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“Come on,” she said.

Joel grabbed a volume that looked promising. Man and Rithmatics: Origins of Power. He handed her the book and they left. Ms. Torrent gave them another dissatisfied glance, but reluctantly checked the book out to Melody. Joel let out a deep breath as they walked out onto the green.

Melody handed him the book, and he tucked it under his arm. At the moment, however, it seemed far less important than the little slip of paper. Joel had proof that Nalizar was interested in new Rithmatic lines.

Of course, Fitch was convinced that the looping swirl was not Rithmatic. This was really just another suspicious connection—it wouldn’t prove that Nalizar was involved. I need to get that book, Joel thought. If it contains anything like this looping pattern, I’ll have evidence.

That sounded extremely dangerous. Perhaps it would be best for Joel to simply go to Harding and express his concerns. Undecided, he folded the paper and stuffed it in his pocket. Melody walked beside him in her white skirt, binder held against her chest. She had a distant expression on her face.

“Thank you again,” he said. “Really. I think this is going to be a big help.”

“Good to be useful for something, I suppose.”

“Look, about what I said the other day. I didn’t mean it.”

“Yes you did,” she said, voice uncharacteristically soft. “You were only being honest. I know I’m no good at Rithmatics. My reaction only makes me doubly a fool for trying to deny the truth, right?”

“You’re not being fair to yourself, Melody. You’re really good with chalklings.”

“For all the good it does me.”

“It’s a great skill,” Joel said. “You’re way better at that than I am.”

She rolled her eyes. “Okay, you’re laying it on thick. There’s no need to be so melodramatic—I know you’re just trying to make yourself feel better. I forgive you, all right?”

Joel blushed. “You’re an annoying person. You know that?”

“Okay,” she said, holding up a finger. “Now, see, you’ve gone too far the other direction. If you try really hard, you should be able to find a happy medium between patronizing me and insulting me.”

“Sorry,” Joel said.

“Regardless,” she said, “the fact of the matter is that no matter how good I am with Lines of Making, I still can’t build myself a decent defense. One good shot with a Line of Vigor will take me out of a duel.”

“Not necessarily,” Joel said. “You know, for all Professor Fitch’s talk about keening, maybe that strategy isn’t right for you.”

“What do you mean?” she asked, eyeing him suspiciously, apparently expecting another insult.

“Have you ever tried the Jordan Defense?”

“Never heard of it.”

“It’s advanced,” Joel said. “One of the most advanced I’ve ever read about. But it could work. You have to draw a Forbiddance net, then…” He hesitated. “Here, I’ll just show you. You have chalk?”

She rolled her eyes. “Of course I have chalk. During your first year at Rithmatic school, if any professor catches you without chalk, they’re allowed to make you scrub floors for two hours.”

“Really?”

She nodded, handing him a piece. The quad was nearby, and it didn’t look like anyone was using it. Joel rushed up the hill, Melody following. “Hey,” she said. “Aren’t we going to get into trouble for being late to Fitch’s office?”

“I doubt it,” Joel said, reaching the concrete-covered top of the hill. “Fitch was up late last night, and he got interrupted a couple of times this morning. I’ll bet he’s still dozing. Okay, here, watch this.”

Joel set his book aside and knelt, doing a rough sketch of the Jordan Defense. It was an ellipsoid defense, with a line at each bind point to stabilize it. The main feature of the defense wasn’t the primary ellipse, however, but the large cage made from Lines of Forbiddance around the outside. It reminded Joel a little bit of what Herman Libel had tried.

“That boxes you in,” Melody said, squatting down beside him. “You can’t do anything if you surround yourself with Lines of Forbiddance. That’s basic Rithmatics—even I know that.”

“That’s a basic rule of thumb, true,” Joel said, still drawing. “A lot of advanced Rithmatic designs break with early wisdom. The really good duelists, they know when to take a risk. Look here.” He pointed with his chalk to a section of the design. “I’ve made a large box on either side. The theory with the Jordan is to fill these boxes full of offensive chalklings. If you’re good with chalklings, you should be able to instruct them to wait and not attack your own line from behind.

“So, while your opponent is wasting time blasting away at your front, you are building a single overpowering attack. When you’re ready, you let out the burst of chalklings, then quickly redraw that Line of Forbiddance. You use Lines of Vigor to destroy any enemy chalklings that got inside while your defenses were down, then you build another wave of chalklings.

“While you might be slower than your opponent, it doesn’t matter because your attacks come in huge rushes that leave him confused and unable to respond. Matthew Jordan, the one who designed the defense, won a couple of very high-profile duels with this and caused an uproar among academics because of how unconventional it was.”

Melody cocked her head. “Dramatic,” she said.

“Want to give it a try?” Joel said. “You can use my little sketch as a pattern.”

“Probably shouldn’t,” she said. “I mean, Professor Fitch…”

“Come on,” Joel said. “Just once. Look, I got you into the library so you could ogle Nalizar, didn’t I?”

“And get yelled at by him.”

“That was your idea,” Joel said. “Are you going to draw or not?”

Melody set down her notebook and knelt on the concrete. She took out her chalk, eyed Joel’s miniature drawing, then began to draw an ellipse around herself.

Joel began to draw as well. “I’m going with the Ballintain,” he said, drawing a circle all the way around himself. “But with your Jordan Defense, you don’t need to pay much attention to what I’m doing. Just draw as fast as you can.”

She got into it, doing a defensive rectangle around the Circle of Warding, then quickly beginning her chalklings.

Joel drew, hoping his instincts were right. The big weakness in the Jordan Defense was the chalklings. Controlling them in this way was difficult; it was only possible because it was a formal duel, and she could orient them right at her target.

For some reason, chalklings were difficult to control if you wanted them to just wait around. That was why most Rithmatists either sent them out to attack or stuck them on a bind point.

I really need to study more chalkling theory, Joel thought as he finished his defense. Maybe I can get Melody to check out a few books on it.

“Okay,” he said, reaching out to draw a few Lines of Vigor. “This is going to take some imagination, since I can’t make my lines do anything. Pretend that I’m good at drawing Vigor Lines—which I am, by the way—and that each of these is hitting your defense at the same point, weakening it. A well-drawn Line of Warding can take about six hits from a Vigor; a Line of Forbiddance can take ten. When you see where I’m shooting, draw another Forbiddance line behind your first to slow me down.”

She did so, drawing a line.

“Now I have to get through two lines of Forbiddance and one Line of Warding. That means that with this defense, you have about twenty-six Vigors to get your chalklings done. That’s not much time, with how—”

He fell silent as she whipped her hand forward and laid her chalk against the inside of her Line of Forbiddance to release her chalklings.