Off on a Tangent 17.6

Mathematical gift ideas

It is that time of year. No, not final exams, Christmas shopping. But what do you get the math geek on your Christmas list this year? To help you out, Business Insider made a list of some awesome gifts that include a Klein Bottle bottle opener, a telescope for your iPhone, chocolate gaming dice, and math joke t-shirts.

Esty also has a site devoted to gifts for math lovers that includes a Merry X-Math sweatshirt, pi ornaments, and some object that contains a hippopotenuse. (I guess that one is for those where only a hippopotenuse will do!) Head over to Zazzle for prime number mugs, math clocks, and lots of things that say something about math being importanter. (I’m not sure who would buy something like that!) Finally, if you take a trip over to Mathnasium you will find a cutie pi onesie, a hypercube paperweight, and pi cookie cutters.

Remember it is only (the smallest square that can be written as the sum of two squares)-days until Christmas!

Problem Solvers of the Fortnight

Congratulations to Adair Cutler, Alyssa O’Donahue, Britta Johnson, Calvin Gentry, Camryn Zeller, Ce Gao, Chris McAuley, Christian Erickson, Dane Linsky, Dylan Burke, Ford Fishman, Hugh Thiel, Jacob Zoerhof, Jordan Corstange, Julia Loula, Kachikwu Nwike, Kam Wilcox, Karen Quay, Keegan Frisbie, Lydia Meinhard, Mackenzie Green, Mark Powers, Meredith Bomers, Micaela Wells, Mikaelah Snoap, Thomas Kouwe, Valerie Dien, Yizhe Zhang, Zach Wylie, Jonathan Washburn and Zheng Qu – all of whom correctly solved the Problem of the Fortnight in the last issue of America’s preeminent fortnightly mathematics department newsblog.

Problem of the Fortnight

Positive integer n has the property that n – 64 is a perfect cube. Suppose that n is divisible by 37. What is the smallest possible value of n?

Write your solution (not just the answer) on a Christmas decoration of your own making, and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson’s office, room 212 in The Werf, by 3:00 p.m. on Friday, December 7. As always, be sure to include your name and the name(s) of your math professor(s) – e.g. Carrie A. Toone, Professor Carol Ling – on your solution. Good luck and have fun!


Off on a Tangent 17.5 (part b)

Graph Pebbling colloquium this week!

  • Title: Graph Pebbling
  • Speaker: Dr. Airat Bekmetjev, Hope College
  • When/Where: 11 am on Thur, Nov 29 in VanderWerf 104
AbstractPebbling is a process defined on a connected graph. Pebbles are configured on vertices of the graph and are moved along the edges. A pebbling move consists of taking two pebbles from a vertex, placing one on an adjacent vertex, and discarding the other. The objective of pebbling is to put at least one pebble on a designated vertex, called the root. A configuration is called solvable if any vertex can get at least one pebble through a sequence of moves. Pebbling can be used in transportation and communication networks and optimal resource allocation.
Some interesting questions arise: How many pebbles are enough to guarantee that any configuration is solvable ? What are the optimal ways to move pebbles? Can we construct an algorithm that determines if a given pebbling configuration is solvable? In this colloquium we will try to answer some of these questions along with the discussion about the existence of the threshold phenomenon in pebbling. We will see how the solvability of a random pebbling configuration dramatically changes after the number of pebbles reaches a certain value called the pebbling threshold. We will also discuss new results in pebbling related to the two-pebbling property and Graham’s Conjecture on the product of graphs.

Off on a Tangent 17.5

The irrationality of pi will be presented in next week’s colloquium

  • Title: Niven’s Proof of the Irrationality of Pi
  • Speaker: Dr. Aaron Cinzori, Hope College
  • When/Where: Tuesday, November 13, at 11:00 a.m. in VWF 102

Abstract:  Approximations of the value of pi are known to a ridiculous number of digits.  But to actually establish that pi is not a rational number requires not just approximations, but proof.  The first proof that pi is irrational was given by Lambert in 1761, and it required some sophisticated analysis, but in 1947, Ivan Niven gave a remarkably short and clear proof that pi is irrational using only ideas from calculus. (So it is as easy as pi!) We’ll take a detailed look at Niven’s proof as well as peeks at some of  the history of pi in mathematics.  And of course there will be poetry.

 Students participate in MATH Challenge

Hope again had a nice turnout of students participating in the Michigan Autumn Take-Home Challenge this past Saturday. Students competed with other students around the state (as well as other states) working in groups on ten interesting problems. Apparently this test was more difficult than usual (or not as easy as pi), so while we look forward to hearing the results in the near future we may not expect them to be very high scores. The following students competed (grouped by team):

  • Chris McAuley, Quentin Couvelaire, Evan Bright
  • Matt Edkins, Adam Czeranko, Mackenzie Green
  • Evan Mulshine, Ruth Holloway, Trevor Hile
  • Eric Leu, Jackson Krebsbach

Memphis Teacher Residency

David Montague, the Director of the Memphis Teacher Residency (MTR), will be on campus next Wednesday, November 14th to speak in Chapel and recruit for MTR. He is also hosting an Information Session on that same day from 6:30-7:30pm in VNZ 247.

MTR provides a full year residency program preparing teachers for work in low-performing public schools in Memphis.  They offer a Master’s degree in Urban Education (tuition free), a full year internship co-teaching with a mentor teacher, free housing and a monthly living stipend.  Their mission is Christian love expressed in equal education.

Problem Solvers of the Fortnight

Congratulations to Cole Persch, Evan Bright, Holly Denouden, and Caleb Stuckey– all of whom correctly solved the Problem of the Fortnight in the last issue of America’s premiere fortnightly mathematics department newsblog.

Problem of the Fortnight

Consider the sequence a1 = 2, a2 = 3, a3 = 6, a4 = 18, . . . , where an = an-1 . an-2.  What is the largest k such that 3k divides a11?

Write your solution on a paper plate turkey (paper hand turkeys will also be accepted) and drop it in the Problem of the Fortnight Slot outside Professor Mark Pearson’s office, room 212 in The Werf, by 3:00 p.m. on Friday, November 16. We are hoping for a rafter of turkeys to appear in the Problem of the Fortnight slot next week! As always, be sure to include your name and the name(s) of your math professor(s) – e.g. Tom Gobbler, Professor Herb Stuffing – on your solution.  Good luck and have fun!

Elvis was in the building during this week’s colloquium

Off on a Tangent 17.4

A colloquium for the dogs

  • Title: Do Dogs Know Calculus?
  • Speaker: Dr. Tim Pennings, Davenport University
  • When/Where: 11:00 AM on Thursday, November 8 in 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.

What’s going on with this graph?

What’s Going On in This Graph? is a fairly new weekly activity from the American Statistical Association and The New York Times. Each week an interesting graph is shown and students are asked questions like “What do you notice?” and “What do you wonder?”

On the Friday following the release, The New York Times Learning Network publishes a “reveal”—a follow-up that includes the original article, summary of student responses, additional questions students may want to answer, and stat nuggets.

This week’s graph in involves red states and blue states and various voting rights issues. Check it out here.

Problem Solvers of the Fortnight

We had a large escargatoire of snails and huge bale of turtles work on our last problem of the fortnight. Congratulations to Camen Andrews, Barry Bait, Cal Barrett, Meredith Bomers, Josiah Brett, Evan Bright, Dominick Byrne, Jeremiah Casterline, Grace Charnesky, Adair Cutler, Liz Cutlip, Annie Dankovich, Emily Dee, Ford Fishman, Ce Gao, Timothy Hwang, Elizabeth Inthisane, Jackson Krebsbach, Jiangcheng Lu, James Mandeville, Michelle Mathenge, Kianna Novak, Jacob Nurenberg, Megan O’Donnell, Zheng Qu, Karen Quay, Theo Roffey, Hugh Thiel, Hans Veldman, Thomas Vongphrachanh, Fangtao Wang, Tracy Westra, Kamaron Wilcox, Yizhe Zhang, and Jacob Zoerhof – all of whom correctly solved the Problem of the Fortnight in the last issue of America’s preeminent fortnightly mathematics department news blog.

Problem of the Fortnight

Autumn walked into the Peanut Store last week to buy some jelly beans for Halloween. “I’d like a hundred jelly beans,” she told the manager. “I’m sorry. I can’t do that,” he said. “What do you mean?” asked Autumn. “I can’t sell you a hundred jelly beans,” he said, “because my scoops have been bewitched. You see, the purple scoop only scoops to the next largest multiple of 30, the green scoop only scoops to the next largest multiple of 70, and my orange scoop only scoops to the next largest multiple of 110.” “I don’t understand,” said Autumn. “Well, for instance,” the manager explained, “let’s say you had 70 jelly beans. I could increase your jelly bean count to 90 with the purple scoop, or to 110 with the orange scoop, or to 140 with the green scoop.” After musing about this curious situation for a moment, Autumn said, “Okay. If you can’t give me 100 jelly beans, then please give me the smallest number of jelly beans that you could scoop out for me in more than 100 ways.” After thinking for a few moments and scribbling a few calculations on the back of a salt water taffy wrapper, the manager gave her a bag with that many jelly beans in it.

How many jelly beans were in Autumn’s bag?

Write your solution on a scrap of paper and affix to it (by staple, paperclip or glue) a piece of your favorite Halloween candy, and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson’s office – room 212 in The Werf – by 3:00 p.m. on Friday, November 2. As always, be sure to include your name and the name(s) of your math professor(s) – e.g. Reese S. P. Sess, Professor Lemonhead – on your solution. Good luck and have fun!

Just one more thing …

We will leave you with a picture of Prof. Vance and her family taken last weekend during Hope’s Homecoming Donut Run. The staff at Off on a Tangent thinks the run should be renamed Run Torus!, Run!

They just felt like runnin’.

Off on a Tangent 17.3

The intricacies of card shuffling will be explored in the next colloquium

  • Title: Card Shuffling 101
  • Speaker: Dr. Darin Stephenson, Hope College Mathematics Department
  • When/Where: 4:00 pm, Thur. Oct 25 in Schaap 1000

Abstract: We discuss the physical process of card shuffling (by various methods) and ways in which these processes can be modeled mathematically. Interesting mathematical questions include “How many shuffles are required to sufficiently ‘mix up’ a deck of cards?” and “How many shuffles will it take to return a deck of cards to its original ordering?” We will also discuss ways various people have exploited the mathematics of card shuffling to cheat at cards or perform card tricks. We will develop the necessary mathematical background relating to permutations and randomness, and explain some other real-world applications of these mathematical notions.

MATH Challenge

The 2018 Michigan Autumn Take Home Challenge (or MATH Challenge) will take place on the morning (9:30am – 12:30pm) of Saturday, November 3 this year. Teams of two or three students take a three-hour exam consisting of ten interesting problems dealing with topics and concepts found in the undergraduate mathematics curriculum.  Each team takes the exam at their home campus under the supervision of a faculty advisor.

The department pays the registration fee for each team and will provide lunch to participants afterwards. The sign-up deadline is Monday, October 22 at 5:00 p.m.  Interested students can sign up by sending Prof. Cinzori an email at

A group of students may sign up as a team.  Individual students are also encouraged to sign up; they will be assigned to a team on the day of the competition.  For more information, please talk with any member of the Mathematics Department or visit the MATH Challenge website where you can also view old copies of the exam.

U of M Biostatistics Prospective Student Day

The Department of Biostatistics at the University of Michigan will hold a Prospective Student Information Day on Saturday, November 10, 2018. The purpose of this event is to provide information to students who may be interested in graduate study in biostatistics. They expect attendees to be undergraduate and masters students who have identified biostatistics as their interest area, as well as students who are completing an undergraduate degree in math, statistics, biology, or some related discipline, and have not yet decided on their future plans.

At the event, presentations by students and faculty will focus on what biostatistics is and what biostatisticians do, on the job opportunities in biostatistics, and on the admissions and financial support opportunities at the University.

For more information, take a look at their flyer or visit their website (where you can also register).

Problem Solvers of the Fortnight

Congratulations to Holly Denouden, Ce Gao, Zheng Qu, Cole Persch, Hugh Thiel, and Yizhe Zhang – all of whom correctly solved the Problem of the Fortnight in the last issue of America’s premiere fortnightly mathematics department blogosphere news post.

Problem of the Fortnight

Tommy Turtle and a Sammy Snail are at the corner of a cubical aquarium with side length 3 ft.  “Wanna race to the opposite corner?” asks Tommy.  “Well . . . I . . . suppose . . . some  . . . exercise . . . might . . . be . . . nice,” Sammy replies.  Tommy can swim through the interior of the cube at a rate of 6 ft/min, while Sammy has to stick to the sides but can slide along at a rate of 2 ft/min.  If they take off at the same time, how long, to the nearest tenth of a second, does Tommy have to wait for Sammy at the opposite corner (assuming, of course, they both go on their respective shortest routes)?

Write your solution on a cubical aquarium (of any size) and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson’s office, room 212 in The Werf, by 3:00 p.m. on Friday, October 19.  As always, be sure to include your name and the name(s) of your math professor(s) – e.g. Ellie DeLuits, Professor N. Candessent – on your solution.  Good luck and have fun!

Off on a Tangent 17.2 (part b)

Colloquium Next Week Tuesday

  • Title: The P+ P’ problem and George Polya
  • Speaker: Dr. Stephanie Edwards, Hope College
  • When/Where: Tuesday, October 2, 2018 at 11:00 am in VWF 102

Many open problems in entire function theory, specifically, the distribution of zeros of real entire functions, can be tracked back to work by George Polya.  One of these such problems was stated in a Polya and Szego text from the early 1900’s: If P is a real polynomial with only real zeros, find the number of non-real zeros of P^2+P’.  If one removes the hypothesis that P has only real zeros, the problem becomes quite difficult and was not solved until the 1980’s. We will discuss a simple solution to the problem, look at natural questions that arise from the problem and discuss some open questions which have their roots in Polya.

Colloquium Schedule

The following other colloquia are on the schedule this semester. More should be added as the semester goes along.
  • Thursday October 25, 2018 at 4:00 pm, Darin Stephenson, Hope College
  • Thursday, November 8 at 11:00 am, Tim Pennings, Davenport University

Off on a Tangent 17.2

Ice Cream and Fun Recap

The mathematics faculty wants to thank everyone who stopped by to enjoy ice cream sundaes with us on Friday, September 7 at our annual Fall Social event. It was fun to mingle while playing “Would you rather…” and discussing important topics such as footwear, vegetables and time travel!   It’s always wonderful to get to know our students better and hear more about your interests!

The Putnam Exam

The William Lowell Putnam Mathematical Competition, administered by the Mathematical Association of America, is the most prestigious mathematical competition for undergraduates in the nation.  If you are interested in taking the Putnam Exam, you must email Professor Cinzori at by Wednesday, October 10.  The date of the exam is Saturday, December 1. There is both a morning (10am-1pm) and an afternoon (3-6pm) session of this exam.  Participants need to be there for both sessions.  Lunch will be provided by the mathematics department during the break.  For more information about the Putnam Exam visit the website.

Auto-Owners Insurance’s IT/Actuarial Day

Auto-Owners invites you to their annual IT/Actuarial Day, which shows students how their degree can be used in the insurance industry.

The event is on Friday, October 19, 2018 from 10:30 a.m. to 4:00 p.m. in Lansing, MI. Sophomores, juniors, seniors, and recent graduates with majors in Computer Science and Mathematics are invited. Faculty and staff are also welcome! Anyone who wishes to attend should fill out a registration to ensure we adequately prepare for the correct number of attendees.

Register online from now until end of day Monday, October 15th. More information is also available here.

Problem Solvers of the Fortnight

Congratulations to Grace Goszkowski, Maya Hecksel, Peter Le, Isabella Lemus, James Mandeville, Matthew Nguyen, Kianna Novak, Zheng Qu, Karen Quay, Hugh Thiel, Bethany VanHouten, Fangtao Wang, Jonathan Washburn, and Yizhe Zhang — all of whom correctly solved the Problem of the Fortnight in the last issue of America’s premiere mathematics department fortnightly blog.

Problem of the Fortnight

Suppose  is a function that satisfies the relationship thatfor all real numbers  and .  Suppose also thatIf possible, find the following:

  • f(0)
  • f ‘(0)
  • ‘(x)

Staple your solution (not just the answer!) to a pair of Cubs tickets (any of the remaining home games would be fine), and drop it in the Problem of the Fortnight slot outside Professor Mark Pearson’s office (room 212 in The Werf) by 3:00 p.m. on Friday, September 28.  As always, be sure to include your name and the name(s) of your math professor(s) – e.g. Saul Vorecks, Professor Al G. Bragh – on your solution.  Good luck and have fun!



Off on a Tangent 17.1


Welcome to a new edition of Hope College Mathematics Department’s newsletter, Off on a Tangent. While this is not our first edition, it is the first edition that is published on Hope College’s blog network. We published previous editions on our own server (which will soon be put to rest) for the past 16 years and even published hard copies in the years before that. But now it is time for a change. Even though we are moving the process of delivering the math department news to a blog, we will continue to publish an issue about every fortnight (or every other week). In doing so, we will try to keep you up on the news of the department (as well as some interesting math news outside the department), let you know what colloquia will be presented in the coming weeks, and provide you with our famous Problem of the Fortnight.


The first mathematics department colloquium will take place in about two weeks. Here are the details.

  • Title: The Intel Math Experience: An Instructor’s Perspective
  • Speaker: Dr. Erin Militzer, Ferris State University
  • Time/Location: Thursday, Sept 20 at 4:00pm in 1019 Schaap Science Center

Abstract: Intel Math is an 80-hour professional development course in mathematics content for K-8 teachers. The program was adapted from the Vermont Math Initiative developed by Dr. Ken Gross. The course is collaboratively taught by a practicing mathematician and a mathematics educator. One of the goals of Intel Math is that teacher participants deepen their own understanding of math through problem solving.

Intel Math is designed to close the gap between insufficient mathematics training of elementary school teachers and the demands of the contemporary mathematics classroom (Kenneth Gross) and places emphasis on deepening the teacher participants’ understanding of core K-8 mathematics concepts.

In this interactive presentation, I will share what we do during the program, what the teachers accomplish, some of the unique content that encompasses the Intel Math experience, and statistics on a specific study of Intel Math in northern Michigan. Please join me to learn more about what teachers are doing with their Saturdays and summers.

Ice Cream and Fun

Please join the mathematics faculty and fellow math students for the Ice Cream and Fun event this Friday, September 7. It will be located on the Van Andel Plaza that is in front of the Schaap Science Center. In case of rain it will be held in the Science Center’s Atrium. Come enjoy delicious ice cream, some fun games and get to know your fellow math students and faculty.  When, you ask? We will gather at π p.m. and plan for the fun to last until π+1 p.m. Or for those not familiar with such times, 3:14 p.m. to 4:14 p.m.

Student research projects receive national honors

Two projects written by Hope students have been honored in the Undergraduate Statistics Project Competition. The awards, just announced, came from entries in the spring 2018 season. The projects came from students in Dr. Yew-Meng Koh’s Math 311 class.

First prize in the competition went to Alyssa Goodwin, Sam Heilman, and Leah Krudy for their project, “Effects of Color on Heart Rate” and the third prize award was won by Maya Smith and Adair Cutler for their project “Tuition to Test Scores: A Statistical Analysis.”

More information about the contest as well as the winning papers can be found here and Hope’s press release can be found here.

Hope student presents at JSM in Vancouver

The following was written by Noah Kochanski as a reflection of his participation in the Joint Statistical Meetings last month.

This past summer, I was given the opportunity to travel to the Joint Statistical Meetings (JSM) in beautiful Vancouver, British Columbia.  I presented research that I did with my mentor, Dr. Yew-Meng Koh, on Predicting Disease Incidence.  It was extremely nerve-racking presenting to other JSM attendees, the majority of whom were either professors or PhD candidates.  However, I was overwhelmed at how supportive the group was.  We were able to attend a plethora of presentations in Statistics that ranged from how the New York Times uses Statistics to using Statistics in the food industry.  The JSM was extremely developmental for me to network with many researchers as well as broaden my scope of what careers can be fulfilled with a degree in Mathematics and Statistics.  When not attending the conference, we were able to sight-see in Vancouver and explore many local restaurants.  The entire trip was a fantastic experience. I would like to thank the Michigan Space Grant Consortium for their support of my research and to Dr. Yew-Meng Koh for being my mentor this past year.

Problem of the Fortnight

Let 1 = a1 < a2 < a3 < … < ak = n be the positive divisors of n in increasing order.  For example, if n = 12, we have a1 = 1, a2 = 2, a3 = 3, a4 = 4, a5 = 6, a7 = 12.

If n = (a3)3 – (a2)3, what is n?

Odds are, you’ll have fun with this one – even if you don’t get it!  If you do crack this one, drop your solution (not just the answer) in the Problem of the Fortnight slot outside Professor Mark Pearson’s office, which is room number 212 in The Werf, by 3:00 p.m. on Friday, September 14. As always, be sure to write your name and the name(s) of your math professor(s) — e.g. R.U. Shurr, Professor S.I. Yam — on your solution.