ishigami had met at university with whom he felt he could talk intelligently and whose ability he respected.
Over time, however, their chance encounters became less and less frequent. Their paths took them in different directions, one in the math department, the other in physics. A student who maintained a certain grade point average was allowed to switch departments, but neither of them had any desire for such a switch. This is really the proper choice for both of us, Ishigami thought. Each on the path that suits him best. They shared a common desire to describe the world around them with theorems, but they approached this task from opposite directions. Ishigami built his theorems with the rigid blocks of mathematical formulas while Yukawa began everything by making observations. When he found a mystery, he would go about breaking it down. Ishigami preferred simulations; Yukawa’s heart was in actual experimentation.
As time went on Ishigami occasionally heard rumors about his acquaintance. He was filled with genuine admiration when he heard, in the autumn of their second year in graduate school, that a certain American industrial client had come to buy the rights to the “magnetized gears” Yukawa had proposed in a thesis.
Ishigami didn’t know what had become of Yukawa after their master’s program was finished; he himself had already left the university by then. And so the years had slipped by.
* * *
“Some things never change, eh?” Yukawa said, looking up at the bookshelves in Ishigami’s apartment.
“What’s that?”
“Your love of math, for one. I doubt anyone in my whole department has a personal collection of materials this thorough.”
Ishigami didn’t dispute it. The bookshelves held more than just books. He also had files of publications from different research centers around the world. Most of them he had obtained over the Internet, but even so, he thought of himself as being more in touch with the world of mathematics than the average half-baked researcher.
“Well, have a seat,” he said after a moment. “Want some coffee?”
“I don’t mind coffee, but I did bring this,” Yukawa said, pulling a box from the paper bag in his hand. It held a famous brand of sake.
“You didn’t have to go out of your way like that.”
“I couldn’t come meet a long-lost friend empty-handed.”
“Well, then, how about I order out some sushi? You haven’t eaten yet, have you?”
“Oh, don’t worry about me.”
“No, I haven’t eaten yet either.”
Ishigami picked up the phone and opened the file where he kept all his menus from local places that delivered. He perused one briefly, then ordered a deluxe assortment and some sashimi on the side. The person taking the order sounded almost shocked to hear a request for something other than the cheap basic selection usually ordered from his telephone number. Ishigami wondered how long it had been since he had entertained a proper visitor.
“I have to say it’s quite a surprise you showing up, Yukawa,” he said, taking his seat.
“Out of the blue I heard your name from a friend the other day, and thought I’d like to see you again.”
“A friend? Who could that have been?”
“Er, well, it’s a bit of a strange story, actually.” Yukawa scratched his nose. “A detective from the police department came by your apartment, right? Guy named Kusanagi?”
“A detective?” Ishigami felt a jolt run through him, but he took care not to let his surprise show on his face. He peered at his old classmate. What does he know?
“Right, well, that detective was a classmate of mine.”
Ishigami blinked. “A classmate?”
“We were in the badminton club together. I know, he doesn’t seem the Imperial University type, does he? I think he was over in the sociology department.”
“Ah … no kidding.” The cloud of unease that had been spreading in Ishigami’s chest vanished in a moment. “Now that you mention it, I remember him looking at a letter that came to me from the university. That must’ve been why he asked about it. Wonder why he didn’t tell me he was a fellow alum?”
“Well, honestly, he doesn’t consider graduates from Imperial University sciences to be his classmates. Sometimes, I don’t even think he thinks of us as the same species.”
Ishigami nodded. He felt the same way about those in the humanities. It was strange to think of the detective as someone who had been at the same university at the same time.
“So Kusanagi tells me you’re teaching math at a high school?” Yukawa asked, staring directly at Ishigami’s face.
“The high school near here, yes. You’re at the university, Yukawa?”
“Yeah. Lab 13,” he replied simply.
Yukawa wasn’t trying to ring his own bell, Ishigami realized; he didn’t seem to have any desire to boast.
“Are you a professor?”
“No. I’m just futzing around as an assistant professor. It’s pretty crowded at the top, you know,” Yukawa said, without any discernible ire.
“Really? I figured you would be a full professor for sure by now, after all that hype about those magnetic gears of yours.”
Yukawa smiled and rubbed his face. “I think you’re the only one who remembers all that. They never did make a working prototype. The whole thing ended as an empty theory.” Yukawa picked up the sake bottle and began to open it.
Ishigami stood and brought two cups from the cupboard.
“But you,” Yukawa said, “I had you pegged as a university professor, holed up in your office, taking on the Riemann hypothesis or some such. So what happened to Ishigami the Buddha? Or are you truly following in the footsteps of Erdős, playing the itinerant mathematician?
“Nothing like that, I’m afraid,” Ishigami said with a light sigh.
“Well, let’s drink,” Yukawa offered, ending his questions and pouring Ishigami a glass.
The fact of it was, Ishigami had planned on devoting his life to mathematics. After he got his master’s, he had planned to stay at the university, just like Yukawa, earning his doctorate. Making his mark on the world.
That hadn’t happened, because he had to look after his parents. Both were getting on in years and were in ill health. There was no way he could have made ends meet for all of them with the kind of part-time job he could have held while attending classes. Instead, he had looked around for steadier employment.
Just after his graduation, one of his professors had told him that a newly established university was looking for a teaching assistant. It was within commuting distance of his home, and it would allow him to continue his research, so he’d decided to check it out. It was a decision that quickly turned his life upside down.
He found it impossible to carry on with his own work at the new school. Most of the professors there were consumed with vying for power and protecting their positions, and not one cared the least bit about nurturing young scholars or doing groundbreaking research. The research reports Ishigami slaved over e
nded up permanently lodged in a professor’s untended in-box. Worse still, the academic level of the students at the school was shockingly low. The time he spent teaching kids who couldn’t even grasp high school level mathematics had detracted enormously from his own research. On top of all this, the pay was depressingly low.
He had tried finding a job at another university, but it wasn’t easy. Universities that even had a mathematics department were few and far between. When they did have one, their budgets were meager, and they lacked the resources to hire assistants. Math research, unlike engineering, didn’t have major corporations waiting in line to sponsor it.
Ishigami had soon realized he had to make a change, and fast. He had decided to take his teaching credentials and make those his means of support. This had meant giving up on being a career mathematician.
He didn’t see any point in telling Yukawa all this, though. Most people who had been forced out of research had similar stories. Ishigami knew his was nothing special.
The sushi and sashimi arrived, so they ate, and drank a little more. When the bottle of sake Yukawa had opened was dry, Ishigami brought out some whiskey. He rarely drank much alcohol, but he did like to sip a little to ease his head after working on a particularly difficult mathematics problem.
Though the conversation wasn’t exactly lively, he did enjoy discussing their old school days, as well as a bit about mathematics. Ishigami realized how little of the last two decades he had spent just chatting. This might’ve been the first time he had talked this much to another person since graduating. Who else could understand him but Yukawa? Who would even recognize him as an equal?
“That’s right, I almost forgot the most important thing I wanted to show you,” Yukawa said suddenly, pulling a large brown envelope from his paper bag and placing it in front of Ishigami.
“What’s this?”
“Open it and find out,” Yukawa said with a grin.
The envelope held a sheet of paper covered with mathematical formulas. Ishigami glanced over it, recognizing it almost instantly. “You’re trying a counterexample to the Riemann hypothesis?”