Alcuin

Maya knew this for certain because she had finally asked her, on a rainy afternoon when the Library was so quiet she could hear the radiator clicking. "How do you do it? How do you always pick the right one?"

Alcuin had been arranging a small drawer of pamphlets that no one ever asked for. She did not stop arranging them. She said, "The book I picked for you today depends on three things. What you just solved. What you almost solved. And one thing that you don't know is connected to either of them."

"And you can remember all of that?"

"I can remember pieces. The rest I keep written down. But the written-down parts are useless without the connecting parts, and the connecting parts only live in my head." She closed the drawer. "And here's the secret: I haven't finished drawing the map. I'll never finish drawing the map. Every time a reader comes in with a problem I've never thought of, the map grows."

What Maya didn't know — what Alcuin had never told anyone, in fact — was that the map had not been Alcuin's idea.

Alcuin had been thirteen herself, once, working in a different library three thousand miles away. The librarian there had been a man named Mr. Sandford who wore brown vests and spoke softly and would lend a book to anyone who asked. He had taken Alcuin under his wing the summer she started spending all her afternoons there to escape the chaos at home. She remembered him sitting her down once and saying, "There's a thing I want to teach you. It's hard to explain. I'll show you instead."

He had drawn three problems on a sheet of paper. Then he had drawn arrows between them. Then more arrows. Then more problems. By the end of the afternoon the page had been a tangle of arrows and numbers and small notes in his fast, looping handwriting.

"This is one corner of the map," he had said. "I've been keeping this map since I was a boy. The librarian who taught me showed me the same trick. He learned it from his teacher. The map has always existed. It just changes shape because the problems keep changing."

Alcuin had stared at the page, and something in her had gone very still, and she had said: "I want one."

Mr. Sandford had handed her the page.

For thirty years now, Alcuin had been growing her own version of that page in her head. She had added problems Mr. Sandford had never seen. She had drawn connections he would have argued with. She had erased connections — most of which he had drawn — that turned out to be wrong. And she had passed the trick on, in fragments, to a small number of readers who showed signs of wanting one of their own.

Maya was one of those readers, although Alcuin had not told her yet.

The afternoon Maya finally asked the right question was a Thursday in early spring. She had been working on a problem about dividing fractions for nearly an hour and had gone past frustration into a kind of focused quiet. Alcuin recognized this quiet. It was the same quiet she had been in, thirty years ago, when Mr. Sandford had handed her the page.

Maya looked up. "Alcuin. Can you teach me how to make the map?"

Alcuin set down the pamphlets. She looked at Maya for a long moment. Then she pulled out a chair across from her, sat down, and slid a blank sheet of paper between them.

"It's hard to explain," she said. "I'll show you instead."

She drew three problems. Then she drew arrows.

By the end of the afternoon the page was a tangle of arrows and numbers and small notes in Alcuin's fast, deliberate handwriting. Maya took the page home with her. She did not show it to anyone. She put it in the back of her math notebook and looked at it most nights before she fell asleep.