1

In the town of Warren City, Indiana sits Warren City University, one of the Midwest's most curious colleges. A particular curiosity can be found in the mathematics department here, behind a door emblazoned with the text "KARL W. KAIRO, PH. D. PROFESSOR OF DISCREET MATHEMATICS." To the casual observer, of course, this appears to be a typo. But no, the word "DISCREET" was Professor Kairo's own choice, being a play on words and on the fact that he teaches cryptography, among other courses.

Behind this door one will often find Professor Kairo himself. He is known to his colleagues and students as quite eccentric, but overall one hears him described as a friendly, helpful professor. Of course, what one would first notice about the professor is nothing pertaining to his personality, but rather his appearance; Professor Kairo is a cat.

But this is nothing out of the ordinary at Warren City University. You see, Warren City resides in a universe in which humans, in the sense of Homo sapiens, do not exist. Instead, the world is populated by a variety of intelligent animals, with the ability to speak, walk, wear clothing, go to work, and all of the other boring things that humans would do. Dogs, cats, foxes, deer, wolves, sheep, goats, lizards, and many other kinds of creatures fill the gap left by the absence of a single human being.

Professor Kairo, for instance, stands at six feet tall, covered from head to tail in yellow, striped fur. His arms and legs end in paws somewhat less like those of a cat and more like the hands and feet of a human; he can pick up objects and use machines with ease, having opposable thumbs. His ears are cat ears, and he does have the tail of a cat, about three feet long. His eyes are a vibrant green and unfortunately not sharp enough to allow for reading; for this reason, Professor Kairo wears glasses.

Professor Kairo teaches a variety of animals the art of numbers, including cryptography, combinatorics, abstract algebra, and other equally bland-sounding fields. However, he makes an effort to make his class anything but bland for his students; once, he brought a box of donut holes to class to cut them up and make a point about the Banach-Tarski paradox. The students may not have all remembered the Banach-Tarski paradox, but they never forgot the donut holes.

Of course, Professor Kairo has also developed another side to his reputation by holding a second job of sorts; he considers it a hobby, but the local authorities depend on him more than he'd like to admit. Professor Kairo solves crimes. Some combination of his mathematical intuition and his catlike mannerism has rendered Professor Kairo a respectable sleuth, and it is thus fortunate that he stumbled upon an opportunity to discover and use this talent in the first place.

This story begins a few years back, one fall semester on November 1. It was a Monday, the day after a long weekend, and Professor Kairo was in his office in Holden Hall, just as usual. Suddenly, his ears briefly twitched, and he spoke. "Hello, Arthur."

"Hello, Karl." The intruder was Arthur Barclay, a custodian at the school and a friend of Karl's. "I'm surprised you knew I was here. You left your door wide open."

"Have you forgotten, Arthur?" asked Karl, turning his chair around to face the dog-shaped figure. "I'm a cat."

"Right, right, of course. How has Monday treated you?"

"So far, it's a bit ... meh, you know. Monday."

"I can tell. How was Halloween? Did you go out in costume?"

"I would have liked to, but you know, I'm kind of busy these days. Four math classes don't teach themselves. You?"

"Well, I'm not sure if they would have liked to see me creeping around the campus in the middle of the night. You know."

Karl surveyed Arthur's appearance. Arthur was a beige dog of average height and build, with a smile on his face and a broom in his hand. He wasn't necessarily the best-kept, his fur roughly growing to one side or the other in a somewhat haphazard manner. But it was Monday, after all, and Arthur wasn't always the type to worry about making a stunning presentation. Arthur was still a Golden, and a Golden could look like a friendly sort no matter what. Though perhaps in this day and age,_Karl thought, _it's best not to say that. Race relations and all.

"Eh, don't be so hard on yourself," he responded at last. "I mean, yeah, you bring up a good point. But maybe next year you and I can go around together, huh?"

"That would be fun. Well, we'd need an idea for a costume. Two costumes, that is."

"Well, we have a whole year to think of one. In any case, is there a reason you wanted to see me?"

"No, no, not really. Just checking to see how you were, that's all."

"Ah, okay then. Well, I still have a lot more work to get done, you know, papers to grade, so ..."

Before Arthur could respond to this statement, both of their ears picked up the sound of a rather large individual attempting to hold back tears. Karl got up, curious, and walked to the door. The sight that came to his eyes was that of a very sad, gray elephant, slowly treading down the hallway to the office of the department chair. "Ronald?" he said to the elephant, whom he recognized as the president of the university. "What's wrong?" But there was no response, apart from a vigorous shaking of the head as the pachyderm continued to sob his way down the corridor.

Arthur started to follow Ronald Grant down the hallway, but Karl stopped him with a quick word: "No, Arthur. Leave him alone right now. We'll find out what's wrong soon enough."

It turned out that soon enough meant within ten minutes, after which the department chair, a Dr. Fred Douglas, called for Professor Kairo to enter his office for a short discussion. Karl walked slowly past the other math offices, as if uncertain if he wanted to know what news had so deeply afflicted Dr. Grant. But he eventually entered the door and sat down. The room was darker than most of the offices, as Professor Douglas preferred to work in a dimly-lit area, and very quiet, as he could not stand unnecessary noise. Karl also noticed a distinct lack of smell, as even the slightest odor could offend Dr. Douglas's hound nose.

After closing the door, Fred finally broke the deafening silence: "Paul Chlebek has passed away."

"What?" Karl was in shock; Paul had been one of his younger students. "What happened?"

"Karl, I'm not sure how well you'll take the answer to that question." Fred always seemed to be eerily calm in situations like this one. "Nevertheless, I feel you should know: it's believed to have been a homicide."

Karl was in more shock. "But ... wow, okay. I'm ... very sorry to hear this."

"The important thing, Karl, and the reason I called you in here, is that you need to be able to explain this to your students in a way that will reassure them that we want Warren City University to be a safe place. Understandably, your 10:00 graph theory class will be quite shaken to hear."

"Right. Paul wasn't in class today. I had no idea ... wow. This is ... really sad."

"Karl, tragedies eventually strike all of us. Those who remain strong in the face of tragedy are able to make a positive difference."

"Well, I don't think I can remain strong, Fred. This is big."

"Karl," Fred continued in an almost therapeutic tone, "the key is that you don't have to feel_strong. You just have to _look strong, and this will cause others to think you _are_strong. And if you keep putting on a brave face, it will make you brave."

"I hope so. I--can I have some time alone to think about this?"

"Of course. I only wanted to make sure you were informed. That is all."

"Well, one other thing." Karl had a thought. "Have the police--are they looking into this? Have they found anything yet?"

"Not yet, Karl. From what I'm told, the body was discovered only hours ago. We don't have any other news. But we will soon, hopefully."

Karl slowly walked back to his office. "Well?" asked Arthur, who had been waiting in the hall.

"You know Paul? He's one of my students."

"Yeah. I was just talking to him last Friday. He was telling me that--"

"He's dead."

"... Oh. Oh, no ... Well, I should get going."

"Yeah, I'm going to need some time alone as well." Karl stepped back into his office and closed the door, a little too forcefully.

Those who remain strong in the face of tragedy are able to make a positive difference. Karl pondered the dog's words. The goal, it seemed, was "to make a positive difference"--but was merely remaining strong enough? Was merely remaining calm and informative the best Karl could do for the memory of the poor kid? The police hadn't found any clues yet, or so Fred had told him. And though Karl knew they probably wouldn't want him snooping about the crime scene, he could provide a unique insight. Paul had been a talented individual--he had not only been passing Karl's class at the time of death, but passing with flying colors. What Karl had always recognized in Paul was a passion for mathematics--a passion that could perhaps have led to his demise. It was a stretch, but if Karl could contribute to the investigation, he had to do so soon.

2

"So explain to me exactly why you think it's a good idea to drive straight to the crime scene," said Arthur from behind the wheel of his slightly rusted Buick.

"I just feel like there's something I can contribute," replied Karl, in the passenger seat. "You know, being one of his professors and all. The kid had a real interest in math."

"Yeah, but what are we going to do? The police likely won't even tell us anything."

"That doesn't mean we can't get any information. You'd be surprised at what can be learned by simple observation."

"Aren't you concerned about ethics? I mean, the kid has a right to privacy, doesn't he?"

"The dead don't need rights, Arfie." It was an old nickname, from the days when Arthur used to have a strange accent; it had rendered him unable to pronounce certain consonants quite correctly. "What are they going to do, complain to us? Besides, having information isn't a sin. It's what you do with it that matters."

"Whatever. As long as you have some way of getting information without us winding up in jail in the process."

"Hey, here's an example: how did I know where the crime scene is?" Karl said this as they approached the house of the Phi Upsilon Rho fraternity, with yellow police tape surrounding the building. "Nobody told me it was here, but I knew where to tell you to go."

"Well, that's easy," Arthur responded. "He was murdered last night, so it was probably wherever he lives. He lives in this house."

"You see? It's easy."

"But--" Arthur felt there was something wrong with Karl's argument, but he couldn't put his finger on it. Deciding not to question further, he proceeded to park the car.

As Karl walked up to the house, Arthur followed behind him hesitantly. Karl seemed strangely calm and confident the whole time, not like someone who had just lost an acquaintance and was walking toward the scene of the murder. Was there something wrong with Karl? Had he gone insane? Arthur shook slightly at the thought. Finally, the two of them reached the tape at the front stoop, where a police officer was looking into the house. Inside, three more officers could be seen walking about, making observations. "Excuse me, ma'am," Karl spoke to the policewoman, "but is this the scene of Paul Chlebek's murder?"

The officer, a tall, white ewe, gave him an icy stare. "Yes, sir, but we're not willing to let anyone in at this time," she said.

"That's fine, I don't need to get in at the moment. Just, have you found any good leads?"

"Sir," she replied sternly, "we will release any relevant information regarding the case when it is appropriate to do so. Now it would be appreciated if you could please leave the scene, as you're not likely to be of much help, and we're very busy."

"Well, I wouldn't make assumptions," Karl replied, almost smiling. "For instance, I'm one of the professors that Paul had in class. So if it turns out that he had had a dispute regarding some math assignment that ended in death, you know, I might be the only one who knows the relevant--"

"Okay, look, are you _trying_to get on my nerves?" the policewoman interjected. "I highly doubt that any of your information would be of use to us, and even if it were, we would find you out and question you; we wouldn't need you to be waiting there, begging for us to ask you a question."

"Right, right, but at the same time, I might be able to think of things that you wouldn't. For instance, you, um, you ..." Karl searched for something on the officer's person that would reveal some intimate detail about her with a bit of logic. It was a technique he had picked up from a few detective stories Karl had encountered as a child. Finally, an idea came to him: "... you, uh, you have a conservative political outlook!"

"What? I'm sorry, but how is that relevant?"

"Well," Karl continued at a rapid pace, "they say that you can tell by looking at a person's eyes whether they're more left-leaning or right-leaning, politically speaking, and you've been staring me down the entire time. If you were more of a liberal, you would be following my gaze every time I looked in another direction. Of course, there's also the chance that you're just trying to keep an eye on me in order to make sure I don't leap over the tape or anything crazy like that, in which case good job with that, but at the same time it means you're less able to see what's going on behind you, which could be an issue if one of the other officers had just discovered something important, maybe?"

Arthur, meanwhile, was attempting to tug at Karl's arm. Upon his observation that the officer was growling at the two of them, Arthur had decided it was time to get going.

"I'm not trying to get her to help us, you realize," Karl whispered in response to Arthur's gesture. "I'm buying us time. Play along. Well," he continued, his voice back to normal, "I'm realizing that yeah, I am being a bit of a jerk by sticking around, huh? So before I go, I'd just like to take a moment to apologize. Really, I'm sorry for wasting your time, Officer. Right, Arfie?"

"Uh, right!" Arthur chimed in. "We're very sorry to disturb you, Ms. ..." (he glanced at her badge) "Caulfield. Officer Caulfield, that is. Very sorry. We'll be leaving now." He turned away, intent on ditching the scene before both of them were in cuffs. But before he got down the steps, and before Karl made a point to follow him, a voice called out.

"Is that Arfie Barclay I hear?" the voice came from the house. Arthur stopped and turned around in time to see a large, white ram walk straight out of the front door. "Hey, pal! It's been a while, don't you think?"

"Sonny!" Arthur responded. "Where did you come from? I haven't seen you in years!"

"Yeah, well, it's Officer Abel now," Sonny responded, pointing to his badge. "I just finished up training at the police academy, and what do you know, the first thing I get is a murder case. Pretty exciting, huh? Though, you know, sad and all. Still, it beats parking duty. How about you? Still sweeping the floors at WCU?"

"Officer Abel," said Officer Caulfield, "these two have been trying to get information or something, or just irritate me, or--well, I don't know what they're trying to do, but they've been nothing but questions and trouble. I would suggest we don't encourage them?"

"Eh, don't worry about it," Sonny waved it off. "I'm sure Arfie's harmless. Anyway, Arfie, how's life?"

"Well," Arthur responded, "look, I really would rather not discuss life events at the scene of a crime while you're on duty. So how does a coffee on, say, tomorrow afternoon sound?"

"That'd be great. We could maybe go to that new local place, huh? Seagulls or whatever it's called?"

"Sounds good to me," Arthur replied with a smile, his expression clearly indicating that he wanted to wrap things up quickly, before Officer Caulfield sprang on the two of them. "Four o'clock?"

"Yeah, that works. See you then!" Sonny cheerfully yelled as Arthur started to turn around again. Karl saw Officer Caulfield give Sonny a disdainful stare as he turned to follow Arthur.

"Well, that went better than expected," Arthur said once they were finally in the car.

"You know, Arthur, I'd like to say that I knew he was your old friend," Karl replied, "and that I made sure to spend enough time that he would notice you and set up a coffee date. But sadly, that was a happy coincidence."

"So what were you buying time for? Have you figured out who Caulfield voted for in the last five elections?"

"Heh, no. What I _do_know is that Paul's body was found in the front hall closet but has since been taken to Warren General Hospital for an autopsy."

Arthur quickly shot a surprised glance at Karl. "How, exactly?"

"Well, the officers seemed to be crowding around a small room near the front of the house. That wouldn't be very helpful if not for this," Karl said as he pulled up a map of the house on his smartphone. "Thankfully, the ??? website for its WCU chapter has a map of the fraternity house on it. This allowed me to easily identify the room. If we can get back to a computer with Flash support, we can take the virtual tour and see it firsthand. Without the body, of course. Meanwhile, there were no doctors or medical professionals on the site, even though we know the police have been investigating for hours. This means, quite simply, that the body has already been removed from the premises for further investigation. Plus, do you really think Sonny would be hanging around if there were still a corpse in there?"

"No, there's no way he would be able to stomach it. Though to be fair, you knew nothing about Sonny before a few minutes ago."

"Right, well, anyway, the last part, the fact that the body went to Warren General Hospital, is simple. It's the only medical center in the city with a morgue."

"Okay, fair enough. You're smart. I'd expect that from a math professor. What are we going to do about it?"

"I'll tell you what we're going to do. Your friend, Sonny. How bright is he?"

"... No, we're _not_going to take advantage of my friend for being dumb."

"Don't think of it that way. We're getting information, nothing more. Besides, Sonny knows the limits. He knows what he legally can and can't tell us."

"I wouldn't be so sure ... but you bring up a good point."

"All I'm saying is this: you go to your meeting tomorrow and let the conversation go naturally. With any luck, he'll bring up the murder case on his own. If not, well, there's no harm in steering a conversation, is there?"

"I'm not sure if I like how this is sounding more and more like a criminal plot in itself. But if it means discovering some crucial piece of evidence--if we can determine who the killer is thanks to your help--then I take back everything I've said about you, Karl."

"Surely not everything. You've said plenty of good things about me."

"Very funny. Now what are we going to do before tomorrow?"

"Well, we have two options. One, we wait around and think this all over. Two, we drive to Warren General Hospital and see what we can see."

Arthur started the engine. "Hey," he said, half to himself, "it's not like I had anything exciting planned for tonight."

***

They had been in the hospital parking lot for an hour, having seen next to nothing. Arthur was seriously wishing he had picked Option #1. "Okay, I'm really getting sick of this," he finally said. "We've been here an hour, and we've seen nothing."

"On the contrary," Karl replied, "we've seen a lot. Maybe nothing that appears to be useful, but looks can be deceiving."

"I think you're just looking for information when there's nothing there. We haven't seen anyone connected to Paul at all."

"The world is more connected than you think, Arfie. Here," Karl said as he procured a pack of cards from the backpack he had with him. "Let's play a little game. Hmm, I need a surface ... here." Karl pulled a heavy textbook from the backpack and set it between their seats. He then placed the cards, face-down, on the textbook. Closing and covering his eyes, Professor Kairo continued, "Pick twenty cards from the deck, turn them face-up, and then place them back into the deck however you wish. I won't look."

Arthur did so. Thinking this might be some psychological trick, he attempted to place cards in unintuitive places, consciously going against his instinct. The result was a deck containing some packets of five or so face-up cards, with a few single face-up cards scattered throughout. "Okay, I'm done," he responded.

"Now, I claim that without looking, I can divide the deck into two piles containing the same number of face-up cards."

"But ... that's impossible. You'd have to know where I placed them."

"Watch me." Professor Kairo counted to himself, drawing twenty cards off the top of the deck. After flipping them over, he set them aside as a second pile. "Now let's see how I did," he continued, opening his eyes at last. There were fourteen face-up cards in each pile. "You see? There were six cards face-up in the twenty that I pulled away. But flipping that pile, I now have fourteen face-up cards and six face-down, and the fourteen perfectly match the fourteen face-up cards left in the rest of the deck. If you know what to do with the information you have, the holes all fill themselves in."

"Well, that's cheating," Arthur replied. "You didn't tell me you were going to flip the cards."

"And I didn't tell you that I wouldn't. That's the other lesson here: if you can avoid it, never make an assumption."

"Well, that's a cool trick and all, but I don't see how it ties in to what we're doing at this hospital."

"The point is that you can gather information and manipulate a situation while remaining outside parts of it. We weren't allowed into the crime scene, and we wouldn't be allowed in the hospital. But from what we can see, we can draw conclusions about what we can't see. For instance, there are entrances and exits to the hospital that we can't see from here. But in the past hour, I can tell you which people have likely entered or left the hospital from those entrances and exits. Of course, it's not perfect--they could also have entered from the doors we see here, but before we came. In any case, take a look at this."

Karl gestured at a figure exiting an unmarked white van, wearing a business suit. "Who is he, Arfie? Why is he here?"

"Well, I don't know. An unmarked white van seems pretty sketchy to me."

"Yes, but someone sketchy wouldn't be wearing a suit and wouldn't be allowed into a hospital. No, this is someone who wants to be here without drawing attention."

"Is he trying to avoid the media?"

"I would conjecture that he is the media. We'll see in a few minutes in any case."

Sure enough, the figure soon left the building and returned to the van. Two more figures came out, setting up a light and a camera. They handed a microphone to the one in the suit.

"Now, Arfie," Karl continued, "if I were the kind of person with access to expensive technology, I might be able to intercept the broadcast and hear it live, without having to drive somewhere with a television. However, I'm a college math professor. So roll down your window and be quiet."

Arthur did so, and the two of them could faintly hear the reporter's words:

"... hospital where, according to hospital authorities, the autopsy is underway. We have no information from the autopsy as of yet, but we'll keep you informed as soon as we do. In the meantime, eyewitnesses from the crime scene report seeing the body carried off with a large gash in the head, probably from some form of blunt weapon. Police at the Phi Upsilon Rho house are meanwhile trying to wrap up their investigation, hoping to have the fraternity members safely in their beds by Wednesday evening. This is Phil Delgado with WCC News. Back to you, Frank."

Arthur rolled his window back up. "So it was blunt trauma to the head. Nothing unusual."

"At least we've eliminated other possibilities," Karl replied. "Eliminating possibilities is always key."

"So you think we should stick around until they finish the autopsy?"

"That's going to take a while, and most likely it's just going to confirm what we already know. At this point, it's not worth it to hang around here."

"Wait, wait. Then why were we spending an hour here before? Did you know they'd do a news report or something?"

"I wanted to come here to figure out a cause of death. We have that information already. Mind you, at this point I'm realizing we could have stayed home and watched the news."

3

Neither Karl nor Arthur could sleep terribly well that night, for the obvious reason. But the next day, Karl got up as usual and rode with Arthur to work. While he wished he had a car of his own, at the same time Karl knew the value of forming connections--and of saving gas. The class that Karl was dreading, Graph Theory at 10:00 AM, happened. At the beginning of class, he got up in front of his students and prefaced his daily lecture with a eulogy.

"Class," he began, "before we continue with the material today, I think we need to get something out of the way. You are aware, I'm sure, that Paul Chlebek was not in class yesterday and is not today. Hopefully by now you have all heard the tragic news. In my experience, Paul was a great student, and I'm sure you will be able to tell me from your experience that he was a great friend as well. And I think you'll agree with me that even though we all want to take a day off--even though we all want to go home and mourn--it's not what Paul would have wanted. Paul always encouraged his fellow students to study and to perform well, not because he wanted them to score high on tests, but because he wanted them to learn. So the first thing I think we all need to keep in mind is that we aren't here to score high on tests. We aren't here to get the right answers, get the credit hours, and get a little piece of paper with B.S. on it. We're here to learn, and the grades, the credits, and the diploma are all just signs of what we've learned. Secondly, and I'm sure Paul was a great example of this, and this is why his loss hurts our community so badly: learning is best done together. You can pick up a subject by yourself, yes, but only when you're around friends who share that desire to learn--only then do you learn the most deeply. So in order to remember Paul, let's keep this in mind. And I know I sound like I'm being a teacher, twisting Paul's death into a lesson about doing your work in class. That's not what I'm trying to say. I legitimately believe that if Paul could see us right now, he wouldn't want to see us crying. He'd want to see us carrying on without him, continuing to learn so that someday we can accomplish what he never got a chance to do."

There was silence, then after a few seconds, one student raised his hand. "Yes?" Karl asked, motioning for him to speak.

"Dr. Kairo," the student began, "if we're supposed to accomplish great things, then why are we taking graph theory? If we're going to make a positive difference in the world, how is this class going to help us? How is learning in itself going to accomplish anything? We're working with a bunch of dots with lines connecting them, and what are we going to do with these lines? Maybe we can figure out how to color maps with them, but is that really important enough? Would Paul want us to be coloring maps in his memory, or would he want us to be saving lives, or helping the sick, or feeding the poor?"

"Well," Karl began by saying, "every branch of mathematics has an application. Maybe we haven't found it yet, but it's there. Let me tell you a story. In the early 20th century, mathematicians wanted a formalized, logical basis for all of mathematics. They wanted a way to be certain that when they said 'let x be a real number' there was a way to prove that x could_in fact be a real number, and that 'real numbers' even _exist, and so on. Clearly these facts seem obvious to us, but the goal was to create a system in which intuition was never necessary--nothing would be said to be true 'because it's obviously true.' I'm sure you will agree with me that this seems to be a relatively worthless pursuit. Why do we need formality? We all know what we mean when we say 'let x be a real number,' and there's no need to clarify past this. But some thought otherwise, and thus the search for a foundation to all of mathematics took place.

"It wasn't easy; Gottlob Frege, for instance, seemed to think that a set of whatever concepts you could imagine must exist, and Bertrand Russell notified him that this resulted in a paradox, rendering Frege's entire system completely unusable. Finally, Kurt Gödel, in the early thirties, struck a fatal flaw in the entire concept: one, he proved that no useful formal system of mathematics can prove or disprove everything; two, he proved that no mathematical system can be so intuitive as to justify its own usage. One last question remained: could there at least be a way to encapsulate these shortcomings, to put them into a box whose boundaries could be determined? Was there a formula or a set of steps to determine exactly which statements _could_be proven and which were left to the void of futility?

"The answer, it turned out, was no. To add insult to injury, not only are there mathematical statements that can't be proven or disproven, there's no method to determine, on the whole, exactly which these are. The futile game of abstraction was ended. But as I stated before, this is a tale of abstract mathematics suddenly finding relevance. Because in order to prove this fact--in order to recognize that there's no method to figuring out what we can prove--we first had to define what a method_is. We needed a mathematical way of describing the notion of a _set of instructions. And this idea, the idea of talking about instructions in mathematical terms, was what gave rise to the formal definition of an algorithm, or a program, and the mathematical backdrop to all of the computer software that we use in our computers and cell phones and calculators.

"So whenever you look something up on the Internet, the server uses an algorithm to find what you're looking for. And the person who wrote that algorithm knew how fast it was by mathematically comparing it to other algorithms. And algorithms can only be mathematically compared if there's a mathematical definition with which to compare them. And this is how a mathematical 'wild goose chase,' begun at the start of the twentieth century by pure math nerds who didn't need applications to justify their work, ended up giving rise to a wealth of mathematical knowledge instrumental in the development of the devices we all use on a daily basis.

"So does all of our classwork have applications? Yes. Yes, it does. Are all of these applications known? No. But sometime in the future, you may find that graph theory is crucial to some aspect of your life, and you'll be glad to know it. And when that happens, hopefully, you'll remember Paul."

A few students began to applaud.

***

"So tell me, Arfie," said Sonny. "How's life been? Are you still stuck in that janitor gig?"

"Well, I don't think of it as being stuck," Arthur replied. "I mean, I like my job. I get to know the students and professors and stuff."

"Well, as long as you're making enough to get by, right?"

"Yeah, money's not too much of a concern. I've got a little place in the apartments downtown."

"Good to hear. How about love, huh? Got a wife?" Arfie was shaking his head. "Girlfriend? ... Secret admirer?"

"None of the above, sadly. Well, not to my knowledge. You?"

"Ah, don't feel bad about it. Me, I've got a girl, been dating her for three weeks, and already my wallet's almost empty. Say, speaking of which, the coffee's on you, right?"

"Oh, uh, sure. Your job's going well, though? You always wanted to be a cop, I recall."

"Heh, well, I'm excited about it now. For the past 20 weeks I've been bored out of my mind at the academy, but now that that's over I can move on to the real stuff. Right?"

"To be honest, I'm a bit surprised that they assigned you to a murder case first and not some boring patrol job."

"Yeah, so am I, but you know, bureaucracy. They don't care who gets assigned to which job as long as they have X people to do job Y, you know?"

"Makes sense, I guess."

"It's a weird case, though. This guy just gets found dead in the morning, and the weird thing ... well, never mind."

"What?" Arthur was eager to hear potentially useful information.

"Nah, it's not the kind of thing to talk about in a restaurant."

"Well, dang it, now I'm curious."

"Oh, what the heck." Sonny lowered his voice, almost to a whisper, so as not to disturb the other coffee-drinkers. "The weird thing is that we found blood all over the goddamn place. Trails between nearly every room on the first floor of the house. Well, not the bathrooms. But the dining room, the kitchen, the lounge, everything."

"Everything?" asked Arthur, in a similarly low voice. "Like, every room?"

"Every door, in fact. Trails of blood through every doorway between rooms. Two of them in one of the doors. Can't remember which, and I don't think I want to remember. Really gruesome."

"Now why would a murderer leave a trail of evidence all over the house?" Arthur said, half to himself.

"Beats me. I mean, maybe it shouldn't. After all, we're talking about a bunch of drunks at four in the morning."

"Four in the morning, huh?"

"Dang it!" Sonny almost shouted, startling Arthur. "That's supposed to be confidential. Say, don't tell anyone I said that. Eh, who cares, the coroner's report will be out soon enough anyway. We think he died around 3:47 a.m. Man, remind me not to do that again."

"Well, I'll keep it in mind, I guess," Arthur replied, not quite sure what else to say. "Hey, speaking of time, I think I've got to get going. See you some other time?"

"Yeah, definitely. Keep in touch!" Sonny said as Arthur was already beginning to leave the shop.

***

"Good boy, Arfie," Karl said jokingly after Arthur told him the news. They were in Karl's office, his classes over for the day.

"So what does this give us? We know the time of the murder, but we don't have a location."

"Unless there are exactly four rooms on the main floor that aren't bathrooms that have an odd number of doorways connecting to other non-bathroom rooms, and two of these rooms are adjacent. Then we're in luck."

"... I'm sorry, what?"

"I'll explain in a moment. First let me print out the map that I found the other day."

Karl left the room. Arthur sat in the office for a few moments. The clock on Karl's desk ticked, the seconds passing slowly as seconds tend to do whilst one is waiting. Arthur admired a couple of the mathematical posters on Karl's wall, wondering what equations or structures could have led to the dazzling patterns depicted. Meanwhile, through the doorway Arthur could hear the copy room printer whirr to life, print out a single page, and cool down. After a few seconds of silence, there was a moderately loud "YES!" from the copy room. Then, Karl hurriedly walked back into the room, his tail bouncing behind him. "Paul was murdered in the library," said Karl, closing the door.

"And how exactly do you know this?" Arthur replied.

"Well," Karl began, practically panting, "you see ... okay, first, picture a house. You walk through every doorway in the house, and at the end, you're back where you started, and you've passed through each doorway exactly once. Now, can any of the rooms in this house have an odd number of doors?"

"Well ... um ... no?"

"No! Because if you enter a room, you have to leave it, unless it's the room you started in. Every room gets entered as many times as it gets left, so there has to be an even number of doors. Now, knowing this, there are four rooms on the main floor, bathrooms aside, with an odd number of doorways to other rooms. See which ones?" Karl placed the map in front of Arthur.

"Well, there's the library, the kitchen, the dining room ... and the closet."

"Right! So of those four, two of them have to be where the trail started and ended. The other two, well, you said there was one doorway with two trails through it."

"That's what Sonny said."

"So in two of the rooms, we have to count a doorway twice, changing odd to even or vice versa. But there can only be two odd rooms in the end, because the trail has to start somewhere and end somewhere, and every other room is entered and left an equal number of times. The only way to change two adjacent rooms and have two odd rooms left is to change the kitchen and dining room from odd to even, leaving the closet--where the body was found--and the library. See? Graph theory can be useful." Karl slumped back in his chair, short of breath.

"Right, but ... that's mostly coincidence. I mean, you would be totally lost if it were another two rooms. It's only because the layout happened to work out that way that you were able to find the right room."

"But without knowing graph theory, we would have come to this point and been stuck still. It wasn't sufficient_to help us out, I'll give you that, but it was _necessary."

"But how does knowing the location help us at all? Are we going to go there and find the murder weapon with some kid's name engraved on it?"

"Well, you never know. It can't hurt to have more information."

"Right, but why get so excited over it? That's just what I'm not understanding."

"I'm a math professor, and I just used math to learn something I wouldn't have realized otherwise. That's the beauty of it."

"Right, well, we could debate this over and over, but I think the real question is what we're going to do about it. That is, with the information we have now, what are we going to do next?"

"Well, that's a very good question. I suppose we could try to get information by interrogating the fraternity members."

"No. I mean, how? I just feel like you'll raise too many eyebrows if you go around confronting every member of the fraternity."

"I'll make up an excuse. I'll say that it's, um, counseling? Or maybe that there was a test that I think Paul ... no, that's not going to work. Hmm ..."

Arthur and Karl thought for several seconds of silence. Finally, Arthur said "Ah! How about this: you want to make a compilation of all of the best things about Paul's life, in his memory."

"... I like it. Only ... that would imply I actually have to go through with the compilation."

"And? It's a small price to pay for information that you say is going to tell us who killed the guy."

"Right, well, I guess that's the sacrifice I have to make for a friend. And, you know, if this doesn't work out, it'll be a nice thing to have. You know, to remember Paul by."

"Right, right. It really is sad and all, having him suddenly go like this. I'm sure it's worse for you, since you knew the guy a lot better than I did."

"Yeah ... anyway, I should type this up. Send out a message to all of the fraternity brothers."

"Isn't it a bit cruel to invite them into your office to have a talk and then have it turn out to be an interrogation?"

"Don't think of it as an interrogation. It's not. We'll just, you know, talk about Paul, and whatever information comes out of them, I'll have it. Like with Sonny, you know?"

"... I guess. I have a bad feeling about this, but ... whatever. Your idea, not mine." Arthur got up and left the room.

With the ticking clock as his only auditory company, Karl sat at his computer and typed the following message:

Dear Sir,

I am sure that you are as grieved as I am over the passing of our beloved friend, Paul Chlebek. As one of his professors, I knew Paul to be a brilliantly-minded individual with a passion for his education as well as his friendships, and we will all miss him dearly.

I have decided to create a compilation of our memories of Paul, as a sort of list of stories and such. For this reason, I would like for all of you to stop by my office at some point in the next week or two to talk about Paul and what about him made his life a wonderful one, tragically cut short. I will be available during most of the hours that I am not teaching, during the school day. I look forward to having a chat with you between your classes.

Sincerely,

Professor Kairo

_ _

4

"Come in," Karl said in response to a knock upon the door. The door was opened by a thin figure, a gray mouse some five feet in height. "Ah. Joey, I presume?"

"Yes," Joey replied as he limped into the room. "I ... got your message."

"Sit, please." Karl gestured at the seat across from his desk. "I understand this is a hard time. If you need my comfort, I am here."

"No ... no, I'm okay. It's ... well ... you know how it is, we're all a bit shaken."

"Right, right. Well, I'm sure you've heard of my idea. I've heard of your creative tendencies before."

"Creative ... tendencies?"

"Well, you've been the designer of your fraternity's newsletter for years, I've heard."

"Oh, yes. ... Right. So ..."

"So I'm wondering if you want to be in charge of the graphic design of this project. You know, maybe draw a few pictures to illustrate the book? Could you maybe do that?"

"Yes, yes, I'll draw ... some lines. ... You know, make it a bit abstract. You don't want to be too graphic."

"I see what you mean. The key is to make sure that this book is a celebration of Paul's life, not one of his death. What happened in the library that night is something upon which not to dwell."

"Wait, how do you ... where did you hear that Paul ... died in the library?" The mouse seemed timid and nervous ... then again, that seemed like a part of his personality more than a temporary condition.

"Have you heard this?"

"Oh ... yeah, Andrew told me. Andrew Fong, you know him? He's ... the librarian for the chapter. He takes care of the library. So ... he was a bit unnerved to find out ... well, you know ..."

"Yes, yes. Well, you seem a bit shaken about all this. So if you have any stories, you can just email me those or something. You sure you're okay?"

"Yes. Yes, I'm okay. Don't ... don't worry about me."

"Okay. Well, have a good day. ... It'll be better soon, I promise." Karl felt as if he had to say some words of comfort before the kid left. After all, the mouse was practically shaking. Joey closed the door with a click. Well, that was awkward, Karl thought.

***

Another knock on the door, only an hour later. "Come in," said Professor Kairo again. This time, the door was opened by a gorilla, slowly yet steadily.

"Professor Kairo?" said the gorilla. "I was told you wanted to discuss Paul."

"Yes," Karl sighed, "that was the idea. Sit down."

The gorilla sat in the chair opposite Karl's. "First of all," Karl began, "I'm realizing I don't know your name."

"Mike."

"Mike ...?"

"Anderson. Mike Anderson."

"Right, and you're from the fraternity?"

"Yes."

"Okay then." Clearly, Karl thought, he is a man of few words."Well, do you have any fond memories of Paul?"

"Well, the evening of the ... Halloween evening, we were hanging out after the party had died down. Six of us were there, myself, Paul, his brother Pete, Tim, Andrew, and Joey. Everyone else had gone to bed, but we stayed up. We played a couple of rounds of cards. We laughed." Mike had a straight face the entire time he told this story. "Then a few of the guys wanted to go to this bar in town. Rick's, it's called. Now, normally I'm the adventurous type, but I've been to that place before, and it is shady. So I just went to bed, and Paul and Pete followed suit."

"I see. And this is a particularly fond memory?"

"Well, yes. You see, we had a very fun time playing cards. We all laughed, myself included."

"Mike, you seem ... not very enthusiastic. Are you all right?"

"Yes, yes. I am a bit shaken, but that's to be expected. I'm glad that we haven't been restricted from the library this past couple of days. Though now that they think it was the library, the police might want to search there."

"The police think the murder happened in the library?" Karl was puzzled at how they had gotten this information.

"That's what Joey told me. Anyway, I should go. I have a class in ten minutes." Without prompt, Mike stood up and left.

***

The door opened, without a knock. "Sup?" asked the sloth who was peeking inside.

"Hello," Karl said, looking up from his desk. "And you are ..."

"Jeff, man. Jeff ... Sanders," the sloth said, with a pause as if he couldn't remember his last name.

"Ah, yes, I've heard of you. Sit down. Oh, and close the door behind you."

Jeff slowly closed the door and sat down, and Karl immediately noticed a smell coming from the man. Karl didn't have the experience to say with any amount of authority that it was marijuana, but it was marijuana. "Now then," Karl began, trying to ignore the smell, "I'm told you were a good friend of Paul's."

"Well, I guess, man, I mean, it's a frat, we're all friends. We're, like, bros, you know? But yeah, I hung around Paul a lot. Real shame he's dead now. With all the books and all, you know. Because, like, they're trees, right? But they're dead, and so is Paul."

"Books?"

"Yeah, in the library. You know, he was killed in the library, right?"

"... Where did you hear this?" Everyone seems to know already, thought Karl. If I can trace this to the source and figure out who the first person to know was, I might have a murder suspect.

"Well, Joey told me, you know? You know Joey, right? Joey's a cool guy. Kind of ... does art and stuff. Good art."

"Yes, I've talked to Joey. ... Do you have any fond memories of Paul?"

"Well, we were fishing once. Paul and I went fishing out by the lake, right? And, I tried to catch this fish, and it was really big! And Paul, he just turned to me and said, like, 'we've got big fish to fry.' And I cracked up."

"So Paul liked to fish?"

"I guess. I mean, we really only went once, right?"

"... Right. Well, thanks for sharing. I'll keep this story in mind." Karl realized at this point that he should be writing down Jeff's story to use later. He hurriedly scribbled a few notes in his notebook.

"Yeah, I should maybe get going ... I have a class now."

"Now? Wait, now?"

"Yeah, it started, like, five minutes ago, man. Catch you later." Jeff slowly got up and left the room. After pausing for a moment in the hall to remember which way was left, he disappeared to the right side of the door.

***

The next to enter was an owl. "Hello," Karl said at the bird as he entered the room. "You are ..."

"Andrew Fong, librarian for the chapter," the owl stated rather quickly.

"Ah," Karl said. I shouldn't jump to conclusions, Karl was telling himself at this point. Just because everyone's knowledge of the location of the murder can be traced to Andrew, and he's the librarian, and the murder happened in the library_, this doesn't tell us anything about Andrew's character._ "Sit down," Karl stated once more, gesturing at the chair.

Andrew sat down rapidly, as if he had somewhere to be. "So, I understand you want to hear about some of the memories I had with Paul," he began.

"Yes, do tell," Karl responded.

"Well, one time we were talking to Jeff, and you know that one trick where you tell someone that their picture is under 'gullible' in the dictionary? I mean, who hasn't heard of that one? But Jeff was never a bright fellow, so on a whim, Paul told him that his picture was in the dictionary, under the word 'gullible'--and he totally bought it!" Andrew chuckled a bit. "We got a good laugh out of that one."

Karl tried to laugh. "I'm sure that was great. It is a bit sad now, though, you know, now that he's gone."

"Right, right, of course," Andrew responded in a voice that didn't sound incredibly sad. "But, you know, life works that way. Or rather, life stops working that way. Murder, of course, is terrible, don't get me wrong. I would rather Paul had lived a nice, long, life and all."

"Andrew--" Karl paused for a split second, unsure of whether he wanted to ask the question. "Andrew, do you ... have any idea who could have killed Paul?"

"No, no idea. I mean, Pete told me it happened in the library, so maybe that helps? I mean, of course I'm the librarian, so that calls me into suspicion, but it wasn't me. Well, you might not believe me, but I was out of the house when the murder happened."

"Where were you? ... No, no, I mean, that's not ... I don't want to stir up bad memories."

"It's perfectly fine. I was at a bar. Tim, Joey, and I went to this place called Rick's."

"Right," Karl said, almost interjecting, then deciding he should shut up in case Andrew was about to reveal key information.

"It's a bit of a sketchy place, but worth a visit if you like to sample some of the stranger beers. I--I mean, not like I would do such a thing--I'm only 20, you see, and--"

"Don't worry," Karl said quickly. "I'm here to hear stories, not to go after a fraternity boy for drinking underage. I mean, you're in a fraternity; if we believe the stereotypes, this isn't a surprise."

"Right, right, sorry, sorry. I ... really should get going, though. I have places to be."

"I'm not holding you in here. See you some other time." Karl sighed after Andrew had closed the door. An interesting fellow, of that there was no doubt. But Karl had to be very careful to not make undue accusations.

***

There was another knock on the door. "Come in," Karl said again. A large lion opened the door, walked forward with a confident stance, and sat. "Hello," he began. "My name is John Lloyd, and I am the president of our Phi Upsilon Rho chapter here at Warren City University. I am told you want me to come in to discuss some of the memories with which Paul Chlebek was able to grace our fraternity prior to his untimely demise."

"... Yes, that's it," Karl replied after a second to take in John's statement. "Continue."

"Yes, well, as you might imagine from the fact that, as I said, I am the president, I have talked with Paul many times throughout his time with us as a member of the Phi Upsilon Rho fraternity. As such, I have quite a few stories to tell."

"Right, well, please, go ahead."

"I am sure that there are many stories I could tell about Paul, so I am not sure exactly which would be most fitting. Perhaps you can give me some sort of criteria as to what exact type of story you are looking to publish in this journal, and I will be almost definitely be able to remember a story that fits the requirements you are seeking."

"Well, okay. I mean, I don't care about too many specifics. So whichever story you like is fine."

"Ah, but see, it is hard to pick just one. There have been so many memories, you see, Professor, that it is hard for me to pick out just one without additional guidelines."

"Right, right. Well, hmm. I mean, ideally we're looking for something that demonstrates Paul's character." Karl began to gesture for John to continue, concerned that it might take a while to get a story out of the man.

"Character, ah, but the question then becomes: how do we want to define character? The term can have multiple meanings in different contexts, and indeed, even if you do mean one of these disjoint concepts, the concept to which I believe you refer is in itself difficult to precisely define. So what would be helpful to me is if you could please elaborate on what exactly you're looking for in terms of character."

Karl was getting annoyed at this point. "What I mean by character is ... well, I want a description of Paul's personality. But, you know, a story. Could you maybe tell me a story that allows me to get an idea of some part of Paul's personality? Like ..." Karl continued, worried that if he wasn't incredibly clear about what he meant by a personality aspect, he would merely get another three sentences from John explaining how John still didn't know what Karl meant. "If I didn't know Paul at all, and you wanted to tell me a story about him so that I could understand what kind of person he was, so I would understand why you're sad to see him gone, what would you say?"

"I would probably want to tell that person the story of the interview I gave to Paul as part of the candidacy process in order to determine whether he would be eligible to become a member of the fraternity."

"Ah, okay." Karl waited for John to continue. When John did not, Karl explicitly asked, "So, could you continue? You know, could you tell that story?"

"Ah, of course, you want me to tell this story right now. Right. Well, it was halfway through the second month of Paul's freshman year--this would have been last year, of course, as Paul was a sophomore this year. So this would take place in early October of last year, that is to say, 2029. I was interviewing Paul as a part of the formal process required for all prospective members of our fraternity before they can be given bids. I--actually, you are writing this down so that it may be published later, correct?"

"Yes, continue."

"Well, if this is the case, would it not be easier if I could simply write the story myself and then send it to you electronically? That way, it would take up less of your time, and the compilation and editing of this story could be done in a more asynchronous manner. Does this sound like a good idea to you? I ask this because I do have a meeting coming up in about twenty minutes, and I am not sure if I can do the story justice in that time."

As little as Karl wanted to attempt to condense a twenty-minute story into something publishable in length, neither did he want to force himself to sit still and hear the story. "Okay, that sounds like a good idea," he replied. As John immediately rose from his seat and began to leave, Karl remembered something. "I am sure," he stated, "that you have heard of the location of Paul's death?"

"The library, you presumably mean to indicate? Yes, I was told this information by our vice president Timothy Olson, one of the most faithful members of the fraternity, I might add. In any case, I must be going so that I can have a few moments to find Timothy before the meeting. Oh, well, speak of the devil and he doth appear," John stated as he opened the door to reveal Timothy Olson standing outside. "Timothy, could you do me a favor?"

"Of course," Timothy replied. He was a Boston terrier of average stature, and his large eyes seemed to look at John's with a simultaneous sense of close friendship and utmost respect.

"Remind me to write the story of Paul's interview for the fraternity and send it via email to Professor Kairo. I have a meeting to attend."

"I'll do that. Anything else?"

"Nothing currently. Have a good day, Timothy," John stated as he walked away.

"You too," Timothy replied before stepping into the office and closing the door. "So I assume you're able to hear my story?"

"Yes, do tell," Karl replied.

"Right. Well, I'll tell you the story about how we discovered Rick's."

"Rick's? That's the bar in town I've heard about, right?"

"Yes, it's a very interesting place. Now, this is last year in the spring, right after midterms, and a few of us want to head out on the town to celebrate. I figure we should meet a few people, and where better to meet people than at a bar? So Pete, Paul, Jeff, and I drive around town looking for a bar, and Paul sees this place called Rick's. Now, none of us have seen this place before, and it looks sketchy as heck. But hey, I mean, it's a bar, right? So I point out, I say to them, 'I don't know, it looks sketchy.' And Paul, Paul's smart. He points out that technically, Jeff is the only one of us who's of age, and so we'd probably better go to a place where they're less likely to check your ID. You know, all of the classy, high-end places are going to card everyone who walks in. Oh, wait," Timothy stopped, remembering that he was talking to a professor. "You're not ... you're not going to get us in trouble, are you?"

"Don't worry. If I find that anything in this story is incriminating, I'll try to censor it as well as I can before publication. Continue."

"Right, well, so I say 'Sure,' and we go park and head into Rick's. Now this place is about as sketchy on the inside as the outside. I'm talking, like, there's paint peeling off the walls, and the bartender is this skinny guy called Rick. And Rick asks us 'What can I get for you guys?' Didn't even card us, nothing. So we have a few beers, and eventually, Paul's pretty drunk. So he goes to the end of the bar and starts hitting on this girl--well, that's what he thinks. So he says 'Hey, you look really pretty tonight, what's your name?' And we're all trying not to crack up, because we think Paul's gonna get his ass kicked. Because we can all clearly see it's a guy who happens to look like a girl. And this guy, this guy says 'Mark,' and his voice is, like, down here," Timothy continued, gesturing at his lower torso. "And at this point, we can't help it. We all start laughing, Rick too, and meanwhile we're looking at Mark, thinking he's about to punch Paul in the face or something. And Paul, Paul has this look on his face, realizing his mistake, like he's about to get his ass kicked, because he totally thinks he is. But then Mark--this is the best part--Mark starts laughing too! Turns out Mark was a totally cool guy, and we talked with him for an hour or so before heading back."

"That's ... amusing," Karl replied, wondering if there was any possibility he could censor this story enough to be publishable. "Well, thanks for telling this story, Timothy."

"Call me Tim. Well, anything else you want to discuss?"

"Just ... I mean ... well, out of curiosity, John mentions that you told him how Paul passed away in the library. Where did you hear the rumor?"

"Oh, right. Well," Tim continued, his smile diminishing as the subject shifted, "Mike was telling me about that earlier. Do you know Mike? He's a nice guy, really likes to try new things. I'm all for that as well, you know. I wonder why he didn't go with us to Rick's that night ..."

"Well, in any case, Tim, thank you. I won't be needing anything more. Go ... do your homework, I guess."

Tim nodded in Karl's direction and left the room, closing the door behind him, making sure to close it slowly so as not to make a loud noise. So John heard about the library from Tim, Karl thought, who heard it from Mike, who got the information from ... Joey? Is that right? Karl got out a piece of paper and began to draw a diagram of the situation. Let's see here, Karl thought to himself.If we label each person by their name, here we have Joey, the first person I talked to. Now Andrew told Joey, so let's draw an arrow from Andrew to Joey. Pete told Andrew, so there's another arrow ... I haven't talked to Pete. Who else? Mike got the news from Joey, yes, and then Tim from Mike and John from Tim. Then the only other person so far is Jeff, who also heard about the library from Joey. So that means Joey told the most people ... but presumably the murderer would be the person whom nobody told, which would be Pete right now. Ah, yes, what we have here is a directed graph, a rooted tree, I guess, and the root is Pete. So when I ask Pete where he heard about the murder, he won't know what to say ... assuming nobody has lied to me in the process, which is probably the case. I'll need to talk to Pete before I can say anything conclusive, though.

_ _

5

"Hey," the next guest said, opening the door himself. "You're Professor Kairo?"

"Yes, please, sit down," Karl said once again.

This entrant was a hyena, with a strange grin on his face. He wore a pair of wide glasses, a red T-shirt, and blue jeans. "So," he said as he sat down, "you're looking for us to tell you stories about Paul. Is that right?"

"Yes, go ahead. Oh, wait, I'll need your name."

"Nick. Nick Handey. Anyway, so it was April Fools' Day last year, and you know, that was a Monday, so I wanted to play a bit of a joke, make things interesting, right? So get this. I took a bucket of paint. I went to the store and got a bucket of red paint, and then I opened it. And then I put it above the front door. Do you see where this is going?"

Property damage, that's where this is going, Karl immediately thought. "Well, continue," was what he said out loud.

"So then there I was, inside the door, waiting for Paul to show up, and he did, and he opened the door! He opened the door!" Nick laughed a bit, a cackling sort of laugh. Karl felt a bit unsettled. "And so then I of course started laughing, like that, and everyone was starting to get worried, and I was worried too, because Paul stood up, and he looked furious! I mean, just picture him, there, standing up, dripping with red paint all over his head and chest and all. So I asked him, I asked him whether he was okay. And you know what he said?"

"No, go ahead."

"He said, 'I don't know, Nick. I'm seeing red.' Get it? Seeing red?" Nick cackled again. "So what do you think? Is that a good story or what?"

"Decent. How did you clean up after yourself?"

"Ah, that was no big deal. We just had to get a mop from the closet and clean up the mess. It was an easy kind of paint to clean, so the room looked ship-shape pretty soon. As for Paul, he just took a shower."

"The closet ... was this the closet where Paul died?" Karl intentionally introduced a slight inaccuracy in this question, hoping to be corrected.

"Yeah. Except, well, he didn't die in the broom closet. He died in the library."

"And how exactly do you know this?" Karl asked, jumping at the opportunity.

"Well, I heard it from Jeff ... hey, hey, wait a minute here!" Nick quickly became angry. "You're interrogating me, aren't you? You think I killed him!"

"No, no, no," Karl hurriedly backpedaled. "No, I'm just asking so that--"

"I don't believe you! This is ridiculous! Just because you're a professor doesn't mean you can just interrogate all of the students hoping you'll find some key piece of information! I'm ... I'm leaving, now." Nick got up and stormed out of the room, slamming the door. Karl winced at the loud sound. Well then, he thought. That went well.

***

Another knock. "Come in," Karl said. He was used to this routine by now, so he didn't even look up from the papers he was grading as his new guest entered, closed the door, and sat down. When Karl looked up, he nearly jumped; sitting in the chair in front of him was Paul Chlebek. "... Wait, but ... you're ... you're not ... you can't ..." Karl fumbled for words.

"I understand your confusion," the goat replied. Holding out a hoof, he introduced himself. "Peter Chlebek."

"Oh!" Karl exclaimed, suddenly understanding. "Right, so you're identical twins then."

"Yes. Well," Pete continued, becoming less cheerful, "no. We_were_ identical twins."

"... Right," Karl replied sadly, still a bit jostled from the sudden surprise. "And now Paul is ..."

"Yes. But I'll get by just fine, don't worry."

"Well, I'm sure this is very hard for you. You know, if you don't want to talk, I understand--"

"No, it's fine. Really, I'll be okay." Pete smiled.

"So the two of you entered the same fraternity," Karl stated, not sure where else to start.

"They ended up giving us both bids, yes. I've talked to a few of the brothers, and they said they gave us both bids just because they worried they'd get us mixed up if one of us was in and the other wasn't."

"That makes sense."

"They never told us which one they wanted to have more in the first place. Of course ... I guess now it doesn't matter." Pete seemed strangely stoic about all this.

"Right. What happened to your brother is ... a tragedy."

"I'm not sure if it's that much of a tragedy. I mean, yeah, I suppose so, but it's not like he had it coming. Well, are we using the literary definition of tragedy or not?"

"Let's not argue over specifics. In any case, this isn't literature."

"Literature ... Paul died in the library. Presumably you've heard this?"

"Yes ... where did you?"

"Oh, um, Tim told me. You know Tim? The dog?"

"Yes, he seems like a nice character. Anyway, Pau--Pete, is there any story you feel represents your brother's character well? Presumably you know his character well, being brothers."

"Yes, well, one story that comes to mind is how--"

Pete was interrupted by Dr. Douglas, who opened the door. "Karl," he said sternly. "I need to see you now."

"Can it wait, Fred? I'm in the middle of a meeting."

"I'm sorry, but no. We need to discuss these meetings you've been having. Please, follow me to my office."

"Okay," Karl said, getting up. "Pete, wait here for a few minutes. I'll be back soon, and then you can tell your story. Unless you have somewhere to go?"

"No, no, it's fine," Pete replied, smiling.

Karl and Fred walked to Fred's office. "Is something wrong?" Karl asked, noting Fred's apparent annoyance.

"We'll discuss it in my office, Karl," Fred replied.

When Karl entered Fred's office, he noticed Arthur sitting at the table inside. Karl sat next to Arthur as he heard Fred close the door with a click. "Professor Kairo," Fred began as he sat at his armchair. "You have, over the past two days, been interviewing students from the Phi Upsilon Rho fraternity, asking them to tell stories about their experiences with Paul Chlebek. Is this true?"

"Yes," Karl replied. "I'm hoping to compile these stories into a book, a celebration of Paul's life so that the student body can remember him well."

"Don't lie, Karl. You're _hoping_to interrogate the students and find leads to a possible murderer. This is unacceptable behavior from a faculty member at Warren City University. Is there anything you would like to say in your defense?"

"Of course there is, Fred. Why would I not defend myself? First, to be fair, I did not lie; I am indeed hoping to compile the book I've mentioned, and although I may have ulterior motives, my intentions do include a desire to preserve Paul's memory in the years to come." Karl sighed. "Secondly, I understand that interrogation is something I shouldn't be doing. I am not interrogating the students. We have conversations about Paul, and if questions relevant to the murder come up in conversation, it may be that they give me information that could help me to narrow down the murderer. Thirdly, hopefully you feel that this action is justified, because finding out who killed Paul Chlebek will allow us to properly prosecute anyone involved and create a safer school. If, however, you don't agree with the points I've made and feel that my treatment of the students has been unfair, I'll end the meetings entirely. I apologize if I've caused any offense."

"Karl, we know you're trying to do the right thing. Everyone tries to do the right thing. What I want you to understand is that what you think is right here may not be correct. Warren City University has had only three instances of violent crime in the past fifty years, and two of the perpetrators have been from outside of this school's populace. As such, the chance that Paul was actually murdered by another student is slim, only one in three, if that. I would thus argue that your attempts to get information out of the student body are futile."

That's not quite how Bayesian probability works, Karl thought. "Again, if you want me to stop, I will. I've already obtained all of the information that I think will be useful to me."

"No," Fred replied. "You won't stop entirely. Instead, if you want to preserve your generally positive relationship with the mathematics department, you will continue to listen to the stories of the remaining members of Phi Upsilon Rho and compile the information into a book as you said. You will not attempt to gain any more information about the murder in the process. Are these terms understood?"

"Understood and agreed. If I may ask, Fred, why is Arthur here?"

"I thought his presence would better persuade you to make the right decision. You are both dismissed from my office."

Karl and Arthur both walked outside. "Well then," Arthur began. "Usually if a professor is angry and wants to meet with someone, it's not another professor."

"Right," Karl replied. "Don't worry, though. As I said, I already know what I need to know."

"If you say so," Arthur sighed, walking away.

Karl walked back into his office, closing the door. Pete had left for another commitment, sadly; Karl decided that to ensure Fred was satisfied, he would be sure to contact Pete and see if another time would be better. In any case, Karl had a diagram to draw. He pulled out a pencil and a piece of paper from his briefcase.

First, Karl wrote eight names around the perimeter: Joey, Mike, Jeff, Andrew, John, Tim, Nick, Pete. Then he drew an arrow from each student to the student from whom he heard that the murder occurred in the library. Okay, let's run through this again, he thought. Joey heard from Andrew, Mike from Joey, Jeff from Joey, Andrew from Pete, John from Tim, Tim from Mike, Nick from Jeff, and Pete from Tim. So we have a cycle between Joey, Andrew, Pete, Tim, and Mike. They can't have heard from one another in a cycle like they say, so one of them is lying. That one is the murderer. Well, I can't say that for certain, but that one is an awfully good lead.

I don't know which, though. I've possibly narrowed the murderer down to one of five people. Those five ... wait a minute. Karl opened the notebook in which he was writing down his stories, and a chill ran down his spine as he remembered Mike's story: "Well, the evening of the ... Halloween evening, we were hanging out after the party had died down. Six of us were there, myself, Paul, his brother Pete, Tim, Andrew, and Joey. Everyone else had gone to bed, but we stayed up. We played a couple of rounds of cards. We laughed. Then a few of the guys wanted to go to this bar in town. Rick's, it's called. Now, normally I'm the adventurous type, but I've been to that place before, and it is shady_. So I just went to bed, and Paul and Pete followed suit."_

So here, Karl continued to think, is a story involving these five students and Paul ... the night of the murder. If this story is true, Paul and Pete were in bed, and so was Mike, while Joey, Andrew, and Tim went to a bar. Rick's, a bar where they've been before. Obviously there are other people at bars, meaning we have witnesses. The next step in Karl's investigation was clear.

6

"I like how I've become your chauffeur now," Arthur said sarcastically.

"It's your car," said Karl. "Would you rather I drive it around the sketchy part of the city?"

"I'd rather we didn't have to drive to the sketchy part of the city in the first place."

"Eh, live a little, Arfie. If you never take risks, you never discover new things."

"Well, I'd like to live a little, but apparently I can't, because I have to be your designated driver."

"Hey, I won't be drinking much either. Alcohol hinders concentration. That is, the higher the concentration of alcohol, the lower is that of your brain."

"I just think it's going to seem a bit strange to head over to a bar in town, by ourselves, without an occasion, and not order alcohol," Arthur explained, ignoring the pun.

"Think of it like any fantasy story. What do you do if you're looking for information? You head to the local tavern and ask the bartender."

"And do you think this is a fantasy story, Karl?" Arthur asked. "In real life, you try that and get weird looks, no information, and at worst you wake up a couple hours later in a ditch."

"Well, you can still get alcohol, I guess. Just no hard alcohol. The legal limit isn't zero."

"True ... but ... hmph." Arthur was still displeased that he couldn't have as many shots as it would take to prevent him from being nervous about the entire prospect.

Karl looked out the window. Warren City was a different city at night than during the day, certainly. When the sun was out, it was a friendly place; shops bustled, people greeted one another in the streets, and the mood one generally perceived was one of cheer; it almost seemed like a quaint, small town, except for the population of over two hundred thousand inhabitants. Now, however, it was night, and the moon cast a foreboding light on the darkened streets.

The part of town into which they were now driving was littered with the odd piece of garbage every one or two street corners. Half of the streetlights were broken, and the other half were dim or flickering. A cool wind was blowing, though Karl and Arthur, being in the car, couldn't detect it. The car rolled past building after building of less-respectable establishments: a club with a neon sign flashing "The Knothole," a Chinese restaurant with bullet holes in the windows, a small "inn" that could just as easily turn out to be a brothel.

"Just explain to me, Karl," Arthur continued, "why exactly we have to come here at night."

"Ah, well, why wouldn't you want to come here at night?"

"Well, it's less safe."

"And why is it less safe?"

"Because the cops can't see as well. People do all kinds of stuff at night. And if you're doing your business at night, think of what kind of business that must be."

"Right. Because when you stay up late, you start to get a bit loony. In fact, that's where the word 'loony' comes from. When the moon's out, crazy things start to happen. Inhibitions are lowered."

"And we have to get in on this ... why exactly?"

"Easy. We need information that might be sensitive. The best way to get information out of someone is to wait until it's late enough for their inhibitions to be lowered. And if they aren't desperate enough to tell us the info at that point, we just wait until it's late enough for them to start drinking, and in vino veritas. Oh, there it is." Karl pointed at a building on the right side of the road, "Rick's" above its door as an orange neon sign, which flickered every few seconds. There were a few lights visible through the window, but the white, plastic Venetian blinds made it hard to tell who or what was inside, only that there didn't seem to be much commotion at the time.

"Well," said Arthur as he pulled the car over to parallel park, "this had better not get too crazy."

The bar was a dull, somewhat musty place. A wooden fan hanging from the rafters provided a feeble attempt at clearing the smell in the room, which was half "decaying building" and half "copious amounts of alcohol," neither smell quite pungent enough to disturb the nose on its own; yet the two synergized to create a mixture slightly disturbing to the senses of both Karl and Arthur. The walls indeed had peeling paint, as Tim had mentioned. Behind the bar, the wall was covered in fake wood wallpaper in a feeble attempt to make the environment look more respectable. There were a few tables out in front of the bar, for those who didn't want to sit in a row; however, currently the entire patronage sat on the cheap, cushioned metal stools at the long, wooden surface marking the center of attention.

Behind the bar stood the man that Karl could only assume was Rick. Rick was a tall, skinny fox, his fur an orange that would have been brighter if he paid more attention to hygiene. His whiskers were a bit bent at the ends, as if he had smashed them on something. Overall, he seemed like a bit of a gruff fellow, though Karl was careful not to judge him prematurely. He was currently striking up a conversation with a patron at the end of the bar, a yellow Labrador with a wide smile.

Wait a minute, Karl thought. That's Jimmy! Jimmy was a friend of Karl's father and the nicest guy Karl had ever met. Of course, Karl immediately thought. Who else would be making friends at the weirdest bar in town? Oh boy, he's going to recognize me, isn't he?

"Karl!" Jimmy yelled as his eyes turned in Karl's direction. "Come here, buddy! I was just talking about you!"

Karl smiled, feeling a bit awkward. "Hey, Jimmy, how's it going? Shouldn't you be back home with your family?"

"Nah, Becky and the kids are visiting her mom up in Indy. What about you? What brings you here?"

"Ah, I was actually looking to visit this place. I've heard some weird things about it and figured I'd check it out. Oh, this is Arthur," Karl said as he gestured to Arthur, who was standing next to him, taking in the conversation. "Arthur's become a good friend of mine at WCU. He's on the custodial team."

"You work at the college?" asked Rick, speaking up.

"Oh, I should have introduced you guys!" Jimmy butted in. "Karl, Arthur, this is Rick. He's the owner and bartender here. Really nice guy. Rick, this is Karl. He's one of the math professors at WCU. I play a lot of golf with his dad, so we know each other really well too."

"Ah," Rick said. "You're a professor."

"Yes," Karl replied. "Discreet mathematics."

"So you're probably here because of Paul, huh?"

"Oh, Paul!" Jimmy immediately said. "Yeah, that was really sad news. Good kid, used to come here all the time. God bless him." He bowed his head a bit, clearly meaning that last sentence as a prayer.

"Well, then," Karl said. "Were you guys here ... that night? You know, Halloween."

"Nah, I was trick-or-treating with the kids," Jimmy chimed in. "After that I was tired, so I went to bed. Rick would have been here, though, right?" Jimmy turned to Rick.

"Yeah, I was here," Rick said. "Saw three of 'em from the frat walk in late at night. Paul wasn't with 'em, though."

"Tim, Andrew, and Joey," Karl plainly stated. "... Am I right?"

"Yeah ... how did you know?"

"I've heard the story. When did they get here?"

"It was already late, about two, I think. They stayed here for a bit having drinks, then at about a quarter past three, Joey decided he was going to get off."

"He ... he left?"

"Yeah, he took the bus back. Then, about 15 minutes later, Tim and Andrew decided to call it quits as well. Took another bus. ... Say, you looking to solve this murder?" Rick noticed Karl trying to remember all of this information.

"Oh, oh," Jimmy said, a pained expression on his face. "Just be careful, okay, buddy? I don't want you to get into any trouble. Promise?"

"Don't worry," Karl said. "I'll be fine. I think I should be going, though. It's late."

Arthur breathed a big sigh of relief.

***

Arthur's paw was barely on the ignition key when Karl spoke up. "Do you know? How late is City Hall open?"

"No," Arthur said firmly, turning to Karl. "We are not going to City Hall tonight."

"Well, okay then. Tomorrow?"

"Tomorrow's Friday," Arthur replied. "You're telling me that I have to spend my Friday evening driving around the city with you, and not spending it with some of my friends, or at home watching Netflix, or getting a few drinks at a club, or--"

"Fine," Karl stated, interrupting Arthur's tirade. "I'll head over there myself, then. You have fun."

Arthur started the car, silent.

"I'm so close now," Karl responded. "If I can prove that Tim and Andrew weren't in the building when Paul died ..."

"Then you'll have a steaming pile of circumstantial evidence, yes. And then what are you going to do?"

"It might not mean much in court, but if I can convince Joey that I knew he did it ... then I can get a confession from him."

"So, literally speaking, why don't we not go to City Hall and say that we did? What difference does it make?"

"The difference, Arthur, is that I'm not sure myself if it was Joey. I don't want to interrogate a student if it wasn't him."

"And will you know even if you do go to City Hall?"

"Well, I mean, it's the next step. I might need to do further investigation."

"Karl, I would just say ... don't tear yourself up over this. What are you going to do if it's _not_Joey? What's motivating you at this point? Pride?"

"... I guess, maybe, that's a part of it. But not pride from others. No, what's motivating me now is just ... the thought that if I do this, I'll have done this. Get what I'm saying? If I succeed ... if I solve the case, if I reveal a murderer ... I'll have solved a murder. And I always liked the thought of being a detective."

"I guess so, Karl, but I mean, you're seriously going to chase after that dream now? Look at your life. You've got a Ph.D. and a job teaching at your alma mater. You have friends and hobbies. Isn't this your dream?"

"... You're right, Arthur. This is my dream. I never really ... I'm often not thankful enough for the life I have right now. ... But I'm so close that I can't stop now. I have to solve this. For Paul."

"... Well, okay then. But I think, Karl, that this is where I have to bow out, if only for now. I just feel like I'm getting caught in something I don't want to be involved in."

"That's fine, but ... promise me something?"

"Go ahead."

"Promise me that you'll be there if I need you? I mean, I didn't need you to come to this bar, and I didn't need you to talk to Sonny. But if I do need another head, or another set of paws ... if it's crucial ... you'll be there?"

"Of course. Just give me a call if you're in trouble."

"... Thanks. I'll try to avoid having to take you up on that ... but no promises." This is about to get very complicated, Karl thought.

7

The wind on Friday afternoon sent a chill through Professor Kairo's fur as he rode his bicycle through the streets. It would probably be to his advantage to take the bus, to get an idea of how the public transit had been working lately, but Karl had ridden the buses many times before; he knew that generally speaking, they ran on time, certainly at two in the morning. Warren City was unusual in that its bus system ran 24 hours a day; the large number of college students, combined with the fact that there were quite a few clubs, meant that the taxpayers saw little reason to be against the idea. Besides, Warren wasn't expansive enough to have a subway system, and the cabs tended to run on the expensive side, especially for someone with mountains of student loan debt, like so many of the city's residents had.

No, today was a day to ride a bicycle. It was a bit windy, yes, and it was the fifth of November as well, so temperatures in Indiana were starting to drop. Heh, thought Karl, they're going to shoot fireworks in Britain tonight. Regardless, leaves were only beginning to fall from the trees, and for the most part, they were still clad in brilliant reds and yellows. The city itself was bustling, people moving from one place to another, living out their Friday afternoons. Karl always loved to ride. The rush of wind through his fur, combined with the beautiful scenery, made him feel almost as if he were flying.Sadly, cats can't fly, he thought. Not naturally, at least. But we find our ways.

Karl felt a sense of quiet awe as he approached City Hall. Warren's primary government building wasn't the largest Karl had seen by far, but it had fascinating architecture nonetheless. Karl didn't nearly know enough about architecture to give it a name, period, or anything of the kind, but the building had a great round dome with pillars at its base, just as you'd expect of a large government building. Of course, Karl wasn't here to take in the grandeur of municipal government. He was here to find out a crucial bit of information, and for that he simply had to head to the information desk.

The information desk, of course, was amidst all of the grandeur of municipal government. After all, Warren City had to advertise itself. Thus they made sure that if any tourists came looking for information, they would find it in the middle of a large building, decorated with ornate carvings and impressive statues. Above the entrance, they would see, as Karl now saw while he parked his bicycle in a nearby rack, a stone carving reading "CITY OF WARREN, INDIANA, ESTABLISHED 1816." Then, entering, they would find themselves inside the large dome, whose interior was covered not only with stone carvings but also decorated, stained wooden panels. They would immediately, or at least inevitably, notice a large column in the center of the dome, upon which was a plaque reading "WARREN CITY, INDIANA / MAYOR, RICHARD J. BLUEGLID." Then, finally, searching with their eyes, they would find, as Karl now found, a desk to the right, cut into an alcove in the outer wall of the dome, with "INFORMATION" written upon it.

Behind this desk now sat a white rabbit with a bored expression, scrolling through some sort of content on a computer. Her ears perked up at once upon noticing Karl's presence; Karl was the only other one in the grand dome at the moment. "Can I help you?" she eagerly asked.

"Yes," Karl replied. "I'm just looking for a comprehensive schedule of the bus system."

"A comprehensive schedule? Well, you can usually just find out the information you need at individual stops. We do have a full schedule of all of the buses, but we don't give it out to people."

"Well, could I ... why not, if I may ask? That seems a bit counter-productive."

"... Do you use the buses?"

"Yes, I ride the bus quite often."

"And do you ever need a comprehensive schedule of the bus system?"

"... You have a point."

"We decided a few years ago that the only reason anyone would want to see the whole schedule at once would be to plan some sort of elaborate prank. And we don't want that. So here we are." The rabbit turned back to her computer screen for a brief moment.

"Well, hm. That's a bit unfortunate. You see, I was ... I need to know how long it would take to get from a certain bar to the college fraternities, late at night ... it's just a question about one specific route. Well, no, I guess I need to know about every route from one point to the other, at least within the timespan of about half an hour ... okay, if we construct this as a weighted directed graph, then ... no, but that doesn't take into account the start/end times. Well, maybe if we add nodes to represent the ..."

"What are you talking about?" asked the rabbit.

"I'm trying to express mathematically the exact amount of information I need in order to solve this. Well, I guess the amount of information doesn't matter--I'm not looking for entropy or probability--I'm trying to find the right premises from which to make my conclusion. You see, I'm a professor of mathematics at Warren City University, and--"

"Look, how about I just hand over the schedule tables for a few minutes, and you can sit down and look them over? That okay?"

"Oh--oh, yeah, I guess that works." Karl was still trying to figure out a way to model the bus schedule mathematically. Ah! he thought as the rabbit handed him an official-looking book. We can have a vertex for every bus's departure from a stop. Then when a bus arrives at a stop, we have an edge whose weight is the travel time plus the time to the next departure. We have an edge at each stop, Karl continued to think as he sat down, from each departure to the next, to represent waiting at that stop for some amount of time. With the problem reduced to path-finding on a weighted digraph, we can simply apply Djikstra ... but wait, it's acyclic ... is that the most efficient? Hmm ...

"Well, well, look what we have here," said a cool voice that echoed through the hall. Karl looked up, his thought interrupted. The sound of dress shoes on tile, previously ignorable, was now more than obvious to Karl's ears, and he was quickly able to deduce its source: a tall, dark blue dragon, in a black business suit, was approaching from one of the hallways that extended from the dome. "Doctor Kairo, the professor who thinks he can be a detective." Approaching Karl, the dragon raised a clawed hand, offering to shake Karl's paw. "I'd introduce myself, but I'm sure you know my name."

"Y-yes, you must be Mayor Richard Blueglid," Karl said, accepting the handshake. The mayor's grip was quite firm, to the extent that Karl worried if his paw would be crushed, only to pull it back completely unharmed.

"Please," the mayor responded, "you can call me Jake."

"Jake? But ..."

"It's my middle name," Jake responded, lifting his hand to dismiss the question before Karl could finish it. "I always thought Richard sounded a bit too old-fashioned. I'm only thirty-seven, after all."

Well then, Karl thought. He's not that much older than I am. The mayor had always seemed old and prestigious, probably just because of his position. "Well, then ... Jake," Karl responded. "I see you've heard about my ... my investigation."

"Yes, your glorious quest to discover the murderer of Paul Chlebek. I've heard all about it."

From whom? Karl thought.

As if he were psychic, Jake continued. "You know how news spreads these days. Fred Douglas met with you, I understand, and after doing so, he reported to Ronald Grant. Ronald Grant mentioned it in today's meeting with the school board, and I, the town mayor, have the ... pleasure of having to sit in on many of their meetings."

"Right, of course," Karl responded, trying to sound like he was in control.

"Now, Karl--can I call you Karl?--if you would follow me to my office, please." Jake began to walk down one of the tile hallways.

"Y-yes, that's fine," Karl answered, not having much time to react. Finally, he grabbed the book, stood up, and began to follow Jake to the mayor's office.

As the two of them passed marble columns, stone steps, and elaborate wooden doors, Karl mulled over this situation. It's not often, he thought, that you hear about someone, and you see the person in pictures, and then you actually get to see that person, not in a picture, but standing in front of you. And it's very rare for him to actually initiate the conversation, greet you by name, and shake your hand. I must be more well-known than I thought.

Finally, the pair reached the door above which "MAYOR'S OFFICE" was written. Jake held the door open for Karl, who stepped in and sat down in a cushioned chair. The back wall of this room was mostly windows, which gave a bright view of the park behind City Hall. The wall on the right side was home to the largest bookshelf Karl had seen outside of actual libraries, while on the left wall hung various portraits of former officials. Directly in front of Karl was a large, oak desk, full of paperwork.

"Now then," Jake began as he closed the door and walked to the black leather chair behind his desk. "May I offer you a bite?"

"What?" Karl noticed that Jake was gesturing at a plate of sushi in between the piles of paperwork. "Oh. No thanks, I'm fine."

"Suit yourself," Jake responded as he popped one of the rice rolls into his mouth. "Now then," he continued after leaning back and swallowing the sushi, "let's talk. You're trying to solve the murder of Paul Chlebek by using mathematical methods."

"Yes," Karl replied. "And you're about to tell me that I shouldn't," he continued as Jake was opening his mouth.

"No, not at all," Jake replied coolly. "Rather, I have a proposal to make. Are you ... having trouble with your investigation?"

"Well, not really. I mean, I'm close now. I can feel it."

"So close you can taste it, would you say?"

"Yes. But ... I'm not sure if I should be doing this anymore."

"Dr. Douglas doesn't want you to continue, fearing that it might affect your relationship with your students in undue ways, no?"

"Right, and now even Arfie ... he's a friend of mine. Works as a custodian at the school. Anyway, he's starting to give up on me as well."

"Let me tell you this, Karl. I have taken quite an interest in this case. When I heard you were working on it, I was ... intrigued." Jake's wings momentarily fluttered as he finished this sentence. "And ... of course, the police haven't been helpful to you, have they?"

"They have ... Jake. They have, but they don't know it," Karl replied, smiling with half of his mouth.

"Well, I am the mayor. And the mayor is the head of the municipal executive branch, and the executive branch ... carries out the law, no?"

"You're proposing that you can tell the police to help me," Karl said. "I'm not sure if that will be necessary."

"Karl, why do you want to solve this crime?" Jake asked, trying another line of questioning.

"As I was saying to Arfie ... I ... I just want to be able to think that I did it."

"You want to be a hero ... a detective."

"Are you ...?" Karl wasn't sure whether to say the question on his mind.

Jake picked it up anyway. "Here is my proposition, Karl. If you can successfully deduce who the murderer of Paul Chlebek is, in a way that produces verifiable evidence, you will become the official detective to the mayor of Warren City. On the one hand, this means we may call on you to solve further cases, and on the other, you will have the full cooperation of Warren City's police force."

"Would this interfere with my work?" Karl asked.

"You're already solving a case right now, without the police's help. Is it interfering with your work?"

"I mean, it's a bit convenient, this case, being on campus. If it were someone in a different area of town, I would think ... you know, it would be a different story."

"Well, in any case, the deal isn't on until you can solve this case for me. I was just ... giving you incentive. After you've had more of a taste of the life of a detective, we can talk."

"Would this be a paid position?"

"The city would grant some amount of compensation, yes. We can discuss details later, after the current tale has wrapped up."

"... Right then," Karl said. "Can I leave? I have to look through this bus schedule."

"You may," Jake replied. "I have ways to contact you if I want to remain in touch. I simply wanted our first meeting to be in person."

"How did you ... never mind," Karl said, getting up to leave the office. How did you know I would come to City Hall? he thought as he left the room, headed back to the dome. Jake simply sat and smiled as he watched the yellow, ponderous cat exit his lair. Things, the dragon thought, just got very interesting.

8

It was not a typical Monday afternoon in Karl's office. He had a few assignments to grade between his classes, but he was finding it hard to work at the moment. The silent tension was broken only by the slow ticking of his wall clock as Karl waited for Joey to enter so he could deliver his ultimatum. Although he was confident by now that Joey was the killer, Karl still had his doubts about whether he would be able to get a confession out of the mouse. He also hoped Dr. Douglas wouldn't kill him for scheduling this meeting. Oh, well, he'll hopefully understand,_Karl thought. _No sense in stopping now.

Finally, after a lifetime of silence, the door opened, and Joey peered in. "H-hello?" he asked.

"Ah, Joey," Karl responded, trying not to frown. "Come in."

Joey slowly closed the door and sat down. "You wanted to see me." It wasn't a question.

"Joey," Karl began. "... Look, I'm really, really sorry. That's what I'd like to say first, that I'm sorry. I'm sorry that I've made you come here, twice now, for what could turn out to be pointless. I'm sorry if this all turns out to be for naught. But ... knowing why I've done this ... I have to take my own actions here, and try to get to the bottom of this ..." Sweat was rolling down Karl's face. "I'll get straight to the point. Again, please, please don't think I want to hold anything against you; I wouldn't want to hold anything against any of my students."

Joey simply sat, waiting for the cat to continue his speech.

"I believe you to be guilty of the murder of Paul Chlebek."

"What? What are you talking about?"

"Your game is over, Joey. I've figured it out. It was you."

"Honestly, I didn't kill him. Please let me ... don't ... I'm sorry, but I don't know who did it."

"Well, perhaps you would like me to explain how I know this," Karl continued. "First of all, I've received word from the police that the body was dragged around the house, through every room on the first floor that wasn't a bathroom, through each doorway once, apart from one doorway, which was traversed twice. By simple graph theory, if the body ended up in the front closet, where it was discovered, it started in the library. You claim to have heard this information from Andrew, who heard it from Pete, who heard it from Tim, who heard it from Mike, who heard it from you. One of you must be lying, as the digraph representing the flow of information must be acyclic. The five of you were talking with Paul on the evening of the murder, and then three of you went to a bar--yourself, Andrew, and Tim. Perhaps you can tell me what happened then?"

"It's all a bit foggy ... I got drunk. Then I decided to leave the bar to head back. I was tired. Tim and Andrew got back a bit later."

"You left at approximately 3:15, according to Rick, the owner of the bar. Tim and Andrew left at 3:30. Paul died at 3:47. Now here," Karl continued as he pulled a large diagram out from his stack of papers, "is a weighted digraph representing the bus schedule. Applying Djikstra's algorithm, we see that, leaving the bar at 3:15, it takes seventeen minutes to arrive at the frat house, allowing plenty of time to commit a murder. Tim and Andrew left at 3:30, and they couldn't make it back until 3:52, meaning that they would arrive late--only five minutes too late. The blood stains would have still been fresh on the floor. I conclude that you were the only one able to have committed the murder, and Tim and Andrew would certainly have seen the aftermath, giving me at least two eyewitness accounts."

"Well," Joey responded. "Is that really a proof? It relies on ... assumptions."

"True, but I figured given enough evidence you'd be convinced."

"Convinced to ... do what? Even if I were the murderer--which I'm not, you can ask Tim to confirm--you don't have a proof. You have a bunch of slight hints, maybe."

"What about Andrew? What will he say?"

"Nothing. He was so drunk that he barely remembers that night. ... I was drunk that night."

"So what you're telling me is that you didn't commit the murder."

"That ... is correct."

"But certainly, you were around when the murder was committed. You arrived back at the house about fifteen minutes before Paul died."

"I can't say for sure. All of these times are approximate, right?"

"But you know it was before the murder?"

"... Yes."

"So who killed him? Tim and Andrew arrived too late, at 3:52 a.m., while Paul died at 3:47. You're not the murderer, so who is?"

"... Tim. Tim killed him."

Impossible ... or not, as the times aren't necessarily correct, Karl thought. They wouldn't necessarily be able to narrow the time of death to within five minutes. "How? Why? Tell me what you know."

"I ... Paul had caught Tim cheating. Tim's a senior and didn't want to fail a class, so he whacked Paul on the head with a textbook."

A textbook in the library, Karl thought. It's at least a plausible story. But this means I have to reconsider all of my former mathematical conclusions. The world doesn't follow strict laws like math does. "And you were there? You watched?"

"I couldn't do anything about it. I was afraid of what Tim would do next--drunk people are like that. So I ran and hid."

"Where did you hide?" At this point, Karl was trying to find inconsistencies or hesitations, something that would give Joey's story away as illegitimate.

"I ran up into the attic. It's cold up there, but not terrible. Especially if you're, you know, running ... from a murderer." Joey was shaking.

"So you ran into the attic, quickly? And you stayed there?"

"I stayed there for a while, then went to my bedroom. I tried to get sleep but couldn't. The police showed up and told us to leave."

"Well, I suppose at this point I'll have to ask Tim what he thinks of your story."

"He'll deny it, of course. Any sane murderer would."

"That may be true only vacuously. Get some rest and keep up with your classes, Joey."

Joey stood up to leave, then slowly walked towards the door. Upon opening it, he was greeted by the stern face of Fred Douglas.

"Karl," Fred began as Joey backed away from the door. "Explain yourself."

"The mouse you see in front of you is the murderer of Paul Chlebek."

"What?" Joey asked. "But ..."

"Don't try to fool me again, Joey," Karl replied. "There's a clear problem with your story. You're the murderer, and because you're so adamant about Tim's involvement, I would go so far as to conjecture him to be an accomplice."

"Karl," Fred spoke. "We have already talked about interrogating students. You understand the implications. Yet you have chosen to continue your actions."

"But I've found the murderer, Fred. You can't deny that. Well, you can, but I have evidence."

"Do you currently have enough court-admissible evidence to convict Joey Turner of the murder of Paul Chlebek?"

"Well, if you give me a day or two--"

"Yes or no, Karl?"

"... No."

"Then I expect a formal apology, written, to my office by tomorrow morning, together with a guarantee that this will not happen again, if you desire to keep your position at Warren City University. And you will need to stay very true to that guarantee if you have any desire to achieve tenure."

"If I were to find court-admissible evidence before tomorrow ... would that change your mind?"

"... Perhaps."

"Then I believe we are done with this conversation." Karl tried to appear guilty and subservient. It wasn't a dishonest expression.

"I look forward to an email message from you tomorrow morning. Goodbye." Fred left the room, Joey following behind him. The door closed.Click.

Karl breathed a long, deep sigh. Court-admissible evidence? There's only one way I can think to obtain that.

***

Arthur was mopping a floor in Holden Hall. "Arfie," Karl began as he approached the dog.

Arthur sighed. "Yes?" he asked, turning to Karl.

"You remember how you said you'd be there if I needed you?"

"Yes?"

"Do you have a set of keys for the Phi Upsilon Rho house?"

"... You can't be serious." Arthur squinted and scowled.

"I'm already in trouble with Dr. Douglas. Unless I can come up with hard evidence, this is all a big mistake. And the hard evidence is going to be in the house."

"Karl, you won't be able to find anything the police haven't. If you don't have evidence, and the police don't have evidence, how are you any better off than they are?"

"Easy. I know who murdered Paul Chlebek."

"No, you don't. That's the point of finding the evidence."

"I do; at least, I have a really good guess. I just have to confirm the guess. You see, Arfie, it's like a big NP-complete problem. It's really hard to find the answer to the problem. But once you have the answer, even if it's just an answer and nothing else, it's far easier to derive the solution."

"Look, why are we even arguing about this? I don't clean the frat house. The frat boys do."

"Then one of them will have the keys. In fact, the way the fraternity works, all of them probably have keys. ... Thanks for the help!" Karl immediately walked off, at a brisk pace.

"I didn't really help you that much!" Arthur called out as Karl walked away.

***

"Thank goodness every student's schedule is online," Karl mumbled to himself as he used his smartphone to pull up Pete Chlebek's schedule. "Drawing in C204 with Dr. Vernon, hm?" I can make it there before it gets out, but only if I hurry, he thought as he nearly ran down the hall. Finally, gasping for breath, he was standing outside room C204. After several seconds of waiting, the bell rang, and students began to pour out.

"Peter," Karl stated firmly as he saw the young goat that looked almost exactly like Paul. Fortunately, this was enough to get his attention.

"Dr. Kairo," Pete said as he faced the cat.

"Peter, listen closely. I think I know who killed your brother."

"You do?" Peter was excited and nervous. "Who was it?"

"I ... I don't want to say until I have hard evidence. Currently I'm going based on conversations and mathematics."

"So do you really know, or are you just ... guessing?"

"I'm pretty sure, but it will be hard to convince the police. That's why I need your help. Do you have a key that will get me into the Phi Upsilon Rho house?"

"What? Oh ... well, I don't think I can give you a key."

"Okay. I'm sorry, Peter." Karl's ears drooped.

"But I can let you in," Pete replied.

"Oh?"

"Y-yeah, just ... you're sure you can find evidence?"

"No. No, Peter, I'm not completely certain of anything. But if there is evidence, I will find it. It's the only chance I have."

"In that case ... come to the house tonight at three in the morning. Everyone else will be asleep. I'll show up at the door and let you in. Three o'clock sharp."

"Does it have to be that late? I have a class to teach tomorrow at ten."

"None of the frat boys will be asleep before midnight, and it'll take a while before they all nod off. If you're dedicated to this case, you won't think twice about staying up late."

"... Right. I'll see you then."

I hope there's evidence, Karl thought. But if there is, if there's any clear sign that Joey is the killer I think he is ... I'm determined to find it.

9

Karl did end up writing that apology letter. Not necessarily a waste of time, he thought as he pulled up to the Phi Upsilon Rho house. After all, building and maintaining relationships is what matters most in life. Besides, if this turns out to be a bust, I'll have several apologies to write in order to pick up the pieces. Somehow I get the sense that this is going somewhere, though.

Karl knocked on the door of the building. It was a cold night, and Karl was hoping that Pete got around to opening the door soon, or else it would soon become unpleasant to wait in the cold. Karl checked his watch: it was 3:00 a.m. on the dot. Fortunately for him, Pete was soon there to greet him. "Quiet," Pete whispered harshly. "We don't want to wake anyone up."

Karl stepped into the house. The blood and such had been cleaned up, and the house was dark. It was eerie to observe the place where, eight days before almost to the minute, Paul Chlebek had been murdered--and here, standing next to him, was Paul's twin brother. "Pete," Karl whispered in a sudden flash of emotion. "I'm so sorry."

"I'll be fine, Dr. Kairo, if we can get this evidence," Pete replied, determination in his eyes.

Dr. Kairo walked slowly to the left, through the door into the lounge. Then, crossing the lounge, he stepped through a doorway to the right, into what he knew would be the library.

A large portion of the space in this room was taken by a table, surrounded by chairs. Upon the table were the stacks of paper that Karl was used to seeing everywhere: homework, exams, and essays. The eastern side of the room had three computer docking stations and a printer, all of which were currently inactive. The western side was occupied by a large bookshelf of textbooks, sorted by course number, as well as files of past exams. A large chandelier hung from the ceiling, which sparkled slightly in the darkness. Beyond that, it was hard to see much of anything, as the room had no windows with which to let in moonlight. "Here we are in the library," Karl said.

"How did you know it was this way?" asked Pete.

"I've studied maps of this place. Do you have a flashlight?"

"What about your smartphone?" Pete asked.

"Ah, yes," Karl replied, taking out his phone and turning the light on.

There were no ghosts, corpses, or blood stains to surprise Karl from the darkness. Instead, he saw what looked to be the same room, just brighter. Karl shone the light up at the chandelier, which dazzled him with refractive crystals. Examining the chain of the large light fixture, Karl saw it to be undamaged. He chuckled to himself a bit.

"What? What is it?" Pete asked.

"Nothing. It's just that in murder mysteries, there seems to be a rule that if there is a chandelier, then it will fall at some point. This doesn't appear to be the case here."

"This isn't a storybook, professor," Pete replied.

"Right, right. Now then," Karl whispered as he looked at his smartphone, "is Joey Turner a good liar?"

"What? You ... you don't--"

"It's to our advantage if you simply answer the question," Karl interrupted.

Fearing, rightly, that Karl suspected Joey of the murder, Pete spoke up. "No. He's creative, of course, but in a slow way. If you ask him to craft a grandiose story, he can get back to you tomorrow, but he's bad with coming up with lies on the spot."

"And ... let's see ... he's taking Drawing, Computer Graphics, Differential Equations II, and Computer Architecture. Clearly a computer scientist, but still trying to be well-rounded. Now then," Karl said as he began to look over the books. "Where would the textbook for Drawing be?"

"There's no textbook for drawing," replied Pete. "It's a workshop class."

"Ah. So it would be better to find the textbook for Computer Graphics first."

"If you're looking for a good lead, try DE II. I remember Joey saying he had a big test in that on Monday."

Who would schedule a test for Monday? Karl thought. Eh, probably Joshua Mitchell. "Okay then," Karl responded, now scanning for mathematics textbooks. On the third shelf from the bottom, on the left side, there was a textbook for Differential Equations II. Karl pulled it out to reveal a dark red stain in the lower corner. "This," he said quietly, trying to contain his excitement, "is why we're here."

Pete gasped. "Is ... is that ..."

"Yes," Karl replied. "This would seem to be the murder weapon."

"But why ... would Joey have his math textbook out at well past three in the morning?"

"Here's Joey's story. According to Joey, Tim was here trying to use this to cheat on a test. Paul walked in and caught him, and Tim whacked him on the head, killing him, while Joey ran away. I suspect this is essentially the correct story, but with Joey's and Tim's names switched."

"So you think Joey was cheating?"

"Planning on it, at least. Paul walked in, and Joey got angry, killing Paul in a drunken rage. Tim saw this and ran away. Andrew was practically blacked out, so he remembered nothing."

"Andrew was there?"

"Yes, those are the four characters in this story."

"No, Andrew wouldn't get blackout drunk. He's the librarian; he would be enraged if anything went wrong in here."

"... Okay, then that was also a lie from Joey. Regardless, we have the evidence now. ... Some details are missing, though. Mind if I snoop a bit more?"

"I mean, go ahead? Try not to get yourself into any trouble."

"Which way is Joey's room?"

"Why on earth do you want to go into Joey's room?"

"I have a hunch."

"You know what, no. You're not going into Joey's room on a hunch."

"... You're right. I'm getting carried away here."

"... Who is that?" asked a voice from the shadows.

Karl turned his flashlight to reveal Andrew Fong, walking into the room with a bloodied knife. Pete gasped, trying not to scream. Andrew raised a wing to cover his eyes, blinded by the flashlight.

"I'm Dr. Karl Kairo from the mathematics department," Karl said calmly. "I've been trying to solve the murder of Paul Chlebek. Perhaps you should sit down and tell us everything you know ... such as why you're carrying a bloody knife."

Andrew slowly placed the knife on the table and raised his wings in surrender. Then, he sat in one of the chairs around the table. Karl sat opposite the owl, placing his phone down so that it would dimly illuminate both of them from below. Pete quickly sat next to Karl, still shaking from the surprise of seeing Andrew.

"Why are you carrying a knife with my brother's blood on it?" Pete quietly but harshly demanded.

"Calm down, Pete," Karl said. "That isn't Paul's blood on the knife. It's Joey's."

"What?" Pete and Andrew both asked at once. Pete was incredulous; Andrew was nervous.

"It all makes sense now," Karl began to elaborate. "You weren't drunk, Andrew. You were right there, in the room, when the murder happened. You weren't about to have anyone spill blood all over your precious library, especially not another member of the fraternity. So you attacked Joey with that knife, in the leg. That's why he was limping when I first saw him, and that's why he was lying to me when he claimed he had run into the attic."

"... Well, I knew getting a Ph.D. meant you were smart, but not that smart," Andrew replied.

"S-so it's true?" Pete asked, still shaking.

"Yes, what Dr. Kairo said is true. At 3:51 I got here from the bar, only having had a few drinks. I saw Joey beating Paul over the head with a textbook, so I lunged at him with the knife. I was able to cripple him before Tim pinned me to the wall."

"So Tim was in on it?" Karl asked.

"Tim told me the story, so that I would shut up about it. This was all John's idea--not the murder, of course, but the cheating ring."

"... Cheating ring?"

"John and Tim were both in danger of failing a couple of courses, so they hatched a plan. The idea was that if two sections of a class had a test, the students from the second section would take pictures of the textbooks and their notes to send to the students from the first section, who would return the favor by texting questions and answers during the exam for the second section. They had quite a few wrapped up in it--Joey, Jeff, Patrick, Taylor ..."

"And if a scandal like that became public--" Karl began to say.

"We could lose our privileges," Andrew finished for him. "We're already on probation for an incident at a party last year, so at this point we'll probably be shut down. No point in trying to get around that now."

"I'll see what I can do," Karl replied, "but no guarantees. The homicide doesn't exactly help."

"It doesn't," Andrew agreed. "I was hoping not to say anything, but you've uncovered the whole story now. I'll help you to clear this up however I can."

"First of all," Karl suggested, "you explain to us why you're here with the knife."

"I came down after realizing that I hadn't gotten a good opportunity to wash the blood off yet."

"Please don't at this point. I'm now going to call the police. I'll show them the book and the knife so they can be analyzed, and I'm also going to turn you over to them for questioning. You'll tell them what you told me?"

"Yes."

"Good, then they won't have to play the audio recording that I'm taking with my smartphone right now. With how strict privacy laws are these days, that probably wouldn't be admissible anyway. Still, if I were you I would stay true to your word." With these words, Karl picked up his smartphone to call in the Warren City police force. "Pete, keep an eye on Andrew while I make this call."

***

Soon, there was a knock on the door. "I'll get that," said Karl.

Heading back to the foyer, Karl proceeded to open the door to a familiar face. "Officer Caulfield," he began.

"Please," Officer Caulfield responded in annoyance. "If we could dispense with the formalities and get to the point?"

"Right. The evidence is in the library. You can follow me." Karl walked the police into the library. Officer Caulfield turned the lights on, and Karl, Pete, and Andrew all instinctively squinted in the bright light of the chandelier.

"Officer Caulfield, I have solved this case," Karl said.

"Sir, I remember seeing you at this house before, and our interaction there was ... unpleasant. So before you give me your speech, I'd like to begin by dispelling any thought you have that you will leave this house on your own and sleep in your own bed tonight. I know that you're up to something, and I will use every means possible to prevent you from causing more trouble than you already have."

"And finding a professor in a frat house at three in the morning with a bloody book in his paw is probable cause enough. I understand."

"In any case, you have the right to remain silent, and I would suggest that you exercise it while we investigate this scene ourselves."

"All of the important evidence is here, don't worry. You'll see that I'm right soon enough."

"Wonderful," replied Officer Caulfield sarcastically. "Officers, please arrest all three of these men. We'll make sure that they don't do any more damage tonight." The officers approached the three characters with handcuffs.

"Sorry, man," said Officer Abel as he closed the cuffs around Karl's wrists, behind his back.

"Don't worry about it," Karl shrugged. "Just as long as I get to call Arfie from the jail. I have a favor to ask of him."

10

"I apologize for inviting you to my office on a Wednesday afternoon," Jake began as Karl sat across from him. The two of them were once more in the mayor's office. "You must still be tired after staying up all night Monday."

"I'll be fine, I've done it before. I just wish the police hadn't interfered."

"Well, Professor Kairo, next time they won't." Jake was smiling.

"You mean to tell me that I've earned your detective position."

"I see no reason to withhold this news. The police have found your book and knife to be what you claim them to be. The trial won't be for another few weeks--justice is never as swift as we'd like it to be--but I, and many others, expect a speedy conviction. Somehow, Dr. Kairo, your little game has paid off."

"Honestly, it was luck."

"Whether it was luck or skill, I must thank you. Your investigation has solved the murder of Paul Chlebek, and I am sure your investigative talents will be useful to me another day. Useful to us all, in fact. Meanwhile, I am sure you would like to have some sort of celebration, or at least something to relieve the stress. I would thus like you and one other person to meet me Friday evening at Baldwin's Steakhouse in downtown Warren. I will pay."

"You're inviting me to dinner." Karl's eyes widened. Baldwin's was known for being the most expensive and luxurious restaurant in the city.

"Consider this your invitation, yes. Any idea who else is coming? Perhaps a lady?"

"Sadly, no. But I'm sure Arthur would love to join us. I ... sort of owe him one after asking him to pay bail at five in the morning."

Jake smiled. "Wonderful. I shall see you two then. In the meantime, expect paperwork to arrive in the mail within a few days about your new side job. It will explain how to ensure that everything goes smoothly in your further investigations. I will call you up myself if I need you."

"Well, goodbye then. See you Friday." Karl got up and turned to leave. But before he was out of the door, he remembered something. "Oh," he said, turning around again. "You said the last time I was here that you wanted our first meeting to be in person. But I was only there because I wanted to look up the bus schedule. How did you know I would be at City Hall?"

"I have my ways," was Jake's simple response.

Karl stared confusedly at Jake for a moment, then decided to drop the question and leave.

***

"Hey," Karl stated. He was calling Arfie to tell him the news.

"... Who is this?" was the response.

"It's Karl. I'm calling from a borrowed phone, remember?" Karl was borrowing the neighbor's phone, as his smartphone had been confiscated as evidence.

"Right, right. What's up?"

"The mayor has invited us to dinner at Baldwin's on Friday evening. Can you come?"

"Wait, what?"

Karl realized that Arfie knew nothing of Jake's involvement in this. "Oh. The mayor, Jake Blueglid. He's met with me a couple of times to discuss the case."

"Oh. Interesting."

"Yes, and he's made me an official city detective ... or something like that. I'm not sure what the position's officially called, but basically, if there's another unsolved case, I might be devoting several hours of my free time to it."

"... That's incredible. Can you still teach?"

"Yes, don't worry. And they'll pay me, too. Look, here's the question I want to ask. Can ... do you want to help me?"

"What do you mean? Be a detective?"

"Yeah, you know, like, Holmes has Watson, that sort of thing. You've been a great help in this case."

"Oh, well, I don't know if I could. Would that even be allowed?"

"What do you mean? I'm sure Jake will be fine with the idea if we ask on Friday."

"... And I can keep my job?"

"Hey, my job's harder than yours, and I'm keeping it."

Arthur thought back to his conversation with Sonny. While he loved his janitor job, it would be nice to have some excitement in his life again. "... I'll do it."

"Good. I'll see you Friday then? Oh, why don't you pick me up so that we can ride together? I think I'm on your way."

"... I guess that works, yeah."

"Cool. See you then." Karl hung up.

***

Paul's funeral took place within the week. Karl and Arthur showed up looking their best; Karl was in his best suit, while Arthur was wearing a polo shirt and khakis. There they saw Pete and his parents and gave their condolences. Pete's mother, at a loss for words, simply walked to Karl and hugged him, showing her gratitude for his work on the case. Pete rolled his eyes at his mother's lack of composure, to which Arthur couldn't help but chuckle to himself.

Karl showed up to the trial on the 6th of December, well after the actual murder but still within the period allowed by the law. Along with other witnesses, including Pete and Andrew, Karl recounted his story. As predicted by Jake, the jury returned with a conviction of second-degree murder for Joey Turner, for which he was sentenced to forty-five years in prison. Tim and Andrew were later also tried for their actions.

Karl was declared to be an Official Investigator of the City of Warren, a position that he suspected to have been entirely created by Jake for him to fill. He still wondered what exactly Jake's motives were; he was an enigmatic personality. In any case, Karl gained a badge to carry with him, allowing him to obtain the support of Warren City's police force if he needed it. The next step for him would be simply to wait for a call from Jake.

The Phi Upsilon Rho fraternity was punished as Andrew had feared; however, they were not disbanded entirely. Instead, the fraternity was simply forbidden from recruiting any students for two years and told they would remain on probation for two more years afterwards. This was sad news for many in the fraternity, but the consensus was that it was, unfortunately, appropriate after what had happened. Joey, Tim, John, and the others in their "cheating ring" were promptly dismissed from the school for academic dishonesty.

Karl was invited to coffee independently by Dr. Ronald Grant, the university's president; Dr. Beau Harvey, the dean of faculty; and Dr. Gregory Werner, professor of history and the faculty advisor for Phi Upsilon Rho. Each of them wanted to talk to Karl about his little "side hobby" of solving the murder. Karl wondered if he ought to write a book about the story, so that people could stop asking him about it. Eh, he thought. I probably won't have time, with my other hobbies and the actual crime-solving in my future.

Pete Chlebek soon became the new president of Phi Upsilon Rho. His first act as president of the fraternity was to invite Professor Kairo for a dinner with the fraternity members. It was the tamest dinner the house had seen in months, as they wanted to give a good image. Karl, of course, insisted that Arthur come with him. It was difficult to explain to the students why one of their math professors was bringing a janitor to dinner, but Karl was happy to explain it to them.

***

Karl walked into Fred Douglas's office after receiving another message asking for a meeting.

"Ah, Karl," Fred said. "Please sit."

Karl closed the door and sat down.

"First of all, I would like to apologize for not taking the time to have this meeting sooner. I've simply been too busy over the last several weeks."

"I understand," Karl responded, hoping Fred would get to the point.

"Karl, I hear that you've accepted a position working for the city government."

"It's a second job. This is still my first job. Think of the other job as more of a hobby."

"Karl, I just want to remind you that if you want to receive tenure in four years, you will be expected to devote a large amount of time to this school and to your position here. I'm not saying that you can't take a second job, but I just want to ensure that your career takes priority."

"As I just stated, this is my first job. I'll try to ensure that there's no interference."

"Good. That is all I need to hear."

"... You called me in here just to ask that?"

"No; no, I also wanted to congratulate you on solving the case. I apologize if I've been a bit ... harsh to you in the process. But what I said still stands. Some of your actions were inappropriate."

"... I did solve the case."

"You did, and it's harder to justify my criticism in that context. Some would say that the end has justified the means. However, I think you and I both agree that you got lucky this time, Karl. In the future, try not to act so unprofessionally."

"... I'm sorry."

"You have already apologized to me in writing, and I accept your apology. I merely wanted to remind you that the events that have taken place since then do not entirely nullify it."

"Is that all?"

"No, one last point remains. My family keeps asking me about you and your detective work, and as such, I would like you to come to my house for dinner this Saturday evening."

"... Can Arfie come with?"

"If you so desire. I'm sure my kids will be happy to meet both of you." Fred smiled.

Karl smiled as well. "All right, Dr. Douglas. I'll see you then."

"Please, you're a professor, and I'm a professor. Call me Fred."

Karl merely nodded, got up, and left the room. Well, it seems I've cleared the issues in that relationship, he thought to himself. I thought this month couldn't get better.

And this was Karl's sentiment as he entered his office and closed the door, with "KARL W. KAIRO, PH. D. PROFESSOR OF DISCREET MATHEMATICS" on it, in Holden Hall at Warren City University. With a professor doubling as a detective, the school was certainly one of the Midwest's most curious colleges.