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.”