Subj: Putnam prep session #2 for eternal mathematical glory ... and donuts, 2 p.m. Fri 4 October
"Hey, Goofusia," said Gallantina. "Did you see this post on the math_majors list? Someone's trying to organize a team for the Putnam competition—here, at SFSU! There's going to be a prep session in Thornton 935 on Friday at 2 p.m. The organizer sounds really desperate—there should be free donuts. Want to come?"
Fraternal twins, the sisters looked so much alike that strangers who didn't know them often asked if they were identical. People who knew them for any length of time never asked.
Goofusia grimaced. "Oh, God, is that that super-hard math competition that guys from MIT win every year, where the median score is zero?"
"Actually, someone not from MIT won as recently as 2018, and last year the median score was nine. But yes."
"Uh-huh. What school was the 2018 winner from?"
"Um, Harvard."
"I'll pass. You should, too."
"C'mon, it'll be fun!"
"Gallantina, you don't know what fun is. You're so caught up in your delusional self-image of pro-sociality that you can't even notice what you actually enjoy." Goofusia spoke with a firm emphasis and cadence, "I, am learning math, in order to get grades, in order to get a degree, in order to get a job. So is everyone else in our major. So are you. That's the only possible reason—the only human reason. You just can't admit it to yourself—"
"That's not true!"
"—and you're so fanatically devoted to maintaining your false self-image as some intrinsically motivated student of the cosmos that you're willing to torture yourself with more schoolwork that doesn't even benefit you. You are not going to score points on the Putnam—"
"I might!" said Gallantina steadfastly, suddenly turning away from three walls of the room to face the remaining one and looking past Goofusia as if to speak to someone else. "With dedication and practice, and with the help of all the lifelong friends I'll make in TH 935 at 2 p.m. this Friday October fourth!"
"Spare me. What does prepping for an impossible exam even look like?"
"Well, the idea is that before the meeting, I and others will prepare at home by trying problems from the 2011 competition with however much time we choose to spare for the task, and then at the meeting, we'll compare answers and discuss the published solutions."
"If any of you losers even come up with any answers to compare."
"We might! I've already made some partial progress on the first problem."
"You don't have to tell m—" Goofusia tried to say, but Gallantina had already begun to read:
A1. Define a growing spiral in the plane to be a sequence of points with integer coordinates P0 = (0, 0), P1, ..., Pn such that n ≥ 2 and:
• the directed line segments P0–P1, P1–P2, ..., P(n−1)–Pn are in the successive coordinate directions east (for P0–P1), north, west, south, east, etc.;
• the lengths of these line segments are positive and strictly increasing.How many of the points (x, y) with integer coordinates 0 ≤ x ≤ 2011, 0 ≤ y ≤ 2011 cannot be the last point, Pn of any growing spiral?
"Two thousand and eleven?" Goofusia asked disdainfully.
"They like to work the competition year into one of the problem statements. I think it's cute," said Gallantina. "Anyway, I started thinking about the minimal growing spiral—one step east, two steps north, three steps west, &c. The x-coördinate steps are 1, -3, 5, -7 ..., the y-coördinate steps are 2, -4, 6, -8 ..., the x-coördinate net endpoints are 1, -2, 3, -4, 5 ... and the y-coördinate net endpoints are 2, -2, 4, -4, ... There are more possible spirals besides the minimal one, of course, but we can already see there are patterns in what endpoints are possible."
"You're wasting your time," said Goofusia. "Precisely because the question asks about all possible growing spirals, you're not going to learn anything by examining particular cases. You can immediately see that any point with an x-coördinate less than the y-coördinate will do: just take x steps east and y steps north."
Gallantina was beaming.
"Wh—what are you smiling at?"
Gallantina nodded, still beaming.
Goofusia scowled. "Whatever," she said, and turned to leave, then stopped. "So ... what's the answer?"
Gallantina shrugged. "We haven't finished solving it yet. But if it turns out to be beyond us, I'm sure they'll tell us in TH 935 at 2 p.m. this Friday October fourth."
Goofusia shook her head. "I couldn't possibly. I have an exam this week, and a lot of homework."
"But you don't specifically have anything else going on at 2 on Friday? They're notoriously hard problems, and everyone is busy. There'd be no shame in showing up and eating a donut without having successfully solved anything at home."
"No, I mean that's not who I am. I'm not like you. I'm a student at SF State, not—not the cosmos!"
Goofusia left. Alone, Gallantina addressed the fourth wall again. "Is that who you are?"