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“The conic has its focus at the origin, right? And it has to satisfy these conditions.” Aubrey pointed to where the textbook said, parabola, directrix y = 2. “So what do you want to do?”

A silence unfolded, and kept unfolding.

“You want to figure out what y is, right?” Aubrey said.

“Yeah.”

“Like this-does that make sense?”

“Yeah.” I nodded. “Definitely.”

“And then you plug this in here.”

“Okay,” I said.

“Why don’t you try the next one?”

For a while, I was looking at the problem, I really was. But then I found myself thinking about Gillian Hathaway, and about whether she and her boyfriend Luke said I love you to each other. How did you even know if you loved another person? Was it a hunch, like a good smell that you couldn’t identify for sure, or did a time come when you had evidence? Was it like walking through a house and once you’d crossed a certain threshold, that was love and you would never turn back? Maybe you’d go into other rooms, you’d fight or even break up, but you’d always be on the other side of love, after and not before it. My interest in couples felt anthropological-even liking Cross, even wanting to hear from Martha that she could imagine me dating him, I myself could not imagine us together. Not as a daily presence in each other’s lives, two people who had conversations and made out and sat next to each other in chapel. When I thought of Sin-Jun and Clara-and I did so often-what was hardest to wrap my head around was how they’d been a couple while living in the same room. How had they known when to fool around and when to just sit at their desks doing homework? Hadn’t it been either too intense, too tiring to always be around the person you wanted to impress, or else too familiar? Maybe in such close quarters you gave up hope of impressing them and sat there picking your earwax and not caring if you looked cute. But didn’t you lose something there, too? If that was what people meant by intimacy, it didn’t hold much appeal for me-it seemed like you’d be fighting each other for oxygen.

Aloud, I said, “Do you think Gillian is pretty?”

“Lee, please concentrate,” Aubrey said.

“Gillian Hathaway,” I said. “Not Gillian Carson.”

“She’s fine. If I were you, I’d start by isolating x on this one. And what information are they giving you about x?” But Aubrey was blushing, a hot shade of pink that blossomed in his face and spread down his neck.

“Really pretty or just medium pretty?” I said.

He turned to me. “I’m not doing your homework for you.”

“I didn’t ask you to.”

“And if you don’t grasp these concepts, you’re not going to pass the final exam.”

“Actually, it’s better than that,” I said. “If I don’t pass the exam, I’ll be spring-cleaned.”

“What are you talking about?”

“It’s when they kick you out but they wait-”

“I know what spring-cleaning is,” he interrupted. I was slightly impressed-I hadn’t known until after I came back for the beginning of sophomore year and Alfie and Maisie were gone. “Who told you this?” Aubrey said.

“Fletcher called me into his office today.”

“That’s terrible.”

“It wouldn’t be your fault.”

“I know,” Aubrey said with such certainty that I wanted to rescind the exoneration. “What are you planning to do?” he asked.

I squinted at him. I was pretty sure he didn’t have much respect for me, but it still seemed like a shitty question. “Well, the school near my parents’ house is called Marvin Thompson.”

“No,” he said. “Lee.” He extended his small hand toward my arm, but then it just hovered there; I think he was afraid to touch me. He withdrew his hand and said, “I meant, what do you want to do about the exam? How do you want to prepare?”

“I’m not sure that it matters how I prepare,” I said. “I mean, being realistic about it.” This felt like a confession of sorts. “Do you think I can pass?” I asked.

He remained quiet for several seconds and finally said, “If you’re willing to work very hard.”

This was worse than if he’d just said no. Because I would put in the time, I would sit in a chair with my math book in front of me, but to actually work-the only choice I’d have would be to start from the beginning of the book. I’d always loved the part in movies when a project, or even a person’s whole life, came together: the montage, set to uplifting music, where you saw the spunky multicultural kids set aside their differences and fix up the old man’s house, straighten the hanging shutters, paint the outside, mow the lawn, and weed the flowerbed; or the twentysomething woman who finally lost weight, dancing through aerobics classes, mopping her brow while she rode a gym bike, with a white towel around her neck, and then at last she emerged from the bathroom all cleaned up, bashful but beautiful (of course, she had no idea how beautiful), and her best friend hugged her before she left for the date or party that would be her triumph. I wanted to be that person, and I wanted the in-between time when I improved myself to glide by just that smoothly, with its own festive sound track. But to really learn precalculus would be laborious and miserable. Plus, it still might not work. The only reason my average was as high as 58 was that in March, Ms. Prosek had let me do a special project for extra credit, and I’d made a timeline of women mathematicians through the ages: Hypatia of Alexandria, b. a.d. 370, inventor of the astrolabe, died when Christian mob attacked her with broken pottery; Emilie du Châtelet, b. 1706, French aristocrat and author of Institutions de Physique, dated Voltaire. I’d made the final woman Ms. Prosek herself, pasting a photo of her from the school catalog onto the poster board and writing next to it, Valerie Prosek, b. 1961, precalculus teacher and inspiration to young math scholars everywhere. Ms. Prosek had hung the timeline in her classroom, above the chalkboard, and she’d given me an A plus.

“If I were going to work very hard,” I said, “where would I begin?”

“It wouldn’t hurt for you to review some of the basics of graphing equations. I can create some problems for you.” Aubrey wrote down several strings of numbers in my notebook, then pushed it toward me. The first one read:

3xy = 5

2x + y = 5

This should not have been hard, I knew. He’d said himself that it was basic. But I had no idea what to do. And to admit my ignorance would be to truly reveal just how far behind I was.

“Actually,” I said, “I’m thinking-I’m sorry I just made you write those up-but maybe it would be better for me to concentrate on the homework for tomorrow. Because that will already take me a long time, don’t you think?”

Aubrey hesitated.

“I’ll take those problems back to the dorm and work on them there,” I said. “Thank you.” I turned back to the textbook and read aloud the next problem: Write out the form of the partial fraction decomposition… Maybe this way, if Aubrey heard my voice, it would seem like I was participating. And it worked, I could sense him caving. All our sessions were like this-the warm-up, the persuasion, the part where Aubrey capitulated and just did my homework for me, after stating that he wouldn’t. But even then, we went so slowly, with him narrating his progress, asking me questions, waiting while I made guesses, many of which weren’t even in the right category-the answer Aubrey would be looking for was an irreducible quadratic factor and the answer I’d give was seven.

Though sometimes I needled him, or acted lazy, I had figured out early on how Aubrey liked me best: trying but not getting anything right. Or maybe: not getting anything right but trying. Either way, the other person’s reaction was the only thing that ever counted to me-numbers were spiky and indifferent, but a person was warm and breathing, potentially swayable. I often messed up with people, it was true, but it rarely happened because I was reading them wrong; it was because I got nervous, or because I could see too clearly that I was not what they wanted. And, in fact, it was in falling short that I truly excelled. I might fail to be what the other person sought, but as a failure, I’d accommodate them completely-I could be obsequious or truculent, sad or earnest or utterly silent. If I’d had to guess, I’d have guessed that Cross knew I had a crush on him, and that in not trying to talk to him, in only every so often making eye contact with him and waiting a beat before looking away, I was acting just the way he’d think a girl who liked him but whom he didn’t like back should act. And I might get spring-cleaned, but I’d be able to go out with everyone on my side-Aubrey, Martha, Ms. Prosek, and even Dean Fletcher, all of them rueful and sympathetic.