The key to efficient algorithms is information.” It was Professor Drecker’s mantra, barked at the cadets at the start of every Police Algorithms class, ferociously enough to sear itself permanently into Frank’s memory. “A good algorithm depends on finding the structure in the data and using it. It depends on information.”

Frank smiled to himself at the memory as he turned onto Three Bit Lane, a rutted dirt road lined with a combination of seedy bars and upscale coffee shops. He nodded politely to a pair of passing knights, who clanged as they jittered past in their armor, and made a mental note to grab a Triple-Bold Espresso before he left. First he needed information, something to help guide his search. He knew exactly where to start.

Glass Box Billy would be in one of the establishments by now, sitting quietly and listening to the wisps of conversation drifting through the room. People didn’t mean to say things around Billy; they simply didn’t notice he was there. Billy had been blessed with a single notable talent: utter inconspicuousness. Whatever he tried, there was something about Billy that meant people just didn’t notice him. Maybe it was his pale skin or his small physique; maybe it was his exceptionally mundane taste in clothing. Whatever it was, Billy had long ago decided to put his one talent to use by eavesdropping, collecting information, and selling it to anyone who would buy.

Frank eyed the eight storefronts hunched together in Three Bit Lane and wondered which one Billy would have chosen. He ran through half a dozen search algorithms in his head, but it was pointless. Frank didn’t have any information to go on. Billy could be in any one of the bars or coffee shops.

He’d have to use an exhaustive search—simply try all the possibilities until he found Billy. It didn’t sit well with him. Years of the detective and private investigation game had taught him that there was almost always a better algorithm than exhaustive search, and he hated resorting to something so inefficient.

Grumbling, Frank started his search. He walked into the first bar on the street, The Absolute Value.

“I don’t want any trouble, Abe,” said Frank, holding up his hands. “I’m just here to see Billy.”

“Well, Billy ain’t here,” said the barman.

Frank almost smiled in relief. “Then I’ll be on my way,” he said.

Abe gave a curt nod and watched Frank leave, his hand still under the counter.

Frank took a couple of deep breaths and shook his head in the cool air. Abe held a grudge longer than anyone else Frank had ever met. Then again, Frank had arrested four of his siblings.

The next establishment on the street was The Brazen Boolean, a modern coffee shop decorated in typical Boolean style—stark black and white. The inhabitants of the City of Bool were renowned for their fanatic devotion to the absolute concepts of logic, viewing everything as either True or False. They made good witnesses. As the only Boolean café in town, The Brazen Boolean was a haven for expats. After all, either you were a Boolean or you weren’t.

Frank popped his head in the door and asked everyone in general, “Is Billy here?” There was a brief silence as twenty pairs of eyes carefully scanned every inch of the cafe. Booleans wouldn’t answer a question until they were absolutely sure.

“No,” came the precise reply.

Frank continued his exhaustive search.

The third and fourth shops proved equally fruitless, although significantly more pleasant. The bartender of the Constant Const greeted Frank warmly and invited him in to reminisce about the good old days together, which was odd considering Frank had only met him the month before. And the crowd of the Daring Double, a notoriously loud wizards’ hangout, cheered at each new arrival and sang happily over their steaming mugs.

Frank found Billy in the fifth shop, the Exponentiated Expresso. It was by far the loudest and tackiest coffee shop on the street, but it managed to draw the most devoted following on account of its triply caffeinated beans. On a good day, every table would be packed with jittery people who seemed to think the key to a good conversation was volume.

This morning, the Exponentiated Expresso hosted a comparatively subdued crowd. Only a handful of tables were occupied, and most of those by lone coffee drinkers who shook and mumbled quietly to themselves.

Billy sat at a central table, leaning awkwardly toward a nearby conversation. Nobody seemed to notice him. Frank had even missed him on his first scan of the room.

“Billy!” Frank called.

Billy jumped up guiltily. “Frank?” He grinned, happy that someone had acknowledged him, and sat back down. “Pull up a chair.”

“I’m looking for some information,” explained Frank as he took a seat across from Billy.

“Could be that I have some,” said Billy. “I have such a hard time remembering these days,” he said, glancing toward a long-empty mug that probably wasn’t his.

Frank signaled the barista, who soon placed a fresh mug on the table. “Remember anything about a theft at the police station?” Frank asked Billy.

Billy’s eyes widened and he flinched. “A robbery, you say?” he asked unconvincingly. His eyes darted around the room, but, as always, nobody paid him any attention.

Frank laid two gold pieces on the table, ignoring the sour feeling in his gut. He couldn’t afford to spend this type of money, especially without knowing if he was paying for a lead or idle gossip. But he’d known this wasn’t going to be cheap. He leaned in close. “Two nights ago,” he said quietly, “the thieves took a whole pile of documents.”

“Doesn’t sound like the sort of thing that would be healthy to remember,” said Billy. He eyed the gold pieces. “Afraid you’re asking the wrong guy, Frank.”

“That’s gold,” Frank growled.

“Sorry. I can’t help you,” Billy said. He surveyed the room again before adding, “Even if I did know something about a robbery, it’s the sort of thing I would try to forget. Even if I did know something small, like who might have helped with logistics, it’s not worth the risk of waking up to find my shoes packed with yak dung.”

Frank stared, but Billy had gone silent. For someone who made a living sharing information, Billy had an odd habit of not saying things.“Yak dung?” asked Frank.

Billy nodded, but didn’t offer any more.

“You couldn’t be more specific, could you?” asked Frank. “Are we talking about Northern or Southern yaks?”

“Does it matter?” asked Billy. “The point is that if I knew anything about who arranged transportation, I wouldn’t remember it. Especially not if those people happened to have a large farm about five miles out of town where they could easily make someone disappear. And very doubly especially if the family that owns the farm has a history of illegal activity and an unhealthy sense of humor. Nope. It definitely wouldn’t be healthy to remember anything in that case.”

“Too bad,” Frank said with a smile. “Maybe next time then.” He nodded toward the coins. “Incentive to remember things in the future.”

With that, Frank stood and strode from the Exponentiated Expresso. He turned left and continued up the street. Once off Three Bit Lane, he could swing around and make for Crannock’s farm—the only farm remotely matching Billy’s description.

As he passed the Faulty Register, he noticed a shadow dart into a nearby alley. He cursed under his breath, but kept going. Of course he had a tail already; the captain hadn’t exactly been discreet about his visit.

But by the time he left the city and was on the rough dirt lane to Crannock’s farm, he found himself in a good mood. Billy hadn’t given him much, but even a little information could mean the difference between an efficient search algorithm and an exhaustive one.

Food inspectors,” Frank called out as he and Officer Notation stepped up the narrow gangplank and onto the ship. At Frank’s instigation, Notation waved her badge in a blur, too fast for anyone to read.

“Food inspectors?” asked a crew member. “We aren’t transporting any food.”

Frank looked the man over. He wasn’t an officer or hired security, probably just a sailor who had taken charge while the officers were away. It wasn’t uncommon. Smugglers rarely employed guards to watch their ships. It drew too much attention.

Frank turned on the sailor, growling out his words. “We’ll see about that. I’m told there’s a load of rotting eels on this dock, and I intend to find them.”

“Eels?” The sailor was clearly out of his depth.

“Rotting eels,” Frank shot back. “We’re going down to check the stores.” Then, without waiting for a response, he strode to the hatch leading below deck.

Notation hurried after him.

“We don’t have much time before they get the captain. We need to find the logbook,” Frank said as he climbed down the ladder. The logbook would contain a manifest of items shipped and a list of ports visited. The manifest would be fake, of course. Smuggling ships never documented their true cargo. But with any luck, he could read between the lies and find a clue.

Officer Notation found the logbook at the back of the hold and pulled it out. Frank checked the cover and swore:

Manifest and Log of the Retry Loop

Captain: A. James

Home Port: Usb

Owner: Vinettees Shipping Group LLC

After months of successfully avoiding the Vinettees, Frank had walked right onto one of their ships. He found himself reflexively scanning the hold for hidden henchmen, stashes of weaponized farm equipment, or evidence of a slug racetrack. Frank discarded the last possibility—everyone knew slugs wouldn’t race on a ship; it had something to do with being surrounded by large quantities of saltwater.

He shook his head and focused on the problem at hand. Frank had to find a clue before the Vinettees knew he was on the ship, or he might not get back off again. He turned to the end of the book and started flipping toward the front, one page at a time.

“What are you doing?” asked Notation.

“Looking for the last entry,” said Frank.

“One page at a time?” asked Notation. “There’s got to be a thousand pages. Why don’t you use binary search again? We just used it two minutes ago.”

Frank paused. He wasn’t looking for a specific page number, but he could still use a binary search to find the last entry. He would just refine the search bounds depending on whether the current page had text or not.

“Okay. Binary search,” he agreed.

Pages 500 and 501 were both blank, so Frank knew the last written page was on or before 499—his new upper bound. After another midpoint computation, he flipped to page 250. Again it was blank.

“Looks like a new book,” added Notation. “Good thing you didn’t keep going from the back.”

Frank didn’t bother replying. With a lower bound of 1 and an upper bound of 249, he computed a midpoint of 125. This time he found writing, so he adjusted his lower bound accordingly to 125.

“187,” supplied Notation before Frank could finish the midpoint computation in his head. He turned to 187, again finding writing and adjusting his lower bound.

“218,” said Notation. The pages were blank, so Frank adjusted his lower and upper bounds to 187 and 217, respectively.

“202,” said Notation before Frank had even finished adding the upper and lower bounds.

“How are you doing that so fast?” asked Frank.

“Practice,” she replied. “We used to have binary search competitions at the academy whenever we needed a break from studying. I was undefeated.”

Frank shook his head. “Sounds like a wild time,” he muttered.

Pages 202 and 203 were filled. “210,” supplied Notation.

At page 210, they finally found the last entry, detailing the Retry Loop’s last voyage. “What now?” asked Notation.

“We search for an interesting package or port. On the last voyage, they made about 70 entries. We’ll have to scan through them.”

“Exhaustive search?” asked Notation. “Can’t we use something more efficient? Aren’t the entries sorted by pickup and delivery time?”

“The sorting doesn’t help us here,” answered Frank. “We don’t know the time. Sorted data only helps when it’s sorted by a useful dimension. They didn’t bother to sort by suspiciousness. Go figure.”

“Oh. It’s the Weather Records problem,” said Notation.

“What problem?” asked Frank.

“It’s an illustration of how sorting data by the wrong value doesn’t help a search,” explained Notation. “Professor Drecker gave the example of finding the coldest day in the last 10 years. If the logs are sorted by day, you can use binary search to efficiently find any specific day. But that doesn’t help us to find the coldest day, so we’d still have to scan through all the data. I hadn’t expected to see such a clear example outside of class, though.”

“Welcome to the real world,” said Frank. “Out here, you have to check when the structure in the data is helpful to you and when it isn’t. Don’t worry, it’s a common rookie mistake.”

He could see Notation bristle at his words and tried not to enjoy her reaction too much. Every rookie came out of the academy thinking they knew it all, and every one had a lot to learn. Notation was getting off easy with a lecture. His own education with binary search had involved hours of scooping through barrels of pig waste while he questioned his career choice.

After about three minutes, they found their only clue. The Retry Loop had recently made two suspicious stops, at the Port of Mudwall and Frayed Cable Island. Even for smugglers, these were strange destinations. The Port of Mudwall boasted little trade beyond its outlying mud farms. And Frayed Cable Island was even more desolate; the small, rocky island possessed only a single building—the now abandoned Iron Ring Prison.

“There,” said Frank, pointing. “That’s where they took your documents. Either the Port of Mudwall or Frayed Cable Island. They probably dropped the documents off at one and picked up the payment at the other.”

“How do you know?” asked Notation. She looked skeptical. “Shouldn’t we consider all the ports as—”

Frank cut her off. “No time to check them all.” He didn’t elaborate. He was using his own brand of algorithm now, the heuristic searches that had gotten him in trouble with the captain in the past. But he had a gut feeling, and Frank had learned to trust his gut.

“Are you sure that—” Notation began, but was cut off by noises above them.

Frank couldn’t make out the words, but he could recognize the tone clearly. Trouble was on its way.

“So . . . that didn’t work,” supplied Socks. Frank ignored him and studied the locking mechanism. It was a standard carved keypad with buttons labeled 1, 2, 3, A, B, and C in a single ordered row and an enter button beneath.

“We’ll have to do this the old-fashioned way,” said Frank.

“Wasn’t kicking down the gate the old-fashioned way?” asked Notation.

Frank ignored her as well. “Socks, do you know any magic lock-picking spells?”

“No,” Socks protested loudly. “Those are illegal!”

“How about something to weaken the lock? Or maybe the hinges?” asked Frank.

“You want me to help you destroy property?” Socks looked aghast. “That’s worse than lock picking. Do you know how much trouble—”

“Search spells, then? The Spell of All Combinations or the Spell of Breadth-First Search?” Notation interrupted. She’d heard enough on the topic of proper and improper spellwork after Frank had casually inquired about the feasibility of replication spells on gold coins—a use of magic that fell firmly on the wrong side of both Socks’s and her own ethical line.

“I’ve used the Spell of Breadth-First Search a few times,” Socks answered. “My real expertise is binary search trees, but I’m familiar with a range of computational techniques. Once I—”

“Will breadth-first search work on the lock?” interrupted Frank. Over the years, Frank had worked cases with a handful of wizards of varying levels of respectability. He’d seen at least a dozen different lock-picking spells but had never seen a door opened with an explicit breadth-first search.

Notation smiled. “Definitely! It’s a bit abstract, but I saw a similar problem recently in my Police Algorithms course. When you think about it, a code lock is just a search problem; you enter a string of characters to open it. The search space is all possible strings that can be made from those characters. Every string is a valid search option, from a single character like 1 or A to complex sequences like ABC123CBA321. The search target is the one string that opens the lock.”

“But we don’t even know how many characters we need,” protested Socks. “The lock could have a 30-character combination.”

“That’s why she suggested breadth-first search,” said Frank, thinking aloud as much as addressing Socks’s concern.

“I don’t understand,” said Socks.

Notation quickly picked up the explanation. “You see, breadth-first search expands outward from a starting point, exploring along a frontier of solutions. Naturally it will try the shorter solutions first.”

“Huh?” asked Socks, now looking confused to the point of panic. “I thought breadth-first search used magic lists. I’ve always used a magic list. Isn’t it just a magic list?”

“Yes,” agreed Notation. “Breadth-first search maintains a list of options to try next if the current option doesn’t work. The algorithm is basically a loop that keeps pulling options from the front of the list and adding new options to the back. On each iteration, we pick a new option to try from the front of the list. And, if that’s not what we want, we check if there are any new options reachable from the current one and add those unexplored options to the back of the list.

“You start at a single point in the search space, in this case at a password of length zero. Then for each password you try, you add new search possibilities to the end of the list. In this case, each time we try a password, we’ll add all single-character extensions to the list. For example, here we know the password can only contain the characters 1, 2, 3 and A, B, and C. Once we’ve tested 3A, we’ll add 3A1, 3A2, 3A3, 3AA, 3AB, and 3AC to the end of our list.”

Socks screwed his face up in concentration, then asked, “How do we know which options to add?”

“Think of it like a tree of possibilities,” suggested Notation. “Each branch, or node, is a password from our list, like 3A. The neighboring options are the nodes under it—the passwords we would get by adding one more character to the end. Breadth-first search progresses down each level of the tree before moving onto the next.”

“Since we add the new, longer passwords to the end of our list, we try all the short stuff first,” Frank threw in. “Now, can you do it?”

“This isn’t a proper use of—”

“Come on! Really?” interrupted Frank.

“It’s basically a lock-picking spell,” responded Socks.

“Yes. That’s exactly what it is!” shouted Frank.

“Forget it,” said Notation, throwing her arms up in frustration. “If he doesn’t feel comfortable picking the lock, we’re not going to change his mind by yelling.” She turned and studied the stone wall itself, which stood at least 10 feet tall. After a moment she continued, “Frank, if you give me a boost, maybe I can climb over.”

Frank gave the wall a skeptical look. Despite having been abandoned for years, the wall lacked the large cracks and rambling vines that often aid mountaineering efforts on old castle walls. The workmanship was impressive. Someone had taken real pride in building this wall; you could tell from the artistic way the metal spikes twisted as they jutted up. Those little details took effort.

“Maybe. It’s pretty high, though, and those spikes look awfully sharp,” he said.

“It’ll be just like the obstacle course at the academy,” said Notation. “Aside from the hard-packed ground, the lack of handholds, and the large metal spikes, that is.”

“Those’ll probably add some excitement,” Frank offered.

“Shut up and give me a boost, Frank.”

“No. No. I’ll do it,” said Socks hurriedly. “I’ll use the Spell of Breadth-First Search. I’ll need something for the list, though. A roll of parchment, perhaps?”

Frank and Notation looked at each other. “No can do, kid. Use the ground; it’s muddy enough.”

“Oh. Yes. Of course.”

A few minutes later, the lock began to glow. “Here we go,” said Socks.

The word enter glowed briefly, followed by a clicking noise. But the gate remained locked. The spell had tried the first password, which was nothing at all. Next a series of numbers and letters appeared in the mud:

1, 2, 3, A, B, C

Frank could picture the tree of possible passwords that the list represented.

An instant later, the number 1 glowed, followed by enter. Again the gate clicked, but didn’t open. The list on the ground changed, showing the new list of passwords to try, branching out to the third level of the tree.

2, 3, A, B, C

11, 12, 13, 1A, 1B, 1C

But these were added to the end of the list. The search itself continued on the current level, trying 2.

The password 2 didn’t work, and the list grew again.

3, A, B, C

11, 12, 13, 1A, 1B, 1C

21, 22, 23, 2A, 2B, 2C

Again the tree branched out with new possibilities, but the search still worked its way along the current level, trying all one-character passwords before moving deeper.

In other words, the search explored the full breadth of each level before moving on to deeper levels.

The search finished the first level, trying the passwords 3, A, B, and C, before Socks broke the silence. “This could take a while.”

Frank nodded, eyes fixed on the ever-growing list of numbers. “Notation, why don’t you scout around the front?”

“Okay,” she agreed, her expression betraying great relief. Rookies didn’t tend to handle stakeouts well. Sitting still for hours on end with nothing to do wasn’t something you could teach at the academy, although Professor Cloud’s Philosophy of Law Enforcement lectures came close.

Five minutes after Notation left, the lock gave a loud click, and the gate swung open noisily on its well-rusted hinges. The list in the mud faded as the search algorithm completed.

“1111,” said Frank, without a trace of surprise. It often paid to keep the codes simple enough for the henchmen to remember.

He used a stick to write the code in a patch of mud and circled it twice. Even a rookie couldn’t miss the message. Then he turned to Socks. “Let’s go.”

Two steps into the prison, and Frank knew they had walked into a maze. The old computational prisons used to rely on their bizarre structure as much as on guards. Potential escapees think twice about sneaking through a door when they don’t know what lies on the other side: freedom or the guard’s breakroom.

“How about some light?” suggested Frank.

“Oh. Right,” agreed Socks. He muttered an incantation and a bluish flame flickered from the end of his staff, lighting up the completely unremarkable room.

The square room, rough stone walls, and heavy oaken door were enough to confirm what Frank already knew: the entire structure was a grid of rooms, each with doors to only some of its neighbors. They would have to navigate from room to room. But since they didn’t know which rooms had doors between them, they would have to search out a path as they went.

“Time for another search,” he said.

“A search?” asked Socks. “For what?”

“The papers, of course,” responded Frank. He had no doubt that the papers were stashed here. An abandoned prison provided an ideal location for stashing stolen goods, clearly surpassing the more commonly used warehouse. Arguably, the only better location would be an abandoned castle—provided it had a moat. The question now was whether they could find the documents and then, if they did, whether the documents would provide any valuable clues.

“Not another breadth-first search,” protested Socks.

Frank considered the idea. In theory, breadth-first worked fine on a grid. Each state of the search space was a grid square. Once you explored one grid square, you could add its unexplored neighboring squares to your list of things to try. Frank could clearly picture the search propagating out over an empty grid, like a wave moving across the water.

However, breadth-first search had one major drawback in the physical world—an excessive amount of backtracking. Since you were always adding items to the end of the list, the next square to explore could be annoyingly far away. Even on an empty grid, without walls blocking your path, you could find yourself hiking back to the other end of the search space.

It was the type of unnecessary movement that Frank made it a policy to avoid.

“No,” said Frank. “Too much backtracking. We’re better off going depth-first here.”

“Depth-first search. Depth-first search,” Socks mumbled to himself as though willing the spell into his memory. “I—I don’t think I remember—”

Frank waved him off, and strode confidently down the corridor. “We don’t need a spell for this one. I’ve been doing depth-first searches through buildings since you were in diapers.”

“No backtracking with depth-first search then?” asked Socks.

“There’s backtracking with most search algorithms. But backtracking in a depth-first search is better suited for walking.”

“Um . . . I see.”

“No, you don’t,” said Frank bluntly. “If you don’t know the algorithm, just ask. Pretending to know algorithms is a recipe for disaster. I’ve seen too many rookies tripped up due to bad searches. Good kids, like you.”

“Okay. What is depth-first search?” Socks asked.

“It’s a simple algorithm,” explained Frank. “Basically we explore deeply down each path. We go down one path until we hit a dead end. Then we backtrack to the most recent path that we didn’t take and try that. We’ll stop when we find the target.

“In this case, we’re going to use clockwise ordering. Whenever we have multiple options, we’ll try north, east, south, then west—avoiding paths we’ve already tried, of course. We’ll use the same ordering at every intersection, so we’ll always prefer going north if we can. But in this case we have only one option, so we start by going south.”

Even as Frank spoke, they reached their first decision point. Frank surveyed the options. They had come from the north, so he chose east—the next unexplored direction in his ordering. Before leaving the intersection, he retrieved a piece of chalk from his pocket and made a small mark on the wall.

“Now we backtrack to the last room, room 5, with the mold that looked like a horse,” explained Frank as they retraced their steps. This time they chose the only unexplored option from room 5, heading west. Unfortunately, they immediately hit another dead end—an empty room that sported a complex floral pattern of green and blue fuzz.

They backtracked through the most recent intersection whose options had been exhausted until they had a new option at room 4. The eastern option was a dead end, and they’d already explored the northern option, so this time they went south.

They ventured through two new empty rooms (8 and 9), differentiated only by the occurrence of a large stalactite of orange mold, which they stayed as far away from as possible. Orange mold was not known for its structural stability. After hitting another dead end, they found themselves retracing their steps all the way back to the first intersection in room 2.

“What if we miss it?” asked Socks in his now-standard worried tone. “Or what if we end up in a loop? We could be stuck forever!”

Frank groaned. “Listen, kid. This isn’t my first time depth-first searching. I know what I’m doing.”

“But loops.”

“Why do you think I’m marking the walls?” asked Frank. “If we avoid taking passages that we’ve already explored, we avoid going in loops.”

Frank had learned that lesson during a Police Algorithms exercise. With the whole class watching, Frank had done six loops of the hedge maze before he heard another student loudly joke, “There he goes again.”

They explored deeper into the maze, following snaking paths and backtracking at dead ends.

Then, in room 23, they found a small cell packed high with rolls of parchment and stacks of ledgers.

“We found it!” said Socks enthusiastically. His staff’s flame cast a flickering blue glow through the room.

Frank felt the hairs on his neck rise as he took in the scene. He compared the height of the stacks with the mountains of paperwork he had completed through the years and did some quick calculations. The captain had never been shy about dumping paperwork on him, but Frank had still never seen anything like this. There were even mold-stained pages at the bottom of the stacks. Everything felt wrong.

Frank walked to the nearest stack and pulled off a sheet of parchment: a notice on the proper use of duck fences. The date and station number marked it as belonging to the stolen files. The next sheet, listing noise complaints in the Port of West Serial, also came from the stolen collection. It appeared equally random and unhelpful.

He knelt down and pried open a gap near the bottom, yanking a ledger free. The pages were spotted with a trio of mold-butterflies, but Frank could clearly make out supply lists for the castle guards. This ledger could have come only from the castle itself. He grabbed another book and found castle guard rotations for last November.

“This is wrong,” he muttered. “There’s too much here. There’s castle ledgers as well.” Frank shifted to an adjacent pile, starting again at the top.

“Is there a pattern?” asked Socks, as though he had just noticed the extent of the document piles.

“I—” started Frank, but he pulled up short as he opened another ledger, entitled Transfer Requests. Four pages had been torn from the middle of the ledger.

“Very strange,” said Frank, flipping through the undamaged pages. “This could be—”

Frank was cut off as Socks stumbled toward him, flailing for balance. Behind him, Frank could see motion in the gloom. It wasn’t until he heard the rusty shriek of the door’s hinges that he realized what was happening.

“Door!” Frank yelled as the junior wizard fell into him.

The two of them tumbled to the ground. The door slammed. A loud click sounded as the lock engaged. Socks’s staff, which had been dropped in the commotion, spun lazily into a tall stack of dry parchment. The staff’s blue flame seemed much larger than Frank remembered.

Frank lay stunned on the stone floor as he watched the papers ignite.

What happened?” asked Officer Notation, who stood by the gate. Frank studied her as he tried to catch his breath. Was she concerned? Confused?

“We were attacked!” Socks blurted out. “We were trapped in a cell and everything was on fire! But I used a metal weakening spell so we could escape.” He looked quite pleased with himself.

“Attacked?” asked Notation. “Who attacked you? Did you see them? What did they look like?”

“No,” admitted Socks. “He snuck up behind me.”

“Frank?” asked Notation, turning to Frank.

Frank shook his head. “All I saw was Socks flying at me.”

“I bet he was big,” offered Socks. “A giant thug. And he was stealthy. Maybe a trained assassin.”

Frank rolled his eyes. “Sorry, kid. He was an amateur. Professional assassins don’t lock people in cells and run away.”

“But the fire,” said Socks.

“Your staff started the fire,” Frank reminded him. “You dropped it on the papers.”

“Papers?” asked Notation. “Did you find the logs? Do you know what they were after?”

Frank and Socks looked at each other. Notation glanced from one to the other. Finally, Frank spoke, “The logs are gone. Our junior wizard here dropped his staff and poof—fire everywhere. Any clues are gone now.”

Socks turned a deep shade of red and stared at the ground.

“Gone?” asked Notation. “Everything’s gone? Are you sure?”

“Yeah,” said Frank. He nodded toward the smoke trickling from the door.

“What about the person who attacked you?” asked Notation.

“I didn’t get a look at him,” said Frank. “I don’t suppose you saw anything?” he asked. The question came out sharper than he’d intended, but after being attacked, getting trapped in a burning room, and fleeing a darkened prison, he didn’t feel like pulling any punches.

“No,” she said calmly. “There was nothing around the front.”

“No tracks near any of the doors?” asked Frank. “Anything that could give us a clue about our attacker?”

Notation shook her head. “Nothing,” she said. “It looked as though no one had been around the other side in months.”

Frank nodded, but didn’t speak. Something felt wrong. Either the attacker had cleverly slipped by Notation through the same gate they had used, or she wasn’t telling them something. How long had she been away from the gate? And why had she stayed outside? Frank decided not to press the issue. “All right. Let’s head back to the boat.”

“Now what?” asked Socks as they made their way toward the water.

“Time to backtrack,” said Frank. “No more clues here.”

“Backtrack to where?”

“The open clues,” Frank answered. “We investigate the leads we still have left.” He paused for a moment, weighing his options. “I think it’s time we parallelize the search.”

“Really?” asked Notation.

“Parallelize?” asked Socks.

“It means we split up and explore different parts of the search space,” answered Notation. “Parallel algorithms divide up work and do that work in parallel—at the same time, that is. For example, the work might be distributed over different people. In this case, we can divide the leads into three sets. Then you, Frank, and I can each take a set of leads. We can investigate different leads at the same time, allowing us to work almost three times faster.”

“But,” objected Socks, “I’m not an officer or a private investigator. I don’t know what to do. Shouldn’t I stay with one of you?”

“No,” said Frank. “I’m still not sure what’s going on, but I have a feeling we’re working with limited time. Whoever we’re after knows we’re on the case now and knows we’ve tracked them this far. If they’re smart, they’ll start destroying the rest of the evidence.”

“It’ll be late by the time we get back to Usb,” noted Socks.

“We can split up tonight and meet at my office tomorrow morning,” said Frank. “That should give us enough time to follow leads and possibly grab some sleep.”

“Okay,” agreed Notation. “How do we divide up the work?”

Frank knew the key to an efficient parallel algorithm was making sure that the benefit of using multiple workers was worth the cost of dividing up the work. Parallelizing work involves a certain amount of overhead. The problem needs to be broken into pieces. Each worker has to get a task, get ready to do it, and then actually do it. And then at the end, the work has to be recombined. Parallelizing a simple task can sometimes be more costly than just solving it outright. However, when the problem gets large enough, parallelization can greatly accelerate an algorithm.

“Easy,” said Frank. “Socks, I need you to talk to your wizard friends. Ask them if they know about a group called the League of something. The thugs on the boat said they were working for a league before Rebecca Vinettee interrupted them. Judging by previous cases, the name will be something evil like the League of Power-Hungry Maniacs or the League of Darkness. Evil leagues tend not to be subtle with their names. Find out everything you can about this group.”

“That’s not much to go on,” Socks complained.

“Notation,” continued Frank. “I need you to pull all police transfer records for the last six months.” While he didn’t like the idea of leaving this lead to Notation, she was the only one who could get the records easily. If he tried to collect them himself, he would be met with suspicious looks and a small barricade of paperwork. The capital police department used paperwork with the same determination and effectiveness as roadblocks.

“Transfers?” asked Notation, clearly surprised. “Why?”

“Call it a hunch,” Frank lied. “We’ll meet at my office tomorrow morning and combine our information.”

“What about you?” asked Notation. Her voice now carried a note of irritation. She obviously knew Frank wasn’t telling her the whole story.

Frank gave her an innocent smile. “I have to go shopping.”

As the TCP Flyer slowly cruised back to Usb, Frank found an out-of-the-way corner on the deck and sat down to think. This was the part of the investigation he hated most, when promising leads started drying up or, in this case, burning up. The loss of a crucial clue always filled Frank with a sense of dread—like he was running one step behind. Frank forced the doubts from his mind and refocused on the clues he did have. The voyage to Usb would give him time to comb through what he had seen and find connections he had missed.

He closed his eyes and took a deep breath.

“Oh. Sorry. Are you sleeping?” asked Socks.

“No. Thinking,” Frank said, and congratulated himself for not yelling. After all, the boy had saved his life.

When Socks didn’t say anything more, Frank prompted, “What do you want, Socks?”

“Um . . . I was curious about the search,” Socks responded.

“How so?” said Frank.

To Frank’s dismay, Socks walked over and sat next to him.

“Do you think we’ll find the criminals?” asked Socks.

Frank shrugged. “We’ve still got some good leads,” he offered.

“But do you think we’ll be in time?” asked Socks.

Alarm bells went off in Frank’s head. He swiveled and stared hard at Socks. “Time for what?”

Socks nearly fell backward. His eyes darted around, as though searching for an appropriate answer. “Whatever they’re planning?” he finally stammered.

Frank didn’t buy it. “What else do you know?” he asked.

“Nothing,” replied Socks. “At least, nothing concrete. It’s just speculation. Not mine—my mentor Gretchen’s. She has good insight into these types of things, though.”

“Which is?”

“I really shouldn’t say anything. It’s just speculation.”

“Which is?” Frank growled.

“She thinks that whoever is behind this is going to attack the castle in a few days.”

Frank leapt to his feet. “Why didn’t you mention this sooner?” he shouted.

“It’s just speculation,” repeated Socks.

“No. Not entirely,” said Socks. “It’s based on the stolen mask. Magical artifacts are most effective during the full moon, which is in two days.”

“What exactly does this mask do?” asked Frank, starting to pace anxiously.

Socks hesitated for a moment. “It’s an incredibly powerful artifact,” he started. Upon seeing the angry look in Frank’s eyes, he sped up. “It’s officially called the Mask of Combinatorial Looks. It was lost hundreds of years ago during the Great Slug War. Everyone thought it was destroyed until Princess Ann recovered it during one of her quests. She found it—”

“What does it do?” prompted Frank.

“It allows the wearer to look like anyone else. The scholars believe that it uses a massively parallel search. Each feature runs its own search to find the best match. The nose will transform into a perfect match of the target’s nose. The eyes will transform into—”

“A perfect disguise,” offered Frank.

“Yes,” said Socks.

Frank cursed. “And the castle? Why does Gretchen think they’ll attack the castle?”

“She didn’t say,” admitted Socks. “Maybe that part was a guess,” he added without any real conviction.

Frank didn’t believe it either.

“I’m sorry I didn’t mention it earlier,” offered Socks. “Since there isn’t any hard proof . . .” He trailed off, looking miserable.

“What else aren’t you telling us?” asked Frank, staring down at Socks.

Socks thought about the question for a long while before answering. “I think that’s it.”

“Everything?”

“Everything I know,” Socks qualified.

Frank took a deep breath and looked up at the sails, wishing they were fuller. The wind had died down within the last hour, and the TCP Flyer seemed to be barely inching toward their destination.

He ran the timeline of the next few days through his head and wondered if they had enough time. Even with three of them searching in parallel, there was no guarantee they would cover enough ground. Worse, they couldn’t even begin the parallel search until the TCP Flyer docked. Until then, they were all stuck on the boat.