By D. Aviva Rothschild

"Messy," said Police Inspector Narishkeit as she surveyed the scene of the murder. The chalk outline of the body of world-famous mathematics professor and puzzle-maker Loy Gevalt lay in the center of his circular living room. A red trail that led in and out of the room attested to the fact that Dr. Gevalt hadn't died right away, but had been able to stagger around his house before loss of blood became too great and he lay down and died.

Impatiently, the inspector shifted her weight from one foot to the other. "What I can't figure out," she muttered to herself, "is why in the world he didn't try to staunch his wounds or even call someone, since he was alive for at least fifteen minutes after he was shot. Instead," and she shook her head, "he went around collecting and arranging rubbish!" She glared at the octagonal piece of cardboard lying on the chalk outline, the offensive medal near it, the piece of yellowed newspaper jammed into the thermostat.

The inspector also rolled her eyes at the odd construction of the living room. Not only was it a perfect circle, but it had 12 hallways radiating out from it at regular intervals. Although there was furniture in the room, the exact center was empty--except for the chalk outline, of course.

Then footsteps sounded in one of the twelve halls that radiated out from the living room, and in stepped Garten Mardner, the whiz-bang trigonometry teacher from UCD. Inspector Narishkeit rolled her eyes; she hated involving civilians in police work, but this case was so baffling that they were willing to try anything.

"Greetings," said Dr. Mardner, and the two women shook hands. "I feel privileged to have been asked to help solve this heinous murder. The world of mathematics lost a great man to a maniac, and I want to make sure that the killer is coterminal with Dr. Gevalt."

"Excuse me?" said the inspector.

"Sorry, just making a fairly bad mathematical joke. Now, why don't you fill me in on what you know."

"Right." The inspector took out her little notebook. "The subject was found by his housekeeper on Tuesday, June 13, at 8:34 A.M. She came into the room looking for him because he was supposed to attend a book signing that day."

"A sine function, I see," mused the doctor. "And afterwards, he might have gone surfing on a sine wave. Would that have been a cofunction? And if it happened regularly, it would be a periodic function."


"Sorry, just going off on a tangent. Please continue."

The inspector gave Dr. Mardner an annoyed glance, then continued to read: "An autopsy showed that the professor had been dead for about 30 minutes at that time. Cause of death was from a bullet wound to the stomach. The killer fired from one of the hallways leading into this room."

"Which one?" asked Dr. Mardner.

"We don't know, for two reasons. One, the killer moved the body after it was shot so that we would not be able to tell the direction the bullet came from. Unfortunately, he or she wore gloves, so there are no fingerprints. Two, the professor came to after the killer left, and he moved around for a few minutes before collapsing back in this spot, which, as far as we can tell, is where he was shot initially."

"Do you have any clues as to the killer's identity?"

"None that we can interpret. The professor had gathered some things that I suppose he thought would tip us off, but so far we can't understand a single one of them." The inspector sighed. "The professor apparently was so dazed by his wound that he could only think to make a puzzle out of his killer's name--he didn't conveniently write the person's name down."

"Are these the clues?" Dr. Mardner made a kicking motion at the things on the floor.

"Some of them. There are others. We left everything as was, just in case."

"Good." The doctor bent and picked up the little medal. It was a Nazi swastika.

"We think perhaps the killer left that there as a calling card," said the inspector. "Professor Gevalt was Jewish, and it's possible that the killer's angle was anti-Semitism."

"Oh, now don't jump to conclusions, especially about angles." Dr. Mardner grinned. "I do believe this is a signal to us to employ the unit circle."


"The Nazis were part of the Axis Powers, weren't they? One uses the unit circle to solve right triangles on a standard X-Y axis."

"If you say so," the inspector said uncertainly. "I'm sorry to be so obtuse, but we've had an acute shortage of mathematical training ever since the Republicans decided it was unnecessary for law enforcement agents."

Dr. Mardner smiled and continued: "My interpretation makes sense, given the construction of this room. And this." She picked up the cardboard and turned it over. "A stop sign. Clearly one of the trigonometric functions for angles."

"Sine?" the inspector said, showing off what little she remembered from college.

"Probably not, given that it was inverted, which would indicate either cosine, cosecant, or arcsine. I lean towards the latter, but without an angle it will have to wait. Now." Dr. Mardner carefully replaced the sign and moved to the thermostat. "What's this?"

"It's a review of one of Professor Gevalt's books. It's a very old review and doesn't relate to anything he's been working on in the last 20 years."

Removing the review slowly, so as not to rip it, Dr. Mardner read the review. "No wonder he cut it out--it's a very good review. One might even call it--complementary."

"Yes, so?" the inspector prompted.

"Why, here's a major clue! The thermostat is in degrees, which can be used for angles as well. I daresay Professor Gevalt set the temperature to some significant angle. However, the placement of this review indicates that rather than take the reading directly off the thermostat, we should treat it as an angle that is complementary to the real angle he wanted to depict--doubtless a small one that couldn't otherwise be programmed into the thermostat." Dr. Mardner looked at the LED readout. "Hm, 72--that means the complementary angle is 18 degrees."

"Just a second, Dr. Mardner," the inspector said hurriedly. "That wasn't the original temperature we found when we came in here the first time. I remember it well--the heat had been turned all the way up, and it smelled just horrible in here. So we turned it down."

Dr. Mardner's face fell. "I see. You wouldn't happen to remember what the temperature had been set to?"

"As a matter of fact, I do--it was at 88 degrees. That's a number that's hard to forget."

"Wonderful! Now, I wonder what a 2 degree angle could relate to?"

The inspector crossed the room to stand over the chalk outline. "Here's a guess--the angle at which the bullet entered the professor's body. We weren't able to get a precise measurement, but we estimated that the bullet entered at an angle less than 5 degrees from where the gun was held when it fired."

Dr. Mardner slapped her forehead. "Of course, of course! One of Professor Gevalt's most amazing talents was being able to estimate the angle of anything--even, it seems, the bullet that ultimately killed him."

"Here are more helpful facts," said the inspector. "The bullet entered him at a point precisely 3 1/2 feet from the ground and was traveling downward. Also, from the amount of damage done and the depth of the bullet, we've estimated that it traveled 14 feet before striking him."

Dr. Mardner rubbed her hands together. "I do believe we can derive some figures from this information." She whipped out a blank piece of paper and drew what looked like a right triangle on top of a rectangle, entering in data where she had it. "Now, let's see--when we determine this length--"she tapped the side of the triangle adjacent to the angle "--we'll know precisely how far away the killer was standing, and when we determine this"--she tapped the side opposite the angle, "we'll know how high the gun was held, which will give us a clue as to the gunperson's height."

She busied herself with sine and cosine calculations and soon reached the two conclusions: "The killer stood exactly 13.99 feet away and held the gun .49 feet higher than the 3.5 feet, so the gun was 3.99 feet in the air. Almost a straight line, but not quite."

Inspector Narishkeit hopped up and down most unprofessionally. "With this information we can eliminate those suspects of heights that would not let them comfortably shoot a gun at that angle!"

Dr. Mardner merely smiled. "But we still need to determine what hallway the professor was shot from and the killer's actual identity. Its fundamental trigonometric identity, if you will.

And don't forget that we still don't know what to do with the stop sign. Are there more clues around, perhaps as to the killer's motive?"

"Well, see for yourself. Nothing has been touched, so you might find something useful."

For the next ten minutes or so, Dr. Mardner cruised around the professor's home, not picking anything up but taking careful notes on what she saw. Finally she returned to the inspector, who was sitting in an overstuffed chair, resting her feet. "I found a few interesting things, including one I half expected." She showed the notebook to the inspector, who read:

Four small pecan pies balanced on top of three more in the oven

A little Noah's Ark-shaped salt shaker next to the oven controls; the oven was set to 500 degrees

A legal document on the professor's desk

A piece of paper with an angle on it; the angle's points were labeled FGH, with the G at the vertex

"I think we've got this sewn up," the doctor beamed.

"How so?" the inspector asked.
"Well, first, the pies tell us in which hallway the killer stood."

"They do?"

"Yes. Really, this was somewhat simplistic of Professor Gevalt! He simply means 4 pi over 3, which corresponds to an angle of 240 degrees, or--"Dr. Mardner swung around to point at the southwest part of the room, "--that hallway, whose opening corresponds to the terminal side of the angle. Very terminal, in this case," she added.

Inspector Narishkeit rushed into the hallway to look at it more closely. She came back excited. "There's a tiny bit of dried mud fourteen feet away back there! No wonder the killer wanted us not to know where he or she stood--we can chemically analyze the mud and trace its origin!"

"Exactly. The rest of the clues, plus the stop sign, should pinpoint the killer. First, I looked at the angle. Its significance puzzled me for a few minutes, until I remembered that the side on which the angle starts is the initial side of the angle."

The inspector's eyes widened. "You know, we saw that angle and wondered if the letters meant anything, but the professor knew so many people... besides, no one we know of associated with him has the initials FGH. FEH, sure, and one FCH, but that's too clearly a G to suspect FCH. Besides, she's in Italy."

"You came closer than you realized. If you just take the initials on the initial side, you get GH, or perhaps HG."

"Of course!" the inspector cried. "Harry Greenaway! The two were rivals at Boston University!" But then her excitement faded. "But there's also Greta Harris, and Grant Hayes, and Harvey Greenbaum, and the professor's brother, Hy Gevalt, and his estranged wife Helen Gevalt. It's a narrowing, for sure, but we need proof. A motive would really help."

"We have one," Dr. Mardner announced proudly. "At least, I believe it will suffice." She moved her finger up to the entry for the ark on the stove. "Here is a situation similar to that of the thermostat, except that rather than using degrees, we must figure the angle in radians."

The inspector looked at the entry. "Can you have a 500 degree angle?"

"Of course! Now, let's see, a 500 degree angle works out to (500pi/180), or 8.7266 radians."

"How is that significant?"

"By itself, it's not. But take a look at this." Dr. Mardner showed the inspector the legal document, which was a copy of a loan agreement. "Note the amount requested."

"$87,266! But I thought the professor was a wealthy man."

"Not according to the Western Journal of Mathematics gossip section. It seems our famous professor had gotten himself into several bad stock deals and needed this loan to keep from losing his house. Now, I want you to take a look at this signature here..."

"Hy Gevalt!"

"Right--he cosigned this loan, hence the meaning of the inverted stop sign. Now we know that Professor Gevalt mean the cofunction of sine rather than the inverse or reciprocal identities. Here's your motive: Hy Gevalt was convinced that his brother would default on the loan, leaving him stuck with the payments. Rather than risk that, he probably felt killing his brother would get him off the hook somehow. I wouldn't be surprised if someone has taken out a large insurance policy on the professor; you might also question Helen Gevalt."

With a quick nod, the inspector ran out of the room to make a phone call.

Filled with the satisfaction of a job well done, Dr. Mardner sat down in the overstuffed chair and flipped out the footrest, stretched out with a sigh. "Maybe I can talk the math department into giving me a raise for today's work--sort of a wangle of elevation."

Copyright 2000, D. Aviva Rothschild

About the Author

See the "About the Editor" page. I wrote this story back in 1995 when I was taking a trigonometry class under the assumption I would someday be a programmer. I've lost it all; I can't even remember why some of these jokes are funny. I'd love to hear from someone who knows trig--are these jokes funny?

Tell Aviva Rothschild what you thought of her story!

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