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“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.”