Rose woke up with a start. One second, nothing, and then the next she was sitting bolt upright as a sliver of window-light slashed across her face.

She kicked off the blankets that had scrunched up near her feet. She’d slept in her cloak, after doing her best to clean it, and the bedding had been too warm on top. Maybe it was unsophisticated. But she’d slept with it every night since she could remember, and wasn’t about to stop now.

She swung her feet over the edge of bed, floorboards creaking as she stood and stretched her fingertips as far as they could go. Her brain was positively buzzing, the fogginess of yesterday a distant dream.

She’d need to look through everything she’d written, she mused as she wandered over toward the window. The curtains were closed, only a tiny strip of light sneaking through the gap between them. Rose threw them wide and opened the glass, breathing in the mountain air.

She had to blink the light from her eyes for a few moments. When she could see, she exhaled slowly, feeling through her cloak for the raven’s feather she’d taken in the catacombs.

The town was beautiful. Two nights ago she’d been too awake, too focused, to really take it in. Yesterday she’d been too calm, too slow, to really feel the press of wonder on her mind.

But the town in front of her was magical. Wood and stone buildings, two or three or even four stories tall, spread almost as far as she could see through the mist of early morning. They were somehow both banal and magical: simple arrangements of common materials, but put together in ways that were strange to her, and ornamented with oddities she couldn’t recognize.

Closer to the tower, the magic grew more and more blatant. Torches and lamps were replaced with glowing orbs. Chimneys disappeared somehow. Buildings began to stretch to truly implausibly heights. Finally, there was en enormous open area. The ground was dark gray stone, perfectly flat, dotted with plazas and seats and amphitheaters and pagodas like a raucous fair carved from the mountaintop. The Agora.

And in the center of the Agora, a dark black tower, stretching so far into the sky that it disappeared into the clouds. Dexter had said that magic wasn’t real, down in the catacombs. That it was tricks all the way down. But nobody could possibly deny that unnatural forces raised that tower.

She brushed her thumb across her feather’s vane. It was silky.

There was a noise behind her, a groan and a thump. She turned to see Dexter sitting up, holding his head in both hands. Dark greasy hair spilled out between his fingers as he massaged his temples.

In the end, Robin had flatly refused to stay in a room with him. Rose had volunteered, over the others’ objections. She didn’t really understand their hesitance. Robin was so small, he was barely bigger than her, and she’d been raised outdoors. The smart money was on her in any sort of scramble.

Besides, Dexter wouldn’t hurt her.

She felt a nagging sense of doubt, and suddenly the feather felt too light in her hand. She tucked it away, darting over to the hallway door and turning the handle.

She opened her mouth to say that she’d go get some breakfast, give him time to change before he joined her. She could offer to bring him some tea. Offer him some words of encouragement. But maybe that was too familiar?

Dexter growled, digging his fingers into his face. Rose slipped out the door, closing it quietly without saying a word. Things had seemed so much easier, yesterday.

Alexandria was already downstairs when Rose arrived. So was Borodin, oddly enough, upright and wide awake.

He waved animatedly as she approached. “Morning!” he said as she sat down. “How’d you sleep, Rose?” He tilted his head back, taking a long drink, and when he put the cup down Rose saw a trickle of dark liquid leaking from the side of his mouth.

“Fine.” she replied cautiously. Her eyes swept down and across the table, noting two other empty cups next to Borodin, and one half-empty in front of Alexandria, next to the sheet of paper she was reading very intensely.

“Capital! Simply capital. Are you ready to meet the day, then? Ready to set your feet and see the world’s problems smash themselves to bits against you?”

“What are you drinking?” Rose asked.

“Coffee, my good lady, delicious black perfect wonderful coffee. Did you know it’s free, here? Oh! Where are my manners. What a fool. Jenny!”

Borodin bellowed the name loudly enough to make Rose wince.

The serving girl from yesterday popped her head out of the back. It was early enough that the place was empty, aside from the three of them, and the older woman had yet to make an appearance behind the bar. Rose wasn’t even sure the front door was unlocked.

“What are you yelling about?” the server asked in a cross tone that didn’t seem to match her smile.

“Jenny, have you met Rose? Rose, Jenny. Jenny, Rose. She’s our secret weapon. Some mad wizard packed the brains of two masters and the mettle of ten soldiers into a tiny farmgirl from the sticks. Could you see to getting her some coffee?”

Rose wasn’t sure if that was an insult or a compliment, but she felt herself blushing. Oddly enough, Jenny seemed to be blushing a little too. “Er,” Rose said, hoping to ask for tea and maybe a piece of toast instead, but her throat wasn’t quite functioning properly, and the server had disappeared into the back room before she managed to say anything.

“Tone it down a little, Borodin,” Alexandria said primly from beside him. She lifted her own cup and took a measured sip to hide her smile.

“I most certainly will not.”

“What are you reading there?” Rose asked, turning to Alexandria.

“Our itinerary.”

Rose scooter her chair over, and Alexandria angled the paper so she could see. It was a long list of people and places, scrawled in a neat hand.

Right at the top, the very first line read: “Class in Agora, Master Breton, all. Orientation after.”

Rose’s leg began to jiggle. Her stomach felt a little fluttery, for some reason. She felt through her cloak for the familiar shape of her riverstone.

“I’m afraid you’re missing an item, dear sister.” Rose looked up to see Borodin reading the page upside down. Alexandria rolled her eyes. “Am I, Borodin?”

“Yes, you are. Right at the very top, you should budget an hour for waking everyone up and herding us out of–”

Alexandria tried to elbow Borodin in the ribs, but he brought his own arm down too quickly. She huffed, leaning back with her arms crossed. Borodin smirked.

Rose heard the door to the back of the inn creak, and her eyes snapped up to it, too quickly. She was a little on edge, her leg still jiggling. She saw the server come through with food and drink, and turned her attention back to the paper, still staring at that line. Class in the Agora.

There were other things to do, a whole list, but that seemed like the only thing that could possibly matter. It would be the first time she’d had a teacher since her mother. That had been years and year and years ago. What would it be like?

Rose leaned back politely as the server arrived and slid food in front of her. A fresh cup of coffee appeared in front of Borodin, as well. As the girl turned to go, Rose realized she should’ve thanked her. “Thank you!” she called to the girl’s back, her voice a little squeaky.

“Three more meals please,” Alexandria said loudly to the girl’s back, pulling the agenda back in front of her as Rose picked up her cutlery. “Our companions will want to get an early start.”

The server kept walking toward the kitchen, back turned. “Of course hon,” she said, and Rose could practically hear the girl’s eyes rolling from her voice. “Three delicious hot meals coming right up. Do you want me to leave them on the table, or should I put them outside so they can get cold even faster?”

She disappeared into the back before Alexandria could respond. The blonde girl sighed, closing her eyes briefly and holding a hand to her temple. “I don’t know what her problem is with me,” she said quietly.

“It’s a status perception mismatch, sister of mine,” Borodin replied not-at-all-quietly. “You see her as a serving girl, someone there to make life easier for people who matter, like you. She sees you as a dirty ignoramus from the blighted lands outside her magic mountain, here to fumble in ignorance for four years before getting a real job. The tower told someone to tell her mother to tell her to serve you food and drink, but she doesn’t like it, and you acting like you own the place doesn’t help.”

He reached for his new cup of coffee, but Alexandria grabbed it before he did, sliding it in front of herself instead and giving him a warning look. “Three cups is plenty, Borodin. And she doesn’t seem to have any problem with you.”

“Well,” he said, grinning and raising a hand to stroke his bushy beard. “The difference is that I am a mysterious wild man with rogueish good looks–”

“Spare me.”

“Rogueish good looks and an abundance of elan vital. A new and dangerous figure from the savage lands outside her magic mountain. She’s of marriageable age, and what maiden wouldn’t be–”

“Stop. Please, stop, it’s too early for this.”

Alexandria made a warding motion with her hands, and Borodin broke off into a grin.

They were back to their old routine, bantering so loudly and quickly Rose couldn’t seem to squeeze a word in. She rubbed her stone, back and forth. That was fine. It was who they were.

She finished her meal, taking a long drink of the strange boiled water to wash it down, and stood up. She scooped up the extra cup of coffee. “I’ll go try to get Dexter down here,” she said, eyes flickering back to the schedule. They absolutely couldn’t be late.

“Thank you,” Alexandria said, pushing her own chair back. “I’ll go grab the other two. They may even be up already.”

The Agora was beautiful. It was enormous, like a dozen forest clearings stitched together in a giant circle. The ground was flat, polished stone that extended so far it blurred into the early morning mist. The effect was eerie.

In the middle, stretching up and up and up until it hurt Rose’s neck to look, was the great black tower. It took every ounce of willpower she had, clutching her iron bloom like a lifeline, not to dash over and run her hands across it. It was probably rude, or dangerous, and she’d be separated from her friends in this strange place, and class was starting in just a second–

She realized she’d strayed a little toward the tower, and squeezed her eyes hard, forcing her feet to guide her back into the midst of her friends.

Getting out of the inn had been surprisingly easy. Dexter had perked up considerably after a cup of coffee and a few barbed words. Robin was in the middle of a book he didn’t want to stop reading, but Alexandria had convinced him to bring it along rather than stay at the inn.

There was a stony silence between Borodin and Dexter, but whatever was simmering between the two of them, they seemed content to let it lie for now. They walked with Esper as a silent, oblivious buffer between them, and that seemed to be working.

Alexandria led the three of them deep into the agora, to a small amphitheater a mere hundred yards from the tower. Rose knew the word, amphitheater, but she’d never seen one. It was on the smaller side, stacked stone benches for a few dozen people cut into the stone like a tiny valley. There was a set of stairs leading down, and a tall podium carved from living rock at the very center.

Other students were already there, scattered around the benches in groups ranging from one to half a dozen. Rose felt her heart began to speed up as they got close enough to make out faces. There were so many, all of them strange, and all of them must know so much more than her about so many things. Some of them looked almost a decade older than her, although she wasn’t sure whether that made them more or less intimidating.

She switched from her iron to her stone, trying to quiet the tumult that had appeared in her stomach. None of the others seemed bothered. No, that wasn’t true. Alexandria’s neck was tense. Borodin was swinging his arms a little too vigorously. Robin was studiously looking anywhere but in front of them. Only Dexter and Esper seemed calm, and somehow knowing that she wasn’t alone helped Rose quiet her mind.

Really, there was no time to be intimidated. She was here. Studying at the foot of the tower. She wished she could hold her stone to calm her mind, her iron to keep her focus, her new feather to feel the moment flow through her, all at once, and in a moment she’d probably need her wood. There was just too much to process.

They got settled on the benches, in the back, and Rose pulled out her notebook.

Merzhin was late. He at least had the good grace to appear rushed, hurrying down the steps to the lectern. He began unloading things from his pockets as soon as he reached it.

He looked much like Rose remembered him, if a bit more harried, with his robes in disarray. “Hullo, hullo,” he said absently, projecting his voice across the amphitheater with the practiced ease of a professional speaker. “Sorry for the wait, everyone. And good to see the new faces.”

He gave a little nod toward Rose and her group. Heads swiveled, the other students either finally aware of their group, or happy to have an excuse to gawk. Rose slid down a little in her seat, gripping her riverstone, as Alexandria waved merrily to the staring crowd.

“But since I’m late, we had better begin.” Merzhin fished around in his robes, eventually producing a rectangular bar of…shiny metal?

“Today we begin our unit on characteristic values. I like to start this topic off with a little challenge. Anyone who has attended this unit before, please keep your hands down for this first lesson.”

As Merzhin spoke, he casually tore the metal, which didn’t make even a lick of sense. The shiny silverish material must be made of something else. Rose craned her head, but couldn’t get a better view from their seat in the back.

Her curiosity warred briefly against her shyness, and it was a rout. She stood up and began to pick her way down the seats of the amphitheater to the bottom, drawing the occasional curious or annoyed look from the other students. Alexandria whispered something she didn’t catch.

Merzhin peeled away the shiny material, revealing a dark brown substance. As Rose got closer, she saw that whatever the wrapping was made of, it couldn’t be much thicker than a sheet of paper.

“This, for those of you who haven’t seen one, is a bar of chocolate – a popular treat in the tower. It’s divided into squares, as you can see, so as to make it easy to share.” Merzhin broke the bar in two, then broke the smaller piece again, repeating until he had a single square of chocolate, which he popped into his mouth. Rose was close enough now that she could see there were vertical and horizontal lines running across the bar, material removed to perhaps half the total depth, providing clear lines along which to break it.

“Now, this particular bar of chocolate is divided into squares, in a 5x7 grid. It supports one operation, for our purposes today: any bar of chocolate may be broken along any of the marked lines to produce two smaller bars of chocolate.” Merzhin broke the largest piece of chocolate again, emphasizing the motion.

“It seems, at first glance, that this choice of where to break it should matter in some way. I can break my 5x7 bar into a 4x7 and a 1x7, or into a 5x3 and a 5x4. But there is an interesting property, which you can confirm for yourselves: no matter what pattern of breaks I choose, it always seems to take the same number of breaks to reduce the bar to squares. 34, in the case of this particular bar.”

That did seem rather curious, but Rose was still entranced by the shiny wrapping the bar of chocolate had come in. Merzhin had removed it fully at this point, crushing it into a small ball with his hand. Unlike paper, it stayed perfectly in its new shape rather than expanding a little when he let it go.

“Whether living life or playing games, you will encounter situations which at first appear to have near-infinite complexity, but upon further examination have some simple property which will allow you to reason about them easily. We call this a characteristic value. You will need different characteristic values for different different purposes, of course. But there is some property of both the (4x7, 1x7) set of bars, and of the (5x3, 5x4) set of bars, such that both of these sets require exactly 33 more breaks to reduce to squares.”

Merzhin had an odd way of speaking. Casual, but very formal. No, formal wasn’t quite the right word. Precise. He used words to mean exactly what they meant, even if it sounded awkward at first blush.

Rose realized she’d made it to the bottom row of seats, and was leaning forward as far as propriety would allow. She was probably staring. Merzhin didn’t seem put off, though.

“Often when you find the characteristic value for a problem, the entire affair seems trivial. So, here is my challenge to you: can any of you convince me, here, that it will always take me exactly 34 breaks to reduce this chocolate bar to squares? Can you explain why it must be so, so clearly and succinctly that I cannot possibly argue? If so, the bar of chocolate is yours! Minus a small fee, of course.” Merzhin popped another square in his mouth, smiling, and a smattering of polite laughter rose up from the crowd.

Rose’s heart began to race. If she figured it out first, Merzhin would give her the strange silver substance. Well, he hadn’t technically said that, but it seemed likely. She reached for her driftwood, fumbling it the first time in her excitement, and pinched it lightly as she let her mind loose.

Area seemed like the obvious value that was preserved. Breaking a 5x7 bar into any two halves preserved the area. But that wasn’t actually helpful, because–

“Yes, you,” Merzhin said. “With the dark hair. Mr. Brezich?”

“Yes,” said a voice in a strange accent. High-pitched and highly enunciated, from somewhere Rose didn’t recognize. “Jonathan Brezich. The answer is simple, Master Breton. Every time you break the bar, you increase the number of pieces by one. When you have 35 pieces, you’re done. So it must always take exactly 34 steps.”

Rose’s brain skipped, hard, like a speeding wheel hitting a rock. What? How had he gotten that so quickly? It did seem obvious, when he said it that way, but she’d barely even–

“Very good,” Merzhin said. “Come on down and take your prize.”

There were a few halfhearted applause, mostly from Jonathan’s friends. Rose’s shoulders drooped as the boy made his way down and stuffed the remaining pieces into his pockets. She’d had less than a minute to figure that out. He didn’t look that much older than her. Maybe someone had told him the answer?

A tiny flicker of hope flared up in her when he returned to his seat, leaving the crumpled silver ball on the lectern. She stared at it hungrily, even as Merzhin began to speak again, giving several increasingly complicated examples of characteristic values.

Rose remembered she was supposed to be taking notes. She hurriedly filled in her notebook, writing down the examples she’d already heard and filling in new ones as Merzhin spoke. He went on for what must have been half an hour, until finally one a red-winged raven swooped down from the sky to land on his shoulder, squawking in his ear.

He broke off mid-sentence to stroke the bird’s head. “Is it that late already, Archimedes?” he asked. “I suppose I must be going then. I have a little puzzle for you, everyone! A problem to figure out yourselves before our lesson next week.” There was a chorus of pained groans from the crowd, which didn’t make even a tiny bit of sense to Rose.

“Let us say you have an 8x8 grid, all white at the start. At the beginning, you may color any number of squares black. Then, every white square bordered by at least two black squares turns black, repeatedly, until no more can change. Clearly you can color the entire board black by filling in the diagonal. But can you color the entire board black by filling in only 7 squares? And if not, how would you convince me of that?”

Merzhin clapped his hands together. “That’s all! Thank you for coming, and I hope to see you at our next lesson. Could our new students please stay for a moment? We have a few administrative details to square away.”