Rose was alone in her room with a notebook, and everything was right with the world. Fresh air, light, a good night’s sleep, plenty of pages and charcoal. She’d need to leave at some point, to visit the other masters with Alexandria, but that was in the future. She had experience squeezing her thinking in between chores.
Her eyes flickered over to the other bed, rumpled and abandoned. Dexter hadn’t made it back to the inn yet. But she pushed that from her mind, returning her gaze to her notebook where it lay on the table in front of her. Everything was perfect.
The problem was simple. An 8x8 grid of squares, with some number of them colored black to start. After that, any white square bordered by two or more black squares would turn black, over and over until there were no more such squares.
Could she color the entire board with 7 squares? Just thinking about it on the walk home, it sure didn’t seem like it. But could she convince Merzhin of that? Or, well, he already knew the answer. But should she convince him that she’d made a convincing argument for that?
Her first instinct was to break it down into subcases. Prove it for a small grid, then prove that any solution to the larger grid was composed of solutions to smaller grids. But she resisted the urge. She had a hint, which would normally be cheating, but in this case the whole point of the puzzle was to practice a specific technique.
Characteristic values, Merzhin had called them. Some summary of the state of a complex system which let you reason about it easily, even when it underwent a transformation.
She sketched a grid in her notebook, absently, as she considered possible values. In the chocolate example Merzhin had used in class, the characteristic value was just the number of pieces. Would that work? Could she just count the number of black squares?
Alexandria pointed across the agora, and Rose squinted her eyes as they walked. “That’s her,” Alexandria said. “Master Ozma.”
“What’s her first name?”
“Mary. But just call her Master Ozma. Or Master of Truth.”
Rose rolled her eyes.
Mary was wrapping up a small class, barely half a dozen students sitting under a gondola as she spoke. She had a strange construction behind her, an enormous but painfully thin slab of black stone suspended in a wooden frame. As they drew closer, Rose could see that she was drawing white lines on it with a long thin rock held in one hand. Her other hand held a steaming mug with a lid. To keep the heat in? That made a lot of sense; why had she never seen that before?
The woman herself was tall, with gray hair and green irises. Her mouth was covered by an enormous green scarf. Crinkled eyes and dimpled cheeks told Rose that she was smiling underneath it. She was light on her feet, for her age, moving easily along the length of the black slab as she drew on it.
Rose and Alexandria waited a polite distance from the Gondola. After a few minutes, a ringing sound eruped from Mary’s robes. It sounded like a trilling bird made of metal. She reached her hand into a pocket, and the trilling ceased. After that she seemed to wrap up the lesson, students packing their things away as she got in a few final words.
Her green eyes flickered up to Rose and Alexandria as her students began to make their exit, and she waved them over.
“Master Ozma?” Alexandria asked as they approached, sticking out her hand.
“Hm.” the woman replied. She pulled down her scarf and took a sip from the mug she was carrying. Rose winced – the liquid inside looked like it was almost boiling, from the amount of steam leaking out – but the woman didn’t seem to mind.
“Your eyes,” she said.
“What?” Alexandria asked, putting on a polite smile.
The woman walked toward them, looking past Alexandria at Rose. Rose swallowed, and resisted the urge to take a step back.
“Your eyes,” Mary said again, drawing close enough that Rose had to look up a little bit to maintain eye contact. Rose searched the woman’s face, looking for meaning. “You have intelligent eyes, my dear.”
“Um.” Rose felt herself blush a little. “Thank you?”
After a moment’s thought, her curiosity got the better of her. “What does that mean? What makes eyes look intelligent?”
The woman smiled again, the skin by her eyes wrinkling. “Do you have time for a story?”
“Not really,” Alexandria said.
“Absolutely.” Rose replied at the same time.
Mary took a sip of steaming liquid. “I lived many lives, before I came to the tower. I spent some time as a sommelier in Two Forks. Do you know what a sommelier is?”
Rose shook her head.
“We tasted wines. Every year, vineyards would ship us bottles, and we would taste them so as to inform our little corner of the world which were the finest. Wine is complicated, but we learned to taste all the subtleties, all the individual components. If you gave the same vintage to two of us, we’d rate it near identically.”
Mary took another sip from her mug, maintaining eye contact longer than seemed comfortable. Rose’s eyes fell to her feet, then darted back to the slab of rock with the white marks, then past Mary to Alexandria’s. The other girl was tapping her foot impatiently.
“But I was always a troublemaker,” Mary continued. “I took a little bit of the money I’d earned, and put in a special order. I asked a river mrechant to bring me back the strangest wines he could find. Wines from as far as he could manage.”
Alexandria looked like she was in pain, trying to figure out how to interrupt without interrupting. They were on a bit of a schedule.
“When he returned with the next year’s barges, I took the box of eclectic bottles he’d brought, and I gave them to my colleagues to taste. Can you guess what happened?”
Rose had a few ideas, but in deference to Alexandria’s impatience she shook her head.
“Chaos. Ratings were all over. Sommoliers arguing in the cellars, grown men yelling at each other about sweetness and tanins and whether or not there were hints of chocolate in a baked red from god knows where. Do you see what I’m saying?”
Rose shook her head again, and the woman gave her a knowing wink. “Since you’re in such a hurry, I’ll skip ahead. We’d thought we were training to recognize good wine. In fact we were training to recognize similar wines. We knew which vineyards in the area were considered the best, which were old and famous, which cut costs wherever they could. We learned to recognize the wines from these fields by taste, and to give them the scores we’d been taught to. But what we’d learned was narrow and fragile. It was, in the end, mostly memorization.”
Rose was a little lost at this point. What did this have to do with – oh.
“Ah, comprehension! Good, good. You see, when you ask me what makes your eyes look intelligent, I could talk about the details. They’re wide, glistening, with large pupils. They dart around, as though drinking in the world around you, never staying too long in one spot. They look bright, which I’ve never quite been able to define, but isn’t it strange how everyone agrees waht it means? In any case, your eyes look intelligent because I’ve seen those eyes before, and I remember.”
The woman gave her a wink. Rose got the distinct feeling she was missing something.
“Wow, that’s an amazing story, Master Ozma.” Alexandria sidled up next to Rose, and the old woman’s smile deepend.
“I’ve seen you before too, dear. Let me guess, the two of you want to know what you need to do, for me to bless your ascension to the tower?”
Rose and Alexandria nodded.
“I’m afraid I can’t help you. My job is to watch, to observe, and to judge when you are ready. If you knew what I was looking for, it would make my job harder. I’m terribly sorry.”
The woman looked significantly more amused than sorry.
“Perhaps you could give us some advice on our studies?” Alexandria said. “You’ve seen many students in our position, over the years.”
“Or it could be like with the wine,” Rose cut in, eyes returning to the woman’s face. “You didn’t tell them what you were looking for, but you did arrange it so you could learn what you needed. What can we do to help you see if we’re ready?”
Mary took a long sip of her mug, considering. “So in sync, and so at odds,” she said. “I like you two.”
The old woman stood, looking lost in thought. Her eyes drifting to up and away as she stared into the middle distance.
After a few moments, she nodded to herself. She pointed at Rose. “I’d like to see you make a new friend,” she said. Her finger swung over to Alexandria. “I’d like to see you in one of my classes.”
There was an awkward silence. “Um,” Alexandria said eventually, “I realize we’ve already taken too much of your time, but there are others, in our party–”
“I know,” Mary said, nodding. “I know more than you might suspect, Alexandria Mikovich Kastchoff. And I have eyes everywhere. Since you asked so nicely: I’d like to see Dexter finish his little project. The others are of no interest to me at the moment.”
Rose felt the small hairs on the back of her neck rising. Mary took one last sip from her mug, pulling her scarf back up, still wearing that friendly smile that reached all the way to her eyes.
Rose leaned back in her chair, tucking a hand up and under her chin. Counting the number of black squares on the grid hadn’t gone anywhere. The trick had worked with the chocolate because the number had always increased by exactly one, and the goal was to count the number of steps before the end.
Here the goal was something different: to make sure you couldn’t end up in a particular place. She had to say that all arrangements of seven squares had some value, which despite transformations never reached some other value. The value of all the seven-square configurations would have to be different than the value of the fully-colored board, and each step had to either not change the value, or never move it to the value of the fully-colored board.
That was the problem with counting squares. The value for the initial configuration was always 7, and the value for the full board was always 64, but it increased by a difficult to characterize number every step, and she couldn’t see how to bound it.
Really she wanted a value which either never changed, or moved away from the value of the fully-colored board. That would be much easier than bounding how much the value changed.
She nodded to herself. She had a handle on the space she was trying to explore. She reached into her cloak, finding her driftwood and letting it sit lightly on her palm.
They found the Master of Secrets at the rookery, tending to the birds. He was dressed head to foot in gray robes. His hood hung so far over his face it must have obscured his vision. A long gray beard poked out the bottom of the hood, but his steps were sure and steady.
In a way, he was the only person they’d met so far who looked like an agent of the tower. He reminded Rose a little of the old man who’d brought the others from the barge, what seemed like a lifetime ago. Except the Master of Secrets wore his age and his mystery with a quiet dignity. The river guide had been trying too hard, in retrospect.
Alexandria coughed politely into her hand. The old man looked up from his work, his expression unknowable. “Yes?” he asked, his voice quiet.
“Hello,” Alexandria said, somewhat quieter than normal. Now that Rose was thinking about it, the birds in the rookery were all deathly silent. That wasn’t normal for birds, was it? “My name is Alexandria Kastchoff. My friends and I are seeking–”
The Master of Secrets waved a hand in what could plausibly have been construed as a shooing motion, if he’d tried a little harder. “I don’t approve entries to the tower,” he said, turning back to the birds. His voice was unnatural. It was too low and scratchy, like he was pitching it oddly on purpose.
“We spoke with Master Breton,” Alexandria continued gamely. “And he–”
“I have not recommended a group for entry in over a decade. Speak with the others.”
The Master of Secrets bent down and opened a cage full of oddly docile birds. He whistled a complicated tune, and one of the birds hopped forward. He began inspecting it, turning its head this way and that, checking under its wings.
“Do you teach any classes?” Rose tried. “I didn’t see any in the packet.”
“I do not. Anything that could be taught in good conscience isn’t truly a secret, is it?”
“What do you do, then?” she asked. “Just feed the birds and decline to admit people to the tower?”
Alexandria tried to elbow her in the ribs, but Rose barely managed to dance to the side. Was that the first time she’d tried that? Rose got an odd warm feeling in her belly. They must be friends, if Alexandria was trying to jab her in the stomach.
Alexandria’s face was anything but friendly, though, when Rose looked up. “Be polite,” the other girl mouthed.
The Master of Secrets sighed. He patted the bird on the head, then locked the cage and stood slowly. “My duties are none of your business, I’m afraid.”
“What happened ten years ago?” Rose asked, eyes darting to Alexandria to make sure the other girl didn’t get close enough to jab her. “When you admitted some group? And what’s the deal with these birds?”
“Ozma’s going to love this one,” the man muttered under his breath. He turned to look at them, except he must really have been looking at their feet, seeing as how the hood of his robes kept his eyes fully hidden.
“Children. I understand your enthusiasum, but you must in turn understand my role. I am not here to teach you. I am not here to answer your questions. I am not here to chit-chat, and I am not here to help you enter the tower. I keep the tower’s secrets. That is all.”
The Master of Secrets looked them over, raising a hand to stroke his long gray beard. Rose had a million questions she still wanted to ask, but she held them in. It felt like holding her breath. In a moment, they’d come spilling out.
“There is exactly one set of circumstances in which we will speak again,” the old man said, turning his back on them. Rose saw his arm moving, still stroking his beard as he looked away. “If you find a secret of your own, it would be safest to bring it straight to me.”
Rose had moved to the table of their small room, pages torn from her notebook and scattered across the top of it. It was getting dark, so she lit one the candles Dexter had left on it. She wanted to be able to see as many examples as she could while she thought.
Rose felt the light, airy weight of the driftwood in her palm as she scanned across the pages. She was running through paper at an alarming rate. She’d have to get more. Maybe her stipend would be enough, maybe she’d need a job–
No, that wasn’t important now. One of the downsides of feeling her driftwood was that her mind tended to wander. It took more effort than normal to guide its wandering to fertile ground.
She’d broken down the possible types of characteristic values she was looking for, and right now she was focused on characteristic values which could only change in one direction. If she could find one of those, everything else would be easy. She’d just have to convince Merzhin that the value for the fully-colored 8x8 grid was different from the value for any set of 7 black squares, and in the wrong direction. Then she’d be done.
One of the examples caught her attention, as her eyes darted around the table. The simplest possible case. Two black squares diagonal from each other, which in a single step turned into a 2x2 black square. The two shapes looked kind of similar to her.
Why did they look similar? The square kind of looked like the two diagonal squares had had their center point dragged outward. Or like the edges had just flipped upward to encompass another square.
Did that make sense? She wrinkled her brow, trying to express it more clearly to herself. Was this really the right track? Should she even be–
With a practiced flex of her fingers, she swapped the driftwood in her hand for her shard of glass. Why did they look similar? She tried to explain it to herself, as clearly as she could.
Day was fading into evening when Rose and Alexandria found Master Woarck. He was an easy man to find, at least. He was the only master who lived outside the tower, in a personal suite attached to the Refinery.
Rose wasn’t sure if Refinery was a term, or a title, or what, but as soon as she saw it she was sure it was capitalized. It was on the outskirts of town, on the opposite side from where they were staying. The complex was short and squat. It was only a single story tall, and so it had been hidden behind the height of the urban core, even from atop the inn’s roof.
It might not be tall, by the ridiculous standards of this place, but it was big. It sprawled, an enormous collection of building and courtyards larger than her home town, extending all the way from the edge of town to the side of the mountain. It was like a city within a city.
It bustled with people, even this late in the day. Some of the buildings belched smoke. Others rang with the sound of metal striking metal. Everywhere, men and women of all ages scuttled around with the marked air of people at work.
“This is why the town exists, as near as I can tell,” Alexandria whispered in Rose’s ear. “To support this place.”
“What do they do here?” Rose whispered back, not sure why they were being secretive.
“The tower needs processed materials. Steel, coke, aluminum, things like that. And stranger ones as well. Not all of them can be produced on the surface. Those that can are shipped in, on the same river we arrived on. The rest are produced here by an army of failed applicants. The town exists to house and feed them.”
Rose looked around her again. She let out a small breath. What went on in these squat, ugly buildings? What tools and techniques, banished by Ludd to the mountaintop, could she see here for the first time if she darted off and poked her head in?
The more she thought about it, the more it seemed reasonable to take a peek. Merzhin had told them that they should focus on their educations, hadn’t he? That they should learn, and worry about admittance to the tower later? There was so much here, at her fingertips.
Alexandria must have read Rose’s thoughts on her face. The other girl raised her eyebrows, and Rose blushed. “I’m the sure the master would have some useful thoughts on what to explore first,” Alexandria said, patting Rose on the head.
Rose rolled her eyes, batting the other girl’s hand off, but she was right. The master’s personal suite was right on the border of town. They might as well talk to him first. There was still an hour or two of daylight left, she could explore afterward.
The place was easy enough to find. There was a small line of people outside the door. Rose and Alexandria took up position at the back of the line, and Rose stood on her tiptoes, peeking at the people in front of them.
There were only three. A boy about their age, with long hair tied back in a bun, and calloused hands. An older woman carrying a basket. And in the front, a man her father’s age with a bald head and broad shoulders. The man looked back, catching her eye, and she quickly looked down at the ground.
The door in front of the line opened, angry shouting drifting out, too far away to make out into words. A boy barely out of toddlerhood darted out the door and took off running toward the other end of the refinery.
It took almost half an hour to make their way to the front. Other people had queued up behind them in the meantime, and when the long-haired boy finally emerged from the door, Rose was already bored out of her mind. She should’ve just gone to explore the complex.
The boy mouthed something to the two of them that Rose couldn’t parse. Alexandria nodded back to him, and then the two of them were pushing forward through the door, into a cluttered office.
It was a large room. Tables lined most of the walls, covered in seemingly meaningless piles of tools, materials, and papers. In the very center of the room was an enormous mahogany desk. Seated behind it was a red-faced man larger than her father. A young woman sat to his left, holding a notepad and some strange writing implement.
“What do you want?” the man asked, his voice loud. Rose winced at the volume, shoulders shrinking together a little bit.
“Master Woarck?” Alexandria asked, taking a step forward. “It’s a pleasure to meet you. We–”
“Get to the point, girl. Two sentences.”
The woman flexed her hand and began writing, stopping during the awkward pause that followed.
Rose immediately began trying to condense her thousand questions down to two long sentences. That turned out to be a mistake, as Alexandria spoke first.
“What do you want to see from students who enter the tower? What do you teach that we should learn?”
Woarck frowned. “Children. Always thinking of themselves. Take introduction to materials, one of my principles teaches it. I’ll sign your tower forms if he recommends it. Bring any future questions to him. Good day to you.”
“Did you build the catacombs?” Rose asked quickly, before Alexandria could eat up the rest of their apparently limited time. It was the only thing that made sense. “How did you get the tolerances on the walls–”
“GOOD DAY TO YOU!” the man bellowed, his assistant scribbling furiously. Rose winced, and practically ran out of the room, Alexandria following at a more sedate pace a moment later.
She hated when people yelled. Always had. She probably should have gripped her iron and insisted, he might have answered one or two of her questions to get them to leave.
Alexandria sighed. “Well, that’s all of them.”
“What about Mrs. Breton?”
“She actually has a schedule, remember? Like an ordinary person? We’ll see her tomorrow, she might be willing to speak with us after class.”
Rose stared and stared at the shape on the paper. Two black squares, diagonal from each other. They expanded in a single step into a 2x2 black square. Something looked like it was preserved, but what was it?
She absentmindedly drew two squares on a fresh sheet, not bothering to color them black. To turn them into four, she licked her thumb to smudge the middle, then redrew the boundary.
_ _ _ _ |_|_ -> | -> | | |_| -> _| -> |_ _|
Huh. She’d erased four lines, and drawn four lines. A single word floated up from the depths of her mind, slowly but inexorably. Perimeter, she thought.
Her mother used to say that solving puzzles was 50% knowledge, 50% technique, and 1% magic. The saying had always irked Rose, from the moment she was old enough to understand how percentages worked. Her mother had eventually relented. Solving puzzles was 49.5% knowledge, 49.5% technique, and 1% magic.
It was easier to be precise all the time than most of the time. It helped train the part of your brain that noticed when you were letting the edges of an idea get fuzzy in your own mind.
But that wasn’t the point. The point was the 1% of puzzle solving that felt like magic. You could learn all you wanted, you could think carefully and draw out examples and break puzzles down into smaller parts, and at the end of it all you still had to stare at two things which looked similar to you and figure out what the hell was going on.
It was the final, irreducible step; the moment of insight that couldn’t be taught or practiced. And it was sublime.
Perimiter. Rose gripped her glass tightly, careful to keep her fingers away from the sharpest tip. She could see it now.
If you looked at the two black squares as a single shape, infinitesimally small in the middle, the length of its perimiter was 8. 4 for each of the two squares. When it grew into the 2x2 square, its perimiter was still 8. The two edges which had been angling in toward the center point just flipped outward, pointing away from it instead.
She breathed out, slowly. Did it generalize? It did. In order for a white square to turn black, at least two of its edges had to be bordering a black square. After it turned black, those two edges were no longer on the perimiter of the figure, because they were edges separating two black squares from each other. Since it bordered at least two black squares, it bordered at most two white squares, which meant that the square turning black could add at most two new edges to the perimiter of the figure.
The initial squares didn’t have to be touching at the beginning of the puzzle, but the argument still worked. You just summed the perimiters of all the distinct shapes.
She squeaked, bouncing up out of her chair and rocking back onto it. That was it! It was so simple. The length of the perimiter of the black shapes on the board could never increase, only stay the same or decrease, no matter how many steps you took. And 7 1x1 squares had a smaller perimiter than a single 8x8 square. So there was no possible arrangement of 7 squares that colored the whole board.
Rose ran through the answer in her head a few more times, to make sure she didn’t forget it. Actually, there was a better way of saying it: the number of edges with a black square on one side and a white square on the other could never increase. She stood, filled suddenly with a triumphant nervous energy, and stalked around the table.
It was dark, she realized suddenly; the true dark that came after the sun was fully down. The sheer number of flickering lights from the town outside her window had disguised the hour. Her hand was cold and cramped, where it had been clutching her charcoal, and she was alone.
Dexter still wasn’t back.