The Devotion of Suspect X

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 something 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 ended 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?”

“That was quick.”

The Riemann hypothesis was widely considered to be one of the most important unresolved problems confronting modern mathematics. The challenge was to prove a hypothesis proposed by the German mathematician Bernhard Riemann; no one had been able to do it so far.

The report Yukawa brought was an attempt to show that the hypothesis was false. Ishigami knew there were powerhouse scholars elsewhere in the world trying to do this very thing. Of course, none had succeeded yet.

“One of the professors in our math department let me copy this. It hasn’t been published anywhere yet. It’s not a complete counterexample, but I think it’s heading in the right direction,” Yukawa explained.

“So you think that the Riemann hypothesis is wrong?”

“I said it was heading in the right direction. If the hypothesis is right, then of course it means there’s a mistake in this paper.”

Yukawa’s eyes glittered like those of a young miscreant watching a particularly elaborate practical joke unfold. Ishigami realized what he was doing. This was a challenge. He wanted to see just how soft Ishigami the Buddha had grown.

“Mind if I take a look?”

“That’s why I brought it.”

Ishigami pored over the paper intently. After a short while he went to his desk and got out a fresh piece of paper. Laying it down before him, he picked up a ballpoint pen.

“You’re familiar with the P = NP problem, right?” Yukawa asked from behind him.

Ishigami looked around. “You’re referring to the question of whether or not it is as easy to determine the accuracy of another person’s results as it is to solve the problem yourself—or, failing that, how the difference in difficulty compares. It’s one of the questions the Clay Mathematics Institute has offered a prize to solve.”

“I figured you might be.” Yukawa smiled and tipped back his glass.

Ishigami turned back to the desk.

He had always thought of mathematics as a treasure hunt. First, one had to decide where to dig; then one had to determine the proper excavation route that led to the answer. Once you had a plan, you could make formulas to fit it, and they would give you clues. If you wound up empty-handed, you had to go back to the beginning and choose another route. Only by doing this over and over, patiently, yet boldly, could you hope to find the treasure—a solution no one else had ever found.

Therefore, it would seem that analyzing the validity of someone else’s solution was simply a matter of following the routes they had taken. In fact, however, it was never that simple. Sometimes, you could follow a mistaken route to a false treasure, and proving that it was false could be even harder than finding the real answer.

Which was why someone had proposed the exasperating P = NP problem.

Ishigami immersed himself in the problem and soon lost track of time. He was an explorer heading out on safari, a soldier diving headlong into battle, an engine fueled by excitement and pride. His eyes never left the formulas for a moment, his every brain cell devoted to their manipulation.

“Ah!” Ishigami stood suddenly. Report paper in hand, he whirled around. Yukawa had put on his coat and was curled up in a ball on the floor, sleeping. Ishigami walked over, stopped, and shook his shoulder. “I figured it out.”

Yukawa sat up, eyes bleary with sleep. He rubbed his face and looked up at Ishigami. “What’s that?”

“I figured it out. I’m sorry to report that this counterexample is wrong. It was an interesting approach, but there was a fundamental flaw in the distribution of prime numbers—”

“Hold on a second. Hold on.” Yukawa held up his hands. “My brain is nowhere near awake enough to understand whatever you just said. I’m not sure I would understand even after a few cups of coffee, for that matter. To be honest, I don’t know the Riemann hypothesis from a hole in the ground. I just brought that because I thought you’d be interested.”

“But you said you thought it was heading in the right direction?”

“I was just repeating what the professor over in the mathematics department said. Actually, he knew about the flaw in his counterproof. That’s why he didn’t publish it.”

“Oh. No wonder I found it,” Ishigami said, crestfallen.

“No, it is a wonder. I’m impressed. The professor told me that even a top-flight mathematician would never find that error in one sitting.” Yukawa looked at his watch. “And you did it in only … six hours. Impressive!”

“Six hours?” Ishigami looked out the window. The sky was already whitening. He glanced at the alarm clock to see that it was almost five A.M.

“Ishigami the Buddha lives on!” Yukawa cheered. “Some things never change. Which is kind of a relief.”

“Sorry, Yukawa. I totally forgot you were still here.”

“Oh, I don’t mind. Still, you should probably get some sleep. You’ve got school, don’t you?”

“That I do. But now I’m too excited to sleep. I haven’t concentrated on something like this for a long time. Thank you.” Ishigami extended a hand.

“I’m glad I came,” Yukawa said, giving him a firm shake.

“Me, too.” Ishigami nodded. “There’s not much to do here, but feel free to make yourself at home until the trains start running again.”

Ishigami slept until seven o’clock. He slept deeply, either because his brain was tired or from a deep psychological satisfaction, and when he woke his mind was unusually clear.

He was bustling about, getting ready for work, when Yukawa commented, “Your neighbor was up early.”

“My neighbor?”

“I just heard them leaving. A little after six thirty, I guess.”

So Yukawa had been awake.

Ishigami was wondering if he should say something when Yukawa continued, “That Detective Kusanagi I was telling you about says that she’s a suspect. That’s why he dropped in on you, isn’t it?”

Ishigami assumed an air of calm and put on his suit jacket. “He tells you about his cases, does he?”

“Now and then. It’s more like him dropping by to shoot the breeze, then complaining to me about work before I can get rid of him.”

“I still don’t know what the whole thing is about. Detective … Kusanagi, was it? He didn’t give me any details.”

“Well, apparently, a man was murdered. Your neighbor’s ex-husband.”

“Huh. Never would have guessed it,” Ishigami said, his face expressionless.

“You talk to your neighbor much?” Yukawa asked.

Ishigami’s brain went into overdrive. Judging from the tone of his voice, Yukawa didn’t suspect anything. That wasn’t why he was asking questions. Simply brushing him off was an option here. But Yukawa knew the detective—he had to consider that. Yukawa might mention his visit here. Ishigami had to answer.

“I wouldn’t say ‘much,’ but I do frequent the lunch box shop where Ms. Hanaoka—that’s her name—works. Forgot to mention that to Detective Kusanagi, now that I think about it.”

“So, she’s a seller of lunch boxes,” Yukawa mused.

“I don’t go there because my neighbor works there—she just happens to work at the store where I buy my lunch, if you follow. It’s near the school.”

“I hear you. Still, I can’t imagine it’s all that pleasant having a murder suspect in the neighborhood.”

“It wasn’t me she murdered, so I don’t see how it’s any of my business.”

“How very true,” Yukawa said, without a shred of suspicion.

They left the apartment at seven thirty. Yukawa decided not to head for the nearest station, instead saying he’d walk with Ishigami to his school and take the train from there, a route that would save him from having to make a transfer.

Yukawa didn’t speak of the case or Yasuko Hanaoka again. At first Ishigami had wondered if Kusanagi had sent him to get information, but now he decided he’d probably been overthinking the situation. Kusanagi would have no reason to go to such lengths to get information from him anyway.

“Interesting commute you have here,” Yukawa commented. They crossed under the Shin-Ohashi Bridge and began to walk along the slow-flowing Sumida River, past the ramshackle village set up by the homeless.

The gray-haired man with the ponytail was hanging up his laundry. Beyond him, the Can Man was well into his daily routine.

“It’s the same thing every day,” Ishigami said. “This entire past month, nothing’s changed a bit. You could set your watch by these people.”

“That’s what happens when you free people from the restraints of time. They make their own rigid schedule.”

“I couldn’t agree more.”

They went up the stairs just before Kiyosu Bridge, in the shadow of a nearby office building. Seeing their reflection in a glass door on the first floor, Ishigami shook his head. “How have you managed to stay so young, Yukawa? You still have a full head of hair. How different we two are!”

“Not as full as it used to be. And what’s underneath it is slowing down, too.”

“Good thing it was going too fast to begin with.”

While they chatted, Ishigami felt himself growing tense. If they kept on like this, Yukawa would come with him all the way to Benten-tei. He started to worry that this genius observer of the natural world might notice something between him and Yasuko Hanaoka if he happened to see the two of them together. And he didn’t want to fluster Yasuko by suddenly arriving with a stranger.

When he saw the sign he pointed it out. “There is the lunch box place I was telling you about.”

“Benten-tei, huh? Interesting name. The owners must have hoped that Benten, the goddess of wealth, would smile down on them.”

“Well, they have my business. I’ll be buying one there again today.”

“Right. Well, I suppose I’d best be off, then.” Yukawa stopped.

This was unexpected—and welcome, Ishigami thought. “I’m sorry I wasn’t the best host.”

“Not at all. You sell yourself short.” Yukawa narrowed his eyes. “You ever think about going back to the university to continue your research?”

Ishigami shook his head. “Anything I can do at a university I can do on my own. And I doubt any place would be willing to take me at my age.”

“You might be surprised, but I won’t twist your arm. Good luck, Ishigami.”

“You, too, Yukawa.”

“It was good seeing you again.”

The two shook hands, and Ishigami stood on the sidewalk to see his friend off. He wasn’t being sentimental. He just didn’t want Yukawa to see him go into Benten-tei.

When Yukawa had disappeared into the distance, Ishigami turned and walked swiftly toward the shop.