Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
March 31, 2017 Vol. 15, No. 11

Next week's colloquia will explore whether or not dogs still know calculus

Title: Do Dogs Know Calculus?
Speaker: Tim Pennings, Davenport University
Time: 11:00 a.m. on Tuesday, April 4, 2017
Place:  VanderWerf 102

A standard calculus problem is to find the quickest path from a point on shore to a point in the lake, given that running speed is greater than swimming speed. Elvis, my Welsh Corgi, never had a calculus course. But when we played "fetch" at Lake Michigan, he appeared to choose paths close to the calculus answer. In this talk we form a mathematical model and reveal what was found when we experimentally tested this ability.

Title: Do Dogs Know Calculus? Bifurcations at the Beach
Speaker: Tim Pennings, Davenport University
Time: 4:00 p.m. on Tuesday, April 4, 2017
Place:  VanderWerf 102

We will show  that dogs - at least my dog, Elvis - knew calculus. That is, Elvis could find the optimal - fastest - route to a ball thrown into the water some distance down the beach. But what happened when Elvis was positioned in the water and retrieved a ball that was also in the water? When should he swim straight to the ball, and when should he swim in to the shore, run along the shore, and then swim back out to the ball? What is the bifurcation point for the change in optimal strategy and did Elvis find it? The answers will reveal that Elvis was the King of Calculus and more than just a hound dog.

Upcoming Colloquia

We have the following additional math colloquium on the schedule for this semester.
  • Thur, Apr, 20 at 4:00 PM in VWF 102: Dr. Lauren Keough - GVSU

Pi Mu Epsilon induction ceremony held recently

Ten students and professors were recently inducted into the Michigan Delta chapter of Pi Mu Epsilon.  Founded on in 1914 at Syracuse University, Pi Mu Epsilon currently has over 350 chapters at colleges and universities throughout the United States.  Hope College has had a chapter since 1974, the fourth in Michigan.

The purpose of the society is to promote scholarly activity in mathematics among the students in academic institutions.  Students were invited to join based on their GPA in their mathematics courses as well as their overall GPA.  The induction ceremony was held on March 14 at 6:28 p.m. (or pi day at two pi o'clock).  After the short ceremony everyone enjoyed our tradition of eating pie.

The inductees this year were:  Charles Cusack, Ethan Heyboer, Vicki-Lynn Holmes, Russell Houpt, Noah Kochanski, Daniel Kung, Autumn Roth, Jaquelyn Schwark, Ashley Stegenga and Nathan Vance.

Math Department game night enjoyed by students and professors

Problem Solvers of the Fortnight

In the last problem of the fortnight we saw the following exchange between Math Man and Vectoria:

"Hi Vectoria."
"Hey M's.  What's up?"
"Well, I have a question for you.  Your birthday's coming up, isn't it?"
"Yeah, it sure is!"
"Okay, then I have another question for you."
"Oh, c'mon, Double M!  You know when my birthday is!"
"No, yeah, I do.  I wasn't going to ask you that."
"Oh, my bad.  Go on."
"I was going to get you a chocolate bar from The Peanut Store."
"Oh, M's that's sweet.  But usually people keep birthday gifts a secret."
"Yeah, I know."
"Then why are you telling me what you're getting me?"
"Well, the chocolate bar isn't the gift, really.  The gift is this problem I got to thinking about.  I mean, I know you really like math problems, and I figured none of your other friends would give you a problem for your birthday."
"Awww, M's.  That really is sweet.  Thank you.  So what's the problem?"
"Okay.  So the other day at the Peanut Store, the owner -- who's a really nice guy, I have to say -- showed me a special chocolate bar he had in the back room after I told him I wanted to get something special for my friend's birthday.  The chocolate bar had two rows with 10 dark chocolate squares in the top row and 10 white chocolate squares in the bottom row."
"Sort of like a black and white cookie -- but a chocolate bar instead."
"Exactly. Now, I know you have a number of good friends, and you'd probably want to share with them, right?"
"Yeah, probably."
"So, not knowing what kinds of chocolate your friends like, I got to thinking about the different ways you could break it up.  And when I was thinking about it, I realized you'd probably enjoy the problem as much as the chocolate bar.  So here you go.  Suppose you break it entirely into pieces of size 2 x 1 (which have one brown square and one white square), pieces of size 1 x 2 (which have either two brown squares or two white squares, depending on which row they came from) and pieces of size 2 x 2 (which, of course, have two brown squares and two white ones).  In how many different ways can you break the chocolate bar into pieces of size 1 x 2, 2 x 1 and 2 x 2?"
"Hmmm. . . .  That is a fun problem. . . .  Let me play around with it, okay?"
"Sure.  See if you can figure it out before I give you the chocolate bar on your birthday."
"Cool!  Thanks, Double M!"

Help Vectoria determine how many different ways she could break the chocolate bar into 1 x 2, 2 x 1 and 2 x 2 pieces. 

Congratulations to Sydney Les, Philip LaPorte, Keri Haddrill, Sam VanderArk, Erik Johnsen, and Chris McAuley -- all of whom correctly solved the problem posed in the last issue of the newsletter.

Problem of the Fortnight

Figure 1
"Hey there, Mathlete!"
"Oh, hi, Vectoria.  Did you have a nice spring break?"
"Yeah, it was great!  My family and I made a spur-of-the-moment decision to go someplace, and we pseudo-randomly chose to visit Monte Carlo!"
"Wow!  Sounds like kind of a gamble just to up and go halfway around the world, but I bet your vacation was a winner!"
"It was.  One we decided where to go, we were 'all in,' as the kids say these days.  A couple days were cool and rainy, but you play the hand you're dealt, right?"
"Sure enough," Math Man says rather absent mindedly, as his eyes return to the paper he was looking at before Vectoria stopped by.
"What'cha workin' on?"
"Oh, just something," Double M says, "or, rather, s-u-m thing," he says, spelling it out and chuckling to himself at his clever turn of phrase.  "That was just a little joke.  I'm working on a math problem that involves a sum."
"I should have known it would be s-u-m thing like that" says Vectoria, spelling it out to volley his joke back to him.  "What's the problem?  You know I love those things."
"Well . . . A red circle is inscribed in an equilateral triangle of side length w, like this," says Math Man, pointing to figure 1.  "In each of the three corners of this triangle, a smaller equilateral triangle is drawn tangent to the red circle, and a cyan circle is inscribed in this smaller triangle, like so," he says, pointing to figure 2.  "The next step in this process is shown here," says Math Man, showing her figure 3.  "The question is: If this process of drawing smaller equilateral triangles with inscribed circles is continued indefinitely, what is the total area of all of the circles in terms of w?"

Help Math Man and Vectoria figure out the total area of all the circles in terms of w.  Write your solution (not just the answer) on a triangular piece of paper and circle your answer.  Drop it in the Problem of the Fortnight slot outside Professor Mark Pearson's office (Vander Werf 212) by 3:00 p.m. on Friday, April 7.  As always, be sure to include your name, as well as the name(s) of your math professor(s) -- e.g. Julia Szett, Professor C.R. Pinsky -- on your solution.  Good luck and have fun!  

Figure 2

Figure 3

Dave Kung making mathematical music during a Pi Day colloquium

Off on a Tangent