The Game’s Afoot
The following is a synopsis of the first part of an unpublished story. Your task is to solve the mystery and write the rest of the story. Your conclusion must include the mathematics of the solution. It should be written in story form and incorporate the mathematics within the dialogue and other prose as smoothly and naturally as possible. Be creative, but don’t arrest the wrong individual!
It was a dark and stormy night. Holmes and Watson were called to the scene of the murder by Inspector Lestrade of the police. The victim was a wealthy but cruel man. He had many enemies.
The most likely suspects are the wife, the business partner, and the butler. Each has an equally strong motive. Each also has an alibi. The wife claims to have spent the entire evening at the theater across town. She was seen leaving the theater at 10:30 p.m. and returned home at 11:00 p.m., going straight up to her bedroom. Her return was verified by the upstairs maid. The business partner claims to have spent the evening working on papers at the office. His wife and household staff verified that he returned home at 10:30 p.m. The butler was on his night off. He claims to have been at the local pub until 10:00 p.m. The butler returned to his quarters above the carriage house at 10:05 p.m. and did not leave. This was verified by the other servants.
The body was found in the victim’s study. Holmes arrived at the scene at 4:30 a.m. The room was unusually warm and stuffy. One of the police officers went to open a window. Holmes admonished him to delay that action until he had completed his investigation of the crime scene. He instructed Watson to determine the temperature of the body. This was found to be 88.0°F. Holmes questioned the servants as to the room temperature during the evening and learned that the man had liked the room warm and that the temperature in the study was always very near the current 76°F. Holmes asked Watson to take the temperature of the body again at the conclusion of his inspection of the scene, two hours after the first reading. It was 85.8°F.
© 1999 Key Curriculum
Press
Teacher Notes 8
The Game’s Afoot
This is a fun paper to write. The mathematics are relatively simple to compute and the mystery is not too difficult. Writing the story ending as a story, though, and incorporating the mathematics within the dialogue is a bit of a challenge. Students seem to enjoy doing it, though. Trying to explain how Holmes does logarithms and uses the number e without the aid of a calculator gives students a chance to be very creative. Look for smooth incorporation of mathematics, natural dialogue, and, of course, the correct solution to the mystery. By the way, the butler didn’t do it. It was the business partner. Working out the time of death from Newton’s law of cooling, Holmes discovers that the wealthy but cruel man was killed at 10:15 p.m. The only one without an alibi for the crucial time is the business partner.
Sample Student Paper
The Game’s Afoot
Parker Eberhard
Luckily for Inspector Lestrade, Holmes was a former math student of the famous Kamischke duo and he knew what was going on. As Watson continue to dust the area for fingerprints and look for DNA samples on the body, he noticed that Holmes was sitting at the table with a pencil and paper and what seemed to be a small telecommunications center.
“This is no time to be surfing the net!” shouted Watson.
“I’m not,” replied Holmes. “Anyways, I had my e-mail account taken away for sending a chain letter so there is no way that I could surf the net. I’m trying to solve a murder mystery.”
“But how, Holmes?” Watson asked curiously.
“Well, while you are trying to search for clues all over the place, I am using the clues right under my nose.”
“I don’t see anything under your nose but your mouth and your hot, smelly breath,” Watson replied somewhat sarcastically.
“Exactly!!!” shouted Holmes; a wide grin appearing on his face.
“Uhhh, what are you talking about, I don’t understand?”
“It’s all very simple, Watson. See, using my TI CBL unit I can determine the temperature of the dead body; of which it is 85.0°F. I know that two hours ago the body’s temperature was 88.0°F, and that the temperature of the room was kept at 76°F. Do you understand so far?”
Watson nodded his head and signaled for Holmes to proceed.
“Well, using one of the many things that I learned in calculus, Newton’s Law of Cooling (T-TS=(To-TS)e-kt), I can determine the rate at which the body cooled and therefore determine the time at which the body started its cooling process; the time the murder was committed. We can substitute the 88.0°F for to and 76.0° for TS like so:
T – 76°F = (88.0°F- 76.0°F)e-kt
We know that the body’s temperature is 85.8°F right now at time 2 so we can plug this into the equation given us:
85.8°F –76°F = (88.0°F – 76°F)e-k2
With this equation, we can solve for K to find the rate at which the body cooled:
9.8°F = (12°F)e-k2
9.8°F/12°F = e-k2
.817 = e-k
Do you still understand, Watson?”
“Kind of. But what is that e-k stuff?”
“That has to do with natural logs and strange gardeners, but enough of that. You see, with this equation and some help from the calculator, I can find the rate at which the body cooled.” After consulting his calculator, Holmes concluded that the rate of the cooling of the body was:
(ln .817)/-2 = k
.10126 = k
“I can then plug this rate back into my equation along with 98.6°F and come up with the following:
88°F-76°F = (98.6°F-76°F)e-.10126t
12F/22.6°F = e-.10126t
(ln (.531))/-.10126=t
6.25 = t
“This 6.25 is the number of hours that the body had cooled, Watson. Six hours and fifteen minutes before we took the first reading at 4:30 a.m. was when the murder occurred. Therefore, the murder occurred at 10:15 p.m. and was performed by the business man while he was in route from the office to his house.
“You’re a genius Holmes, a genius. How on earth did you learn to do that?”
“Calculus, my Watson. Calculus.”
Writing Assignments A Watched Cup Never Cools
© 1999 Key Curriculum Press