“I wouldn’t call it ‘interest,’ per se. It was just on my mind. I like this business of chipping away at ironclad alibis.”
“It’s less ironclad then simply hard to pin down—which is why we’re working on it.”
“But you have no evidence against her, nothing that would lead you to suspect her yet, right?”
“True enough. But the fact is, we have no one else worth suspecting right now. Togashi didn’t leave much of a trail. He didn’t have a lot of friends, but no real enemies either. That, and doesn’t it strike you as a little bit too convenient that they happened to go to the movies and karaoke on the night of the murder?”
“I see what you mean, but you need to make some logical decisions here. Maybe you should look at something other than the alibi?”
“Don’t feel you have to tell me how to do my job. We’re doing all the groundwork, believe me.” Kusanagi pulled a photocopy from the pocket of his coat where it hung on his chair and spread the paper out on the table. It was a drawing of a man’s face.
“What’s that?”
“An artist’s depiction of the victim when he was still alive. We have a few men around Shinozaki Station asking if anyone saw him.”
“That reminds me—you were saying some of the man’s clothing escaped burning? A navy jacket, gray sweater, and black pants, was it? Sounds like something just about anybody might wear.”
“Doesn’t it? Apparently they have a mountain of reports of people saying they saw someone a lot like him. We don’t know where to start.”
“So, nothing useful at all?”
“Not really. The closest thing we have to a useful tip is one woman who says she saw a suspicious-looking guy wearing clothes like that near the station. An office lady on her way home from work; she saw him loitering there. She called it in after seeing one of the posters we put up at Shinozaki.”
“It’s good to see the people here are being helpful. So why don’t you question her? Maybe you can get something more out of her.”
“We did, of course. The problem is, the man she saw doesn’t sound like our victim.”
“How so?”
“Well, first of all, the station she saw him at wasn’t Shinozaki, but Mizue—one station before it on the same line. That, and when we showed her the picture, she said his face looked rounder than the one in our illustration.”
“Rounder, huh?”
“One thing you come to realize as a police detective is that a lot of our work consists of barking up the wrong tree. It’s not like your world, where once the logic fits, you have your proof and you can call it a day.” Kusanagi busied himself with fishing for leftover chunks of potato with his chopsticks. He was expecting a snappy comeback, but Yukawa didn’t say anything. When he looked up he saw his friend staring off into space, his hands lightly clasped together.
Kusanagi had seen this look before: it was a sure sign that the physicist was deep in thought—though whether the sudden revery had anything to do with the matter at hand remained to be seen.
Gradually Yukawa’s eyes regained their focus. He looked at Kusanagi. “You said the man’s face was crushed?”
“Yep. His fingerprints were burned off, too. They must have been trying to keep us from identifying the body.”
“What did they use to crush the face?”
Kusanagi glanced around to make sure no one was eavesdropping, then he leaned across the table. “We haven’t found anything, but we suspect the killer used a hammer. Forensics thinks the face was struck several times to break the bones. The teeth and jaw were completely destroyed, too, making it impossible for us to check them against his dental records.”
“A hammer, huh?” Yukawa muttered, using the tips of his chopsticks to split a soft stewed daikon radish.
“What about it?” Kusanagi asked.
Yukawa put down his chopsticks and rested his elbows on the table. “If this woman from the lunch box shop was the killer, what exactly do you think she did that day? First, you’re assuming that she didn’t really go to that movie, right?”
“I’m not certain she did or didn’t go, if that’s what you’re asking.”
“Never mind what really happened. I just want to hear some deductive reasoning.” Yukawa made an encouraging motion with one hand while lifting his beer to his lips with the other.
Kusanagi frowned. “Well, it’s more conjecture than anything solid, but here’s what I think. The lunch box lady—let’s just call her Ms. A for short—well, Ms. A gets off work and leaves the shop after six. It takes her ten minutes to walk from there to Hamamatsu Station. It’s another twenty minutes from there on the subway to Shinozaki Station. She takes a bus or taxi from the station to someplace near the Old Edogawa River, which would put her near the scene of the crime at around seven o’clock.”
“And what’s the victim doing during this time?”
“The victim’s heading toward the scene, too. He’s going there to meet with Ms. A. But the victim comes from Shinozaki Station by bicycle.”
“Bicycle?”
“Yeah. There was a bicycle abandoned near where the body was found, and the prints on the bicycle matched those of the victim.”
“The prints? I thought you said his fingertips had been burned off?”
Kusanagi nodded. “After we figured out who the John Doe was, we got some useable prints. What I should have said was the prints on the bicycle matched those we found in the room where the victim was staying. Aha! I know what you’re getting at. You’re going to tell me that even if we could prove the man renting the room was the same one who used that bicycle, that doesn’t mean they were the victim, right? What if the man staying in the room was the real killer, and he used the bicycle? Plausible enough, I suppose. But we also found some hair in his room. It matched the hair on the victim’s body. We even did DNA analyses of both and they were a match.”
Yukawa chuckled. “No, I wasn’t going to suggest that the police had made a mistake identifying the body. I’m more concerned about the idea of him using this bicycle. Did the victim leave his bicycle at Shinozaki Station?”
“No, actually—”
Kusanagi went on to explain what he’d learned about the stolen bicycle. Yukawa’s eyes widened slightly behind his wireframe glasses.
“So the victim went out of his way to steal a bicycle at the station just to go to the scene of the crime? Why not take a bus or a taxi?”
“I don’t know why he stole the bike, but that’s what he must’ve done. The guy was unemployed, after all, without a whole lot of money to his name. He probably wanted to avoid paying the bus fare.”
Yukawa, looking unconvinced, crossed his arms and gave a faint snort. “Well, okay—however he did it, the victim went to meet with our Ms. A at the scene of the crime. Go on.”
“I figure they had planned some sort of rendezvous but Ms. A got there a little early and was hiding somewhere. When she saw the victim approach, she snuck up behind him, wrapped a rope around his neck, and strangled him to death.”
“Stop right there!” Yukawa raised both hands. “How tall was the victim?”
“One hundred and seventy centimeters plus change,” Kusanagi said, resisting the urge to curse. He knew what Yukawa was going to say next.
“And Ms. A?”
“About one sixty.”
“So he was over ten centimeters taller,” Yukawa said with a slight grin, resting his chin on his hand. “You see what I’m getting at here.”
“Sure, it’s hard to strangle someone taller than you. And from the angle of the marks on the victim’s neck, it’s pretty clear whoever strangled him was pulling upward. But the victim could have been sitting. Maybe he was still on the bicycle.”
“Well, I’m glad you had a sad excuse for your scenario ready.”
“Nothing sad about it,” Kusanagi said emphatically, bringing his fist down on the tabletop.
“So what happened next? She took off his clothes, smashed in his face with the hammer she also brought,
and burned off his fingertips with a lighter? Then she set fire to his clothes, and fled the scene. That’s about it?”
“She still could’ve made it to Kinshicho by nine o’clock.”
“Theoretically, yes. But I can’t help thinking you’re grasping at straws. Don’t tell me that the entire department is backing your little scenario?”
Kusanagi’s mouth curled into a frown. He downed the rest of his beer and waved to the waitress for another round, then turned back to Yukawa. “Yeah, well, a lot of the men wonder if a woman really could have pulled it off.”
“As well they should. Even if she did catch him by surprise, it’s not easy to strangle a grown man who is fighting back. And believe me, he would have done everything in his power to stop her. Besides, it would be difficult for a woman of average size and strength to dispose of a grown man’s body after the deed was done. I’m sorry, but I have to join the crowd that thinks your theory is full of holes.”
“Yeah, I figured you’d say that. I don’t much believe it myself. I’m just saying it’s one of several possibilities.”
“Which suggests you have some other ideas. Well, don’t keep them all to yourself. Let’s hear another theory.”
“I’m not claiming that I’ve got much of anything right now. But the scenario I just gave you assumes that the man was killed near where he was found. It’s also possible he was killed somewhere else and then his corpse was dumped there. Truth be told, most of the department thinks that’s what happened. Regardless of whether Ms. A did it or not.”
“It does seem to be the more reasonable assumption. But it wasn’t the one you offered up first. Why?”
“Simple. If Ms. A was the killer, she couldn’t have done it someplace else. She doesn’t have a car or access to one. She can’t even drive. There was no way she could have transported the body to the riverbank.”
“I see. That strikes me as an important point.”
“And then there’s the matter of the bicycle. We could assume whoever left it there did so on purpose to make us think that the murder took place on the riverbank, but then there would have been no reason to go to the effort to put the victim’s fingerprints on it. Especially since they went to the trouble of burning fingertips off the body.”
“The bicycle is a mystery. For a number of reasons.” Yukawa tapped his fingers on the tabletop like he was playing the piano. Then he stopped and said, “Either way, isn’t it better to assume that a man probably did it?”
“That’s what most people at the department think. But I still think Ms. A was involved.”
“So Ms. A had a male accomplice?”
“We’re looking into people connected to her now. She used to be a hostess at a nightclub, after all. There have to be some men in her life.”
“An interesting assumption. I can hear the uproar from hostesses across the country already,” Yukawa said with a grin. He took a swallow of beer, then, a serious look returning to his face, asked to see the illustration again.
Kusanagi handed him the artist’s depiction of the victim. It was a rendering of Togashi as he might have appeared dressed in the clothes they’d found near the crime scene.
Yukawa stared at it intently. “Why did the killer feel the need to strip the body, I wonder?” he muttered.
“To help hide the victim’s identity. Same reason he crushed the face and got rid of the fingerprints.”
“Then why didn’t he take the clothes with him when he left? It’s only because he tried to burn them and failed that you were able to come up with that illustration there.”
“Well, he was probably in a hurry. Or he made a mistake.”
“I agree that you can tell someone’s identity from their wallet or driver’s license, but can you really identify someone from their clothes or shoes? It seems like the risk involved in taking the time to take off and burn his clothes would outweigh any benefits. Wouldn’t the killer want to get away as quick as he could?”
“What are you driving at? You think there’s another reason they stripped him?”
“I can’t say for sure. But if there was, then until you figure out that reason, you won’t be able to pin down your killer.” Yukawa traced a large question mark on the illustration with his fingertip.
* * *
The performance of the junior-year group 2 math class on the year-end exams was appalling. And group 2 wasn’t the only sad story; the entire junior class had done poorly. To Ishigami it seemed like the students were getting dumber by the year.
After he’d passed out the answers, the math teacher put up a schedule for make-up exams. The school had set a lowest acceptable score for each subject, and those students who didn’t reach it wouldn’t go on to the next grade. Of course, they prevented all but the most hopeless cases from failing and being held back a year by making them take as many make-up exams as they needed to pass.
Shouts of protest rose from the class when they saw the grades he’d given them. Ishigami ignored the outcry as usual, but one comment rose above the noise and reached his ears.
“Hey, Teach, aren’t there universities that don’t require a math test to get in?” one of the students was saying. “Why should us guys who are going to those schools have to pass math?”
Ishigami looked in the direction of the student, a boy named Morioka. He was leaning back in his chair, scratching his head and looking around at the other students for support. He was a short kid, but he filled the role of class crime boss—even Ishigami, who didn’t have this bunch for homeroom, knew his reputation. The boy already had a long history of warnings for riding to school on a motorbike, which was strictly forbidden.
“Are you going to art school, Morioka?” Ishigami asked.
“Well, I mean … if I do go to university, it’ll be one without a math exam for sure. Not that I plan on going. Besides, I’m not taking the optional math class next year, so what’s my grade this year matter? Hey, don’t get me wrong, I’m thinking about you, too, Teach. Can’t be much fun teaching with idiots like me in the class. So, I was thinking, maybe we could kind of come to an understanding about this. An agreement between adults, like.”
That last line got a laugh from the class. Ishigami chuckled wryly. “If you’re so worried on my account, then pass your make-up exam. It’s only differential and integral calculus. That can’t be too hard.”
Morioka scoffed loudly. He crossed his legs off to the side of his chair. “What good’s differential and integral calculus gonna do me? It’s a waste of time.”
Ishigami had turned to the blackboard to begin an explanation of some of the trickier problems on the year-end exam, but Morioka’s comment made him stop and turn around. This wasn’t the kind of thing he could let slide. “I hear you like motorbikes, Morioka. Ever watched a race?”
Morioka nodded, clearly taken aback by the sudden question.
“Well, do racers drive their bikes at a set speed? No, they’re constantly adjusting their speed based on the terrain, the way the wind’s blowing, their race strategy, and so on. They need to know in an instant where to hold back and where to accelerate in order to win. Do you follow?”
“Yeah, sure, I follow. But what’s that got to do with math?”
“Well, exactly how much they accelerate at a given time is the derivative of their speed at that exact moment. Furthermore, the distance they travel is the integral of their changing speed. In a race, the bikes all have to run roughly the same distance, so in determining who wins and who loses, the speed differential becomes very important. So you see, differential and integral calculus is very important.”
“Yeah,” Morioka said after a confused pause, “but a racer doesn’t have to think about all that. What do they care about differentials and integrals? They win by experience and instinct.”
“I’m sure they do. But that isn’t true for the support team for those racers. They run detailed simulations over and over to find the best places to accelerate—that’s how they work out a str
ategy. And in order to do that, they use differential and integral calculus. Even if they don’t know it, the computer software they’re using does.”
“So why not leave the mathematics to whoever’s making the software?”
“We could do that, but what if it was you who had to make the software, Morioka?”
Morioka leaned further back in his chair. “Me? Write software? I don’t think so.”
“Even if you don’t become a software engineer, someone else in this class might. That’s why we study mathematics. That’s why we have this class. You should know that what I’m teaching here is only the tip of the iceberg—a doorway into the world of mathematics. If you don’t even know where the door is, how can you ever expect to be able to walk through it? Of course, you don’t have to walk through it unless you want to. All I’m testing here is whether or not you know where the doorway is. I’m giving you choices.”
As he talked, Ishigami scanned the room. Every year there was someone who asked why they had to study math. Every year, he gave the same explanation. This time, since it was a student who liked motorbikes, he’d used the example of motorbike racing. Last year, it was an aspiring musician, so he talked about the math used in designing musical technology. But no matter the specifics of the discussion, which changed from year to year, it was all old hat for Ishigami.
* * *
When he returned to the teachers’ room after class, Ishigami found a note stuck to his desk. It was hastily scrawled, and it read, “Call Yukawa.” With a cell number written below, he recognized the handwriting as belonging to another one of the school’s math teachers.
What does Yukawa want? he wondered, swallowing to clear the sudden catch in his throat.
Cell phone in hand, he went out into the hallway. He dialed the number on the memo. Yukawa picked up after the first ring.
“Sorry to bother you during school hours.”
“Is it something urgent?”
“I guess you could call it urgent, yeah. Do you think we could meet today?”
“Today? Well, I have a few more things to take care of here. I suppose if it was after five o’clock…” He had just finished his sixth-period class, and all the students were in homeroom. Ishigami didn’t have a homeroom class of his own, so he could leave the keys to the judo dojo with another teacher and get out early if he had to.