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."
"Why on earth would anyone want to build a machine that plays games?" Mrs. Sullivan asked.
"Programming a computer to make logical decisions is the first step in replicating human intelligence. If all goes well, Tic-tac-toe is going to help create a better future." Dr. Hopper stood up. "May I use your phone? It's a local call. Dr. von Neumann."
Carolyn's mother seemed to be in a bit of a daze. Guests usually talked about the weather, or how the Phillies were doing that season. "Of course," she said after a moment. "On the table, in the hall. I'll get dessert. Coffee?"
"Please. Two sugars. I'll only be a minute." Dr. Hopper put her napkin down beside her plate and left the room. Carolyn heard her dial, then say, "Johnny? It's Grace. Are we still on for breakfast tomorrow?"
All night, Carolyn tossed and turned, thinking about numbers and Tic-tac-toe-and Vineland. Even though she had never met her father, only seen pictures, she felt like she almost knew the boy in the notebook, who had sworn to rescue his best friend if Bobby was ever in danger. Too late for that.
But now she could rescue Bibber.
The next morning, while her mother was getting ready for a Women's Club meeting and Dr. Hopper was waiting for her taxi, she filled the pockets of her shorts with chalk and pencils and a pen. She tucked a dozen sheets of scrap paper into the composition book that held the secrets of the Lo-Shu Club. She was ready.
By 9:30, the house was empty. Carolyn left a note-Gone to the library.-and headed into the woods. When she reached the stile, she slipped over into the garden. She crept along the far edge until she could see in through the library's bay window.
Good. Bibber was there. He lay on his stomach, moving a line of toy soldiers around a fort made of blocks. She duck-walked along the base of the porch to the front door and tiptoed through to the room of cabinets and animal heads, then quietly opened the double doors to the library.
Bibber looked up, and his whole face filled with a smile. Carolyn put her finger to her lips-Shh.
Bibber nodded. "Why are we being very quiet?" he whispered.
"I don't want your housekeeper to hear us."
"Oh. She won't," Bibber said. "Mrs. Addison is in the kitchen with Cook. She leaves me alone until my lunch." He shrugged. "Unless I make a big noise."
"What time do you eat?" Carolyn asked in her normal voice.
"Lunchtime."
That wasn't much help. But Carolyn figured it'd be at least noon, and that gave her plenty of time. "Do you know how to play Tic-tac-toe?" she asked.
"Uh-huh. I'm pretty good."
"Great." She laid the composition book and two pencils on the table. When she drew the grid on a piece of scratch paper she said, under her breath, "By the sign of the magic turtle."
Carolyn knew how to use numbers for a lot of useful things, but she hadn't known they could be magic until she met Dr. Hopper. If she said that was real, Carolyn believed it. She was at the Institute, with Einstein, the smartest man in the world, so she ought to know.
"You go first," she said.
Bibber drew a tiny X in the top right corner, and she followed with an O next to it. She wanted to put her O in the center-it was the best move-but she also wanted Bibber to win the first game. When he drew his third X in a row, he smiled in triumph. "I told you. I am good at this."
"Yep. So you get to draw the next one."
He did, his tongue in the corner of his mouth, concentrating on making the four lines as straight as he could.
There. They had both drawn the sign that made them members of the Lo-Shu Club. "You are my brother," Carolyn whispered. "I will rescue you from danger." She made an O in the center square, but let Bibber win again.
"Where do you keep Lotion?" she asked.
"He crawls all over the house. The downstairs part. Right now he's under the table."
"Here?" Carolyn moved one of the chairs and squatted down. On the carpet was the biggest tortoise she had ever seen-even bigger than the one in the Bronx Zoo. At her movement, he turned his head and looked at her with a yellow, reptilian eye. "How do you get him to come out?" They couldn't lift him, not even the two of them. He was the size of an ottoman.
"I wait until he's hungry," Bibber said. "We could try and feed him now."
"What does he eat?" He looked as prehistoric as a dinosaur, with scaly front feet the size of salad plates. He might eat anything.
"Fruit and stuff. Leftover salad. Watch." Bibber pulled a cluster of blue-black grapes from his pocket and lay them in a pile a foot away from the edge of the table.
In real-life slow motion, the tortoise rose up and lumbered forward, one leg at a time, until he stood over the grapes. He extended his surprisingly long neck, lowered his head, and mashed the grapes into his mouth in three pulpy bites.
"It's bad to feed him on the rug," Bibber said, pointing to a purple stain that was almost invisible against the elaborate floral design. "Mrs. Addison says."
"I can see why." Carolyn stared at Lotion's back. She'd planned to draw the Lo-Shu numbers onto the turtle with chalk, but she didn't need to. The pattern was carved into nine plates on his shell, the center square marked with five dots arranged like the pips on Monopoly dice, faint traces of red paint in the deepest grooves.
Lotion was already magic.
Now for the tricky part. Carolyn circled the room, chalking √-1 on the four walls, the window frame, the doors, and a side or shelf of each bookcase.
"What are you doing?" Bibber asked.