So it wasn't until after lunch, when her mother went off to the cleaners and the bank and the drugstore, that Carolyn had a chance to get back to the notebook. She sat at the dining room table with a pile of scratch paper, going over her own sums for the third time, checking her work, when the doorbell rang.
The professor was early.
She put down her pencil and went to the screen door. An older lady with blond hair and glasses stood on the front porch in a plain blue dress, a cardigan sweater folded over one arm. Someone from the women's club, raising money for the March of Dimes again?
"Can I help you?" Carolyn asked.
"I'm Dr. Hopper. I believe I'm expected?"
Holy moley. Most of their guests were scientists. Only big brains got invited to the Institute's conferences. But none of them had ever been a lady before.
"Oh. Sure. Please come in," Carolyn said, in her most polite, talking-to-guests voice. "My mother will be home in a few minutes." She held the door open, saw a suitcase, and remembered to ask, "Do you need help with that?" Dr. Hopper was on the skinny side.
"No, thank you. I can manage." She picked up the small Samsonite case and walked into the front hall. "What a lovely home."
"Thanks." Carolyn thought hard. She'd watched her mother check guests in, but had never done it by herself. "Have a seat," she said, gesturing to the dining room table. "Would you like some iced tea?"
"I would. It's rather warm today."
Carolyn went into the kitchen and stood on a chair to get down one of the nice glasses. The lever on the ice-cube tray stuck, and the pieces came out broken, but she didn't think it would matter. She took the sweating glass and a napkin out to the table.
Dr. Hopper had a pencil in her hand, tapping it on the pile of scrap paper. She looked up when Carolyn came in. "I see you're working on the Lo-Shu problem."
Carolyn caught the glass before she dropped it all the way, but it splashed enough to soak the napkin. She set the tea onto the table. "How do you know about that?"
"I'm a mathematician." Dr. Hopper took a sip. "Legend says it was first discovered by a Chinese emperor who noticed the pattern on the shell of a divine turtle, and thought it was an omen."
"A turtle?" Lo-Shu. The light bulb finally went on. Lotion!
"That's the story." She smiled at Carolyn. "Mystical poppycock, of course. But it is the most common variation of an order-three magic square."
"Magic? It isn't math?"
"It's both." She laid the papers flat on the table. "Why don't you sit down, show me what you've found." She was a guest, but she sounded like a teacher.
"It's a square, three across, three up and down, with all the counting numbers, no repeats." Carolyn pointed to one of the diagrams.
4 9 2
3 5 7
8 1 6
"Do you see a pattern in the digits?" Dr. Hopper asked.
"I think so. The top row adds up to 15. So do the others."
"Very good. In this configuration, the sum of every row, column, and diagonal is the same 'magic' number-15."
"What can you do with it?" Sister Liguori was big on using math in real life.
"Not a thing. It has no practical application." Dr. Hopper laughed. "But that's a plus to many of my colleagues. Pure mathematics is about truth, unconnected to everyday life. They create their own perfect worlds, happily-" She turned. "Oh, hello."
"Hello. Welcome." Mrs. Sullivan stepped through the doorway from the kitchen and held out her hand. "I'm Eileen Sullivan. We spoke on the phone." She looked down at the stack of papers. "I see you've met my daughter, Carolyn."
"Oh yes. We've been discussing higher mathematics."
"Really?" Mrs. Sullivan raised an eyebrow. "I hope she hasn't been-"
"Not at all. I'm enjoying myself." Dr. Hopper stood up.
"Well, we try to make our guests feel at home here. Why don't I show you to your room?" Mrs. Sullivan headed toward the stairs. "You're at the back, on the right. A lovely view of the woods this time of year." She looked down at the suitcase. "May I take that for you?"
"No, thank you, I can manage."
Carolyn's mind kept wandering off to a place Sister Liguori had never mentioned-a perfect world where numbers could be magic. But her hands helped make dinner, set the table with the good china, and serve while her mother chatted with Dr. Hopper, who was one of the less boring guests they'd had. She'd been in the Navy during the war, a WAVE, but in a laboratory, not on a boat, working on something called a computer-a machine that could do arithmetic. Now she had a job with an important company, building an even bigger one.
"They're the first machines man has built to serve his brains, not his brawn," she said. "One day children will use something like UNIVAC for their homework, instead of memorizing multiplication tables."
Mrs. Sullivan frowned. "But then they won't learn anything."
"They will. They'll learn how to use numbers, see the patterns and connections. That's what mathematics is all about." She reached into the pocket of her sweater and took out a pack of Luckies. "Do you mind?"
"Not at all. Let me get you an ashtray." She went into the kitchen.
"What's your conference about?" Carolyn asked, leaning forward. She had never told her mother about the numbers making shapes, when they played the cash-register game, because it would sound kind of weird, but she thought Dr. Hopper probably understood.
"Let me see. Tomorrow morning's schedule has papers on combinatorics, twin primes, set theory, and imaginary numbers."
"What, like a make-believe one-fifty blibbity-blips?"
Her mother put the ashtray and a book of matches by Dr. Hopper's plate. "Carolyn! That's no way to-"
Their guest held up her hand. "It's a reasonable conjecture," she said, lighting her cigarette. "But no. Numbers like this." She took a mechanical pencil from the same pocket and drew a figure on the inside cover of the matchbook: √-1. "The square root of negative one."
Mrs. Sullivan stared. "I was an English major, and I only got through algebra, so forgive me, but if you multiply two negatives, isn't the answer always positive?"
"Yes. Every time."
"So the square root of negative one is impossible."
"No, only imaginary." Dr. Hopper smiled. "I know it sounds like mathematical fiction, but it's quite useful in understanding electromagnetics and quantum mechanics."
"I see," Mrs. Sullivan said.
Carolyn could tell that her mother was only being polite, but she wanted to know more about how numbers could be magic and how imaginary things could be useful. Because if the numbers in story problems were about real life, then-
"Is that your topic at the conference?" Mrs. Sullivan asked.
"No, Dr. von Neumann and I are part of a symposium on recent developments in electronic data coding. Among other things, we're going to be discussing one of your favorite games." She turned to Carolyn.
Huh? "Which one?"
"Tic-tac-toe." She tapped her ash into the small glass dish. "A bright young man at Cambridge – England -has programmed a computer called EDSAC to play. The Xs and Os are on a cathode ray display-like the picture tube in your TV."
"It's a pretty easy game," Carolyn said. Why would the brains talk about that?
"Exactly. It's finite, with perfect information."
"What?"
"Sorry. Mathematically, that means there's no luck involved. You can know every possible move, and there are only a limited number."
Carolyn nodded. "Yeah. Nine."
"Not even close. Try 362,880."
"Uh-uh!"
Dr. Hopper smiled again and drew a small tic-tac-toe board on the matchbook. "First move you have nine choices where to put an X, right?"
"Sure. That's what I said."
"Ah, but then the next player has eight choices of where to put an O. Seven choices for the second X, and so on until someone wins. Or ties." She stubbed out her cigarette. "That big number is nine times eight times seven times six-" She waved her hand in the air. "Etcetera, etcetera."