No matter where he went, he took his files with him. There could be no rest, no vacation while there was still progress to be made. And all he needed to work on it was paper and pencils. If only he could have been left alone to do his research, he would have required nothing else from life.
Mechanically, he walked his usual morning route past the Shin-Ohashi Bridge and down along the Sumida River, past the shacks with roofs made from blue vinyl tarps. The man with the long white hair pulled back into a long braid was once again holding a pot over a burner. Ishigami couldn’t see what he was cooking. A mutt with light brown fur was tied up beside him. The dog sat, exhausted, leaning its flank toward its master.
The Can Man was crushing cans and muttering to himself as usual. He already had two plastic bags filled with flattened aluminum balanced next to him.
A little farther along there was a bench. This morning it was empty. Ishigami glanced at it for moment, then dropped his gaze to the ground before his feet, maintaining his steady pace.
He sensed someone walking toward him along the riverbank. Right about now was when he usually met the elderly woman taking her three dogs for a walk, but he could tell from the sound of the approaching footsteps that this wasn’t she. Ishigami glanced up.
A surprised “Oh” slipped from his mouth, and he stopped.
The other person kept on coming. Then with a smile he paused in front of Ishigami.
“Morning,” said Manabu Yukawa.
Ishigami hesitated for a moment, unsure of how to reply. He wet his lips and opened his mouth. “You were waiting for me.”
“Of course I was,” Yukawa replied, still smiling. “Well, maybe not ‘waiting,’ exactly. I’ve just walked here from Kiyosu Bridge. Figured I’d run into you along the way.”
“What ever it is, it must be urgent.”
“Urgent? Maybe. It could be.”
“You want to talk about it now?” Ishigami glanced at his watch. “I haven’t much time.”
“I only need ten or fifteen minutes.”
“Can we talk while we walk?”
“Suits me.” Yukawa glanced around him. “But let’s talk here a moment first. Just two or three minutes will do. How about that bench over there?” He immediately headed toward the bench without waiting for Ishigami’s reply.
Ishigami sighed and followed his friend.
“We walked here once before together, didn’t we?” Yukawa recalled.
“That we did.”
“I remember you saying that the homeless people here had developed routines like clockwork. You remember that?”
“I do. That’s what happens when you take clocks out of people’s lives—I believe that’s what you said.”
Yukawa nodded, satisfied. “Too bad it’s impossible for you and me ever to be off the clock. Like it or not, we’re stuck in the cogs of society. Take them away, and our clocks spin out of control. Or rather, we are the cogs in the clockworks. No matter how much we might think we are off standing on our own, we’re not. It gives us a certain measure of security, to be sure, but it also means we’re not entirely free. I’ve heard that lots of the homeless don’t want to go back to living regular lives.”
“Keep chatting like this and you’ll use up your two or three minutes in no time,” Ishigami said, looking at his watch. “You’re at one minute already.”
“Okay—the world needs its cogs, all of them; and even a cog may say how it gets used. In fact, only a cog may determine its eventual meaning in the system. That’s what I wanted to tell you,” Yukawa said, staring Ishigami in the face. “Are you going to quit your teaching job?”
Ishigami’s eyes opened wide with surprise. “Why do you ask that?”
“I just thought you might be moving in that direction. You’re not planning on being the cog labeled ‘math teacher’ all your life, are you?” Yukawa stood. “Shall we?”
The two stood up and began walking along the Sumida. Ishigami didn’t speak, waiting for his old friend to say something.
“I heard Kusanagi paid you another visit. Checking on your alibi?”
“Yeah. Last week.”
“He suspects you.”
“Seems like it. Though I haven’t the foggiest clue why.”
Yukawa suddenly smiled. “To tell you the truth, he’s a little foggy on that as well. He’s only interested in you because he saw me taking an interest in you, that’s all. I’m sure it’s not my place to tell you this, but the police don’t have a bit of evidence to point them in your direction.”
Ishigami stopped. “So why are you telling me this?”
Yukawa stopped as well and turned toward him. “Because I’m your friend. No other reason but that.”
“Because you’re my friend you thought you needed to tell me about some murder investigation? Why? I have nothing to do with the crime. Why should I care if the police suspect me of anything?”
He heard Yukawa sigh—a long, slow sound. Then his friend shook his head. Something about the sadness in his expression made Ishigami feel nervous.
“The alibi’s immaterial,” Yukawa said quietly.
“Huh?”
“Kusanagi and company are obsessed with picking apart their suspect’s alibi. They think that if they keep prodding at the holes in Yasuko Hanaoka’s story, they’ll eventually find the truth, provided she really did kill her ex-husband. And they think that if you were her conspirator, then all they need to do is check out your alibi, too, and the fortress you’ve built will come crumbling down.”
“Sorry, but I haven’t the faintest idea what you’re talking about.” Ishigami frowned. “Besides, what’s wrong with detectives looking for holes in alibis? Isn’t that what they do? Assuming there really are holes to be found, that is.”
Yukawa’s mouth softened. “Kusanagi said something interesting the other day. He was talking about the way you designed the tests you give to your students, taking advantage of the blind spots created by the students’ own assumptions. Like making an algebra problem look like a geometry problem, for instance. It made sense. It’s very effective for tripping up the ones who don’t understand the underlying principles and just try to solve everything by the book. The student sees what they think is a geometry problem, so they attack it from that angle. But they can’t solve it. They get nowhere, and end up just wasting time. Some people might call it unfair, but it’s a very effective way of measuring someone’s true ability.”
“What are you getting at?”
“Kusanagi and company,” Yukawa said, his face growing serious, “think that this is a question of breaking down alibis. And why not, since their primary suspect has an alibi. Even better, her alibi feels weak. It feels like you could just keep hitting it and eventually it might break. The same thing happens all the time in our research, really. And, time and time again, we find that while we were happily swinging away at a problem, all the while we were completely off the mark. The police have fallen into that very trap. The bait was there and they took it, hook, line, and sinker.”
“If you have concerns about the direction their investigation is taking, shouldn’t you be talking to Detective Kusanagi, not me?”
“Yes, eventually, I’ll have to. But I wanted to talk to you first, for reasons I’ve just stated.”
“Because you’re my friend.”
“Yes. I might also say because I don’t want to lose your genius. I want to clear away all of this distraction and get you back to doing what you do best. You’ve got a precious brain and I don’t want it being wasted like this.”
“I don’t need you to tell me that. I already abhor wasted time,” Ishigami said. He turned away and began to walk again—not because he was worried about being late to his class, but because it had suddenly become too uncomfortable for him to continue standing in that spot.
Yukawa followed. “In order to solve this case, we mustn’t think that the suspect’s alibi is the problem. The problem lies elsewhere. A difference greater than that between geometry
and algebra.”
“So, out of curiosity, what is the problem?” Ishigami asked without looking back.
“It’s not a simple thing, so it’s hard to give you a simple answer; but if I had to sum it up in one word, I’d say it’s a matter of camouflage. Subterfuge, even. The investigators have been fooled by the criminals’ camouflage. Everything they think is a clue isn’t. Every hint they uncover is merely a breadcrumb set in their path to lure them astray.”
“That sounds complicated.”
“Oh, it is. But if you simply change your way of looking at it, it becomes surprisingly simple. When an amateur attempts to conceal something, the more complex he makes his camouflage, the deeper the grave he digs for himself. But not so a genius. The genius does something far simpler, yet something no normal person would even dream of, the last thing a normal person would think of doing. And from this simplicity, immense complexity is created.”
“I thought you physicist types didn’t like talking in the abstract.”
“I can be more concrete, if you like. How are we doing on time?”
“I’m still good.”
“Still have time to drop by the lunch shop?”
Ishigami glanced at his friend before returning his gaze to the path ahead. “I don’t buy lunch there every day, you know.”
“Really? I heard you did. Well, almost every day.”
“Is that your smoking gun that links me to this case?”
“Yes and no. If you were simply buying your lunch at the same shop every day, that wouldn’t mean anything, but if you were going to meet a particular woman every day, that’s something no interested observer could overlook.”
Ishigami stopped and glared at Yukawa. “Do you think because we are old friends, you can just say whatever you like?”
Yukawa met his gaze. Facing him straight on like this, Ishigami could sense the strength behind the physicist’s eyes.
“Wait, are you really angry? I’ve upset you.”
“This is ridiculous,” Ishigami muttered, setting off again. He began climbing the stairs that led up toward Kiyosu Bridge.
“Clothes they think belonged to the victim were found a short distance away from the body,” Yukawa said, following a pace behind. “They were half-burned, in an oil can. They think the murderer did it. When I heard that, I wondered why the murderer didn’t do a better job and burn the clothes completely. The police seem to think it was because he wanted to leave the scene as quickly as possible, but that leads one to wonder why he wouldn’t have just taken the clothes with him to burn someplace else when he had more time. Or maybe he thought they would burn more quickly than they did? Once I started thinking about it, it bothered me. So I tried burning some clothes myself.”
Ishigami stopped again. “You burned your clothes?”
“In an oil can, yes. A jacket, a sweater, some pants, shoes … oh, and underwear. Bought them at a used-clothes shop. I was surprised how much it cost! See, unlike mathematicians, we physicists aren’t satisfied with something until we’ve performed the experiment ourselves.”
“And your results?”
“They burned pretty well, actually, and put off a lot of toxic fumes,” Yukawa told him. “There was nothing left. It didn’t take long at all. Maybe five minutes, tops.”
“And so?”
“So why didn’t the murderer wait those five minutes?”
Ishigami shrugged. He climbed the stairs leading back to the street and, at the top of the stairs, he turned left on Kiyosubashi Road—the opposite direction from the way to Benten-tei.
“Not buying lunch today?” Yukawa asked, as he’d expected.
“I told you I don’t go there every day,” Ishigami retorted, frowning.
“I was just worried about your lunch, that’s all,” Yukawa said, quickening his pace to walk beside him. “They also found a bicycle near the body, you know. Turns out it had been stolen from Shinozaki Station. The suspected victim’s fingerprints were on it.”
“What of it?”
“Kind of surprising to have a criminal who goes so far as to crush the victim’s face, yet forgets to wipe his bicycle for fingerprints. Pretty stupid, really, unless he left his fingerprints on the bicycle on purpose. But why would he do that?”
“I’m sure you’re about to tell me.”
“Maybe in order to link the bicycle to the victim. Clearly, it was better for the criminal for the police to draw that conclusion.”
“Why’s that?”
“Because the criminal wanted the police to assume that the victim had ridden that bicycle from Shinozaki Station himself. And for that, he couldn’t use just any old bicycle.”
“So there was something special about the bicycle they found?”
“I wouldn’t say ‘special.’ It was your typical morning commuter bike. With one exception: it was brand-new.”
Ishigami felt every pore on his body open. It was getting more difficult for him to breathe steadily.
Someone called out, “Good morning!” and he started at the sound. A female student was passing them on her bicycle. She nodded toward Ishigami.
“Oh, hey, morning,” he hurriedly called back.
“That’s impressive. I didn’t think students said hello to their teachers anymore,” Yukawa commented.
“Hardly any do. So why did this bicycle have to be brand-new?”
“If you’re going to steal a bicycle, why not steal a new one—that’s what the police seem to think. But our thief didn’t care about that. He cared about when the bicycle had been left at the station.”
“Because…?”
“The thief had no use for a bicycle that might have been left by the station for days. And he wanted the owner to report the theft. That’s why the bicycle had to be new. Owners of brand-new bicycles don’t usually leave them out on the street for very long and are more likely to go to the police when their bicycle goes missing. Neither of these things was absolutely necessary for the thief to create his camouflage, but either would help, so he chose the course of action most likely to yield positive results.”
“Hmph.”
Ishigami refrained from commenting on Yukawa’s conjecture. He walked on, looking only at the street in front of them. They were nearing the school. The sidewalk was filling up with students.
“Well, this is certainly an interesting story, and I’d like to hear more,” he said, stopping and turning to face Yukawa. “But maybe you’ll let me be for now? I don’t want the students prying.”
“Absolutely. I think I’ve said pretty much what I had to say, in any case.”
“It was interesting,” Ishigami said. “I recall you posing a question to me before. You asked which was more difficult, formulating an unsolvable problem, or solving that problem. Remember?”
“I do. And I have an answer for you. It’s more difficult to create the problem than to solve it. All the person trying to solve the problem has to do is always respect the problem’s creator.”
“I see. What about the P = NP problem, then? The question of whether or not it’s as easy to determine the accuracy of another person’s results as it is to solve the problem yourself.”
Yukawa favored him with a suspicious look, unsure of where Ishigami was leading.
“You’ve given me your answer,” Ishigami went on, pointing a finger at Yukawa’s chest. “Now it’s time for you to hear someone else’s solution.”
“Ishigami…”
“Good day.” The mathematician turned his back on Yukawa and strode into the school, tote bag clutched tightly in his arms.
It’s over, he thought. The physicist had seen through everything.
* * *
Misato sat in uncomfortable silence, eating her apricot pudding. Yasuko wondered once again whether it would have been better just to have left her at home.
“You get enough to eat, Misato?” Kudo was asking. He had been fretting over her all evening.
Misato nodded, mechanically sticking the spoo
n into her mouth, without even a glance in his direction.
You can drag a teenager to a good restaurant, but you can’t make her enjoy it.
They had come to a Chinese place in Ginza for dinner. Kudo had insisted that Yasuko bring her daughter, and so she had dragged Misato along, despite the girl’s protests. In the end, Yasuko had convinced her to come by telling her that it would seem unnatural for them to avoid going out—that it might make the police suspicious.
But now that she saw how worried Kudo was, she regretted it. All through dinner, he had tried a variety of approaches to get the girl to talk, but he had failed to get more than a few terse words out of her all night.
Misato finished her dessert and turned to her mother. “I’m going to the bathroom.”
“Oh. Okay.”
Yasuko waited for Misato to leave, then turned to Kudo, clasping her hands together. “I’m so sorry.”
“Huh? About what?” He looked genuinely surprised, though Yasuko knew it was an act.
“She’s really shy, that’s all. And I think she has issues with older men.”
Kudo smiled. “Don’t worry. I didn’t imagine we’d be great friends by evening’s end. I was just like her when I was a teenager. I’m happy just to have gotten to meet her today.”
“Thanks, you’re too kind.”
Kudo nodded. He fished in the pocket of his jacket, which hung on the back of his chair, and pulled out a cigarette and lighter. He had refrained from smoking during dinner on account of Misato.
“Any developments since we last spoke?” he asked, taking a puff.
“Developments? With what?”
“That investigation.”
“Oh.” Yasuko lowered her eyes for a moment then looked back up at him. “No, nothing. Life’s been pretty normal, actually.”
“I’m glad to hear that. The detectives leaving you alone?”
“I haven’t seen them in a while. They haven’t been to the shop, either. How about you?”