Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
 October 20, 2017 Vol. 16, No. 4

Next week's colloquium will again feature student research

Title: Student research presentations
Speaker:  Mark Powers and Keri Haddrill
Time:  Tuesday, October 24 at 4:00 pm
Place:  VanderWerf 102

Graph Pebbling: Doppelgangers and Lemke Graphs

On a connected graph, we place pebbles on its vertices. By moving pebbles, we explore limits on how many pebbles can be placed in any configuration so that we can place at least one pebble on any vertex in the graph. We have defined doppelgangers on a graph, which allow us to add vertices in a way that bounds how many pebbles are needed in a configuration to still get one pebble anywhere. Using doppelgangers, we can then extend the known set of Lemke graphs, which are graphs that violate the two-pebbling property. Lemke graphs are suspected to be involved in a counterexample to Graham's Conjecture, the most notorious problem in graph pebbling today.

Droning on about Sand Dunes and Machine Learning

Gusty winds expand across the dunes and send sand flying everywhere. They brush past the vegetation grasping at the sand, and together they alter the formation of dunes. The Saugatuck Harbor Natural Area (SHNA) along Lake Michigan has numerous open sand dunes at risk for extinction. Thus, our project’s goal is to map the changes of vegetation over time. The first step in mapping SHNA is comparing the surface reflectance images collected by a drone and field biomass measurements from a few small selected areas. This information allows us to create a convolutional neural network that can approximate above-ground biomass and therefore produce a biomass map for the entirety of SHNA.

MATH Challenge

We'll be holding practice sessions for the MATH Challenge mathematics competition on Monday, October 23 and Monday, October 30 from 8:00 to 9:30 p.m. in 237 VanderWerf.  Please attend if you're interested in competing.

The MATH Challenge will be held here at Hope on Saturday, November 4 from 9:30 a.m. to 12:30 p.m.  This is an intercollegiate regional competition.  If you are interested in participating, please let Prof. Cinzori ( know by Tuesday, October 24.  You can sign up with your friends as a team of up to three people, or you can sign up as an individual and be placed on a team.  More details in Off on a Tangent.

You can find some recent competition problems herehere, and here, and links to more in the sidebar here.

Facebook Messenger now can interpret Latex Code

Latex is a typesetting system that most mathematicians use to write papers, books, and your calculus quizzes. It is used so much because it allows the user to produce nicely typeset mathematical equations as well as other mathematical objects like matrices.

Now you can communicate with your favorite mathematician using Latex code in Facebook Messenger. It does come with some limitations. It only works on the desktop version, not your phone, you need double dollar signs around your text, and the entire message has to be in Latex, not just part of it.

Give it a try! 

Problem Solvers of the Fortnight

In our last problem of the Fortnight, Noah had his workers move 69,489 crates.  The job took nine working days.  Every day after the first day, he put six more workers on the job; and every day after the first day, each of the workers -- by arrangement -- shifted five fewer crates than was the quota for the day before.  The result was that, during the latter part of the period, the number of crates being moved actually began to go down."

What was the largest number of crates moved on any one day?

Congratulations to all those that gave correct solutions: Jincheng Yang, Philip LaPorte, Zheng Qu, Russell Houpt, Andrew Nguyen, Alex Osterbaan, Caleb Stuckey, Yizhe Zhang, Nicholas Lillrose and Karthik Karyamapudi.

Problem of the Fortnight

A red circle is inscribed in an equilateral triangle of side length w (see 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 (see Figure 2).  The next step in this process is shown in Figure 3.  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?

Write your solution (showing all relevant work) on the back of a twenty dollar bill and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson's office (VWF 212) by 3:00 p.m. on Friday, October 27.  As always, be sure to include your name and the name(s) of your math professor(s) -- e.g. Hope Anna Prayer, Professor Wacław Sierpiński  --- on your solution.

Figure 1

Figure 2

Figure 3

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

David Hilbert

Off on a Tangent