Chapter Four Factorial
Scene One: MathemagicThe air smelled warm. Sunlight lay like a blanket on his exposed arms. Denario couldn’t see but that no longer mattered. Guided at first by sound and smell, he poled closer to the shore. He could feel the reeds, mud, and rocks press against the right side of his body through the stalks.
“Any shape bigger than a fist,” he murmured to Torgrim, the dwarf nearest. “That’s how much I can feel.”
“You sound pleased,” said Torgrim.
“Surprised.” From the shape of the nudge under his armpit, he was pretty sure he was feeling a clump of weeds. Rocks pressed in a more definite way.
Jack’s method seemed insane. But it worked better than the accountant had dared to hope. Intimidated by Jack’s expertise, he’d been content to let Jack do the steering for most of the afternoon. He’d thought he was doomed by his lack of boating proficiency. But when he'd grown tired of waiting, he’d decided to test his skills. He’d wanted to understand the suit even though he hadn't believed it could be as simple as this.
It was. The feeling of things came naturally. Knowing what to do about those sensations given his lack of direction and rafting instincts was another matter. Still, those were problems that might be solved in time.
He had already spent a few hours at rest. He'd napped. He'd slapped at insects with no less effectiveness than usual. After a while, his blindfold had filled up with sweat. The pegs under his tunic, even rounded as they were, had dug into his armpits. He suspected the outfit could get annoying at a level he hadn't experienced since his escape from Zeigeburg weeks ago, when he had fast-marched with pebbles in his accounting shoes.
It was hard to move while dressed as a porcupine. But now, after his experiment, he found it reassuring. He could perceive the riverbank. He could feel a tree trunk. That last expectation was confirmed in the next moment when his punt struck roots in the creek bed.
“I'm getting accustomed.” He turned in the direction of his other close companion. “How are you holding up, Ragna?”
“Not too bad,” the dwarf didn't sound drowsy. All of the dwarfs had found work that they could do without using their eyes. Ragna had chosen weaving. Ulf tailored clothes. Borghild carved. Torgrim and Jofried put edges on steel tools.
“Am I the only one bored almost to death?” Brand, the former caravan chief, complained.
“Yes,” said Ragna and Torgrim without pity.
“How many times have you pried off your blindfold?” asked Denario.
“How would you know about that?”
“You got ill. I heard retching.”
“Well, I was trying to figure out when we'll get back to my trade routes. Shouldn't be too far.”
“And looking around while wondering about that made you sick.” Denario hoped the anti-cartography magic was near its peak. From comparing this to Jack's descriptions, it might be. “You swore to secrecy about Jack's methods. Weren’t you serious about the oath?”
“The oath, yes, the method of traveling through magic, no. Come on, accountant, this is insane. We heard a flock of flying frogs overhead. What if they'd been something worse? How can we defend ourselves? Other raftsmen come through here. I wish my troop had run into them instead.”
“Jack knows the methods of the other raftsmen. Or he thinks he does. He chooses not to use them.”
“They can't be worse than blindness.”
“You put your trust in magic charms, I suppose.” Denario was thinking of how Brand had used the tokens of Onuava to track them. He wondered if the caravan master had slyly left out part of the story about his travels through the forest around the lost temple. Could he have used the tokens in some way? Ideas churned in his mind.
“Not at all. Magic never does what you want it to unless you're a wizard.”
“You lied about using accounting, Brand. Now you're lying about magic.” The dwarfs hissed. To them, accusing someone so boldly was nearly a crime. “There were tokens to the goddess Onuava in the treasure you captured. Either you or Mohi had a token to help you find those others. If you hadn't, you'd be dead now.”
“Well done. Smart bastard. But I'm still right about magic and numbers. There's something wrong with using them.”
“That is a strange argument for you to make,” said Borghild to Brand. “A few dwarfs, long ago, argued that particular tools, including magicks, encourage immorality by making lives too easy. But I was not aware than you were opposed to having an easy life.”
Brand made a sullen lack of comment. Denario thought he heard the man fold his arms across his chest.
“A battle between the wisest or most clever of opponents is no more different morally, I think, than a battle between the strongest or the quickest. Are you saying that you didn't outwit your attackers even a little when they attempted to rob you?” Borghild drove the point home further. “Was it all luck, then?”
The middle raft felt the tug of the front one as it moved eastward into the center of the creek. Jack Lasker wasn't ready to set up camp, apparently. Denario ceased feeling the edge of the riverbank through the quills of his suit. Only the tap of his punt against the creek bottom let him know that his leader hadn't taken them out into the depths yet.
After a few minutes, the dwarfs in the third raft shouted to the first that they wanted to land, build a fire, and cook dinner. The first raft shouted back that they were looking for a place to moor. Jack said that he had a method for landing while blind but that he was going to proceed with care no matter how hungry anyone got.
When they were told that the raft master had a method, the dwarfs quieted. For half a day, they'd felt reassured by Jack's smooth navigation. It was almost a parlor trick, albeit one done in the world’s largest parlor, the outdoors. Intellectually, Denario understood how the man could do such a thing, but he also knew that he wasn't up to it on this scale himself. Probably no one except Jack or maybe his father could steer three vessels through sandbars, boulders, and fallen trees.
Jack guided the rafts through the deeps to the Kilmun shore. A while later, he took them back west. Denario could feel the difference. No one else could. He knew that Jack wasn't finding a spot that he liked.
Denario did math in his head for a while, mostly numeromancy. He found himself reviewing the hex codes he understood. He wanted to write real spell, one that the Guild of Accountants knew existed but had never been able to duplicate. Book keepers had reported seeing wizards cast it. They had written down their observations in the guild scrolls. But to date, no accountant had managed to create it, the magic 'echo' command. That was despite the fact it was one of the simplest spells that any wizard knew.
The echo spell detected incoming magic and exposed the hexes in it. It was the means by which wizards studied one another's creations. It also seemed to be how some wizards studied their own work and tweaked their spells. What a marvelous thing the echo hex would be in the hands of a certified accountant! Such a mathemagician could reveal the fundamental equations underlying the world.
The guild had recorded fragments of the spell. While the hexes were numbers greater than zero, the conjuration chopped up those numbers into hexadecimal digits and spit them back to be seen. It was an elegant thing, much like a geometric proof, very formal and very exact. It treated letters as if they were numbers, which made perfect sense to Denario. He didn't find coding letters to numbers and back again to be anything other than a amusing pastime.
He leaned back as best as he could in his suit. He thought to himself, What if I were reckless? How would I solve the problem if I didn't care so much about what other accountants thought? Well, he would try every hexadecimal number combination he could think of in rapid succession. No one here could see his failures. There was no guild master to warn him off. There was no wizard to stop him. He had a ready supply of magic at his fingertips. When would these circumstances ever occur again?
This was the best opportunity an accountant had ever had to figure out the math behind all life. This was it.
With that thought in mind, he started running through all of the hex codes, even the ones unknown to the guild, the total guesses. The first line was a given. That was the one that said while the incoming magic wasn't zero ...
While 'hex twenty-four hex twenty-one hex thirty-thirteen hex thirty'because the number zero, it had been determined, was 'hex thirty' in the only magic the accountants knew. Then he needed the codes for 'shift right' and 'show.' But he already knew the code for showing. He'd used it to show magic stars. So it was really only the 'shift right' he needed to guess. Or was it 'shift left?' How did wizards decide? Did the magic decide for them? Was it hidden from them in their higher-order tongues? Whatever the command meant, the wizards used it as if numbers were mechanical, like balls in an abacus, and they could drop digits with it.
shift rightDenario translated his letters into hexes. It was almost too easy. For a reason known only to wizards, letters emerged as hex numbers in a simple, positional notation. It wasn't a matter of encoding so much as it was adding a hexademical forty-one to the value of the letter you wanted. He could do that in his head.
He mumbled and drew with his finger on the deck, repeating the process over and over. He was sure of most of the spell, especially the parts that had been stolen from wizards over the many years of the guild. For the shift, he started with 'hex 1,' the second hex because there was a 'hex 0' that blanked out other hexes. He already knew that one and he knew 'hex 7,' a sort of magical alarm. He didn't need to try those.
“Um, accountant, what are you doing?” asked Brand.
“Math. Important math.”
“Oh, all right.” Apparently Brand hadn't been the only one worried. Denario could hear the dwarfs closest to him relax. Their sleeves rustled as they returned to their work.
“I'm finishing something that no accountant has completed before.”
“A good thing?” Everyone paused a little.
“It'll be something to write down when we stop.”
He returned to reciting the spell but, by the fifth iteration, he noticed a mistake. He had to back up and try again from the beginning. When Brand coughed, Denario messed up the eighth try. He started to understand why wizards got cranky when they were interrupted. The whole spell, the entire mathematical proof of it, had to be perfect. Distractions were not welcome. Denario repeated the eighth and ninth tries to ensure that he'd given them a valid test. He almost missed hex fourteen. At the end of the cast, he readied himself for a try at hex fifteen. Then he realized that he'd felt something. Had the spell succeeded? How could he know? Sickness be damned. He pulled up his blindfold.
In the air in front of him, numbers scrolled by. He read 0, 0, 7 followed by 0, 0, 7, followed by 0, 0, 7, and so on. Somehow he'd produced the alarm hex. He mumbled a cancel. The numbers stopped.
Where had he gone wrong?
He puzzled over it for a minute and then ran through his incantation once more using the hex fourteen code. Again, he got 0, 0, 7 followed by 0, 0, 7, followed by 0, 0, 7 ... and then a 0, 1, 7. He blinked. The hex was gone, scrolled off into the air, replaced by more zeros and sevens. Had it really happened? Had he seen a hex 17 instead of a hex 7? He waited. He counted.
After a couple minutes, he knew. It was real. Every sixteenth command revealed by his spell was a hex 17. What was that for? He had no idea. But it proved that he hadn't done something wrong and set off a magical bell. No, he had figured out the long-sought-after echo command.
Hex 14 was the shift hex that accountants had been looking for. He had written his spell completely in numbers. It had worked. He would never have been able to do it without coming to this swamp of high background magic.
“Thank Melcurio,” he mumbled. He made the sign of eight.
Truly, he was blessed. He canceled the spell and ran it again to watch the results. They didn't change. There were a lot of hex 7 charms in the air, followed every sixteenth by hex 17 and only occasionally interrupted by other hexes that Denario assumed were part of the background magic.
Lots of alarms. Huh.
He hadn't taken his eyes off of the magic but he started to get a queasy stomach. He pulled the blindfold down over his eyes and wondered.
Why so many alarms?
He snapped off the blindfold to study the landscape. There was wasn't any danger that he could see on either shore. The rafts had left the east side and were drifting west. Ahead of him, Jack stood like the river master he was. His punt guided the rafts in steady strokes. The cat-tail quills of Jack's suit gave him a feel for what lay ahead. The tip of his pole told him a lot about the bottom, no doubt. It was amazing to look at. But Denario didn't stare for long.
What's dangerous? he wondered. Nothing. So why any alarms at all?
He canceled his spell. His fingers dragged across the blindfold and pulled it down. But even though he couldn't see the magic, he knew it was there. Everyone is warned to avoid areas of heavy magic this month, he remembered. Maybe there was a storm on the way. It might be as simple as that. Would that set off alarms? No. After all, danger lay all around them in this ominous and twisty creek. Only the presence of an intruder or some other special circumstance should do it.
Maybe we set off the alarm when we got too close to the temple. If so, no harm would come from it. The temple was long abandoned. There was no one to take notice of the warning.
But what if something eavesdropped in an automatic way? What if magical traps detected geometers in these lands? The spells might set off further hexes. Anti-cartography magic might tune itself according to the alarms in order to be more effective at confusion. Maybe he was feeling a different kind of magic right now and didn't know it.
He thought about the problem for half an hour as Jack in the lead raft took them to the west bank and then east bank. The riverman still couldn't find a landing spot he liked.
Denario listened to the creek. Twice, he heard fish come to the surface. Many times, he heard frogs calling from the water and from the trees above. Once, he peeked under his blindfold. He caught Brand doing the same. The dwarfs sat patiently in their assigned spots and worked on their tasks. The accountant pulled his blindfold down. He leaned back as well as he could in the suit and wondered about the alarm hexes. He'd done good math to discover them. It would be a shame to stop. If the alarms caused spells to be cast, maybe he could halt those. All he needed to do was cast a short loop of a hex.
While the incoming hex is an alarmWith that, he could create magical silence. The curses that he imagined would trigger on those hexes would remain in hibernation, waiting for signals that would never come. That would be perfect.
set the alarm to null
Denario spent a while considering the logic of the spell. Wizards might cast as easy as they spoke but for accountants, naturally, it had to be different. Aside from a smattering of words in magical tongues, hexes were all that any accountant knew. And hexes, perhaps unlike other methods, had to be written or spoken in order. Any letters that were needed had to be created from numbers. He practiced. As he created his formal spell, which took a few minutes, he reflected how lucky he was that a previous accountant had learned how two hexes, thirty-ten and thirty-thirteen, combined to make the hex for 'assign equality.' Without that fellow's hard work, Denario wouldn't have been able to continue.
When he was ready, he pulled up his blindfold to see. He whispered the incantation as he drew the numbers. At the end of it, he felt a difference. He saw a glimmer of zeros in the air before they faded.
It was working. He knew it. He felt so pleased that he squirmed in his seat.
“What did you just do?” said Brand. A few yards away, he writhed. Although a look of pain crossed his face, he didn't touch the cloth strap over his eyes. “Did you spit? I felt something.”
“Nothing like that,” Denario assured him. “Just math. I proved something.”
“What was the conjecture?” Borghild asked. He was one of Denario's best pupils in the group.
“Trade secret. This will be written down for the Guild of Accountants.”
Borghild muttered something in dwarfish. The other dwarfs made grunting sounds of resignation. They were curious but Denario could count on them to respect his craft.
Finally, with the sun long descended below the treeline but not yet turned orange at the horizon, the lead raft swung toward a broad expanse on the Mundredi shore. Denario took the opportunity to watch Clever Jack as he steered by feel. After he scraped his raft's starboard edge along the bank, Jack plucked one of the long reeds out of his suit and swept it through the sand and grass. He kept doing it as he punted left-handed to keep his raft close to the landing. After this had gone on for a minute, he yanked off his blindfold. Denario, now sick to his stomach, put his back on.
“All rafts, hove to shore!” Jack shouted. Denario enjoyed the feeling of the clumps of grass as they pushed the butts of the cat-tail reeds gently into the right side of his suit. As he felt it, he leaned into the sensation and pushed with his pole on the left side of his body.
“Blinders off!” Jack called. “Tie down!”
Denario heard the dwarfs scurry fore and aft. They gathered coils of rope, clambered over the gunwhales, and looked for rocks and roots suitable for hitching. One of the dwarfs behind him mis-stepped and fell into the creek. His friends waded in. Denario didn't. He’d been slow to untie his blindfold. He was in the midst of unbuckling his navigation suit. For every cat-tail reed that he lost in the process, he'd have to spend a minute re-inserting it and maybe whittling it again, too, so he took care. In front of him, Clever Jack unbuckled and untied himself so deftly that it was impossible to copy. He had his suit laid down on the deck, spines up, in less than half a minute. He didn't glance back to Denario. He strode to his stewpot and utensils.
Denario took so long that a pair of dwarfs came to help.
“Thanks, Ulf, Torgrim,” he said as they laid the suit down, cat-tails up to the sky. The ends drooped so that all three of them needed to crouch and step away.
“How did it work, Skilling?”
“Sensations to the sides are good. As to feeling obstacles to the front, I won't have much opportunity for that.”
“Not likely with Master Jack navigating.” They laughed. As his nausea passed, the accountant joined in. The suit was a good trick. His numeromancy had been even trickier. And even better.