Senior presentations 2006
Shaun Boening: Defeating DiMaggio: Probability of Hitting in at least 56 straight Games
Craig Brown: Solving Rubik's Cube.
Darren Clark: A betting strategy for a 7-game series
Ben Graber: Finding Impossible Electric Fields
Ariel Kennell: Euler's algorithm for solving Pell's equation.
Jeremy King: Making Change for a Dollar
Erin Leatherman: Restricted Toppings and Pizza-Deciding Rook Polynomials
Brian Lichtle: Representing Numbers in Other Bases
Christy Maas: A Proof of Heron's Formula
Krista Miller: Maximizing Area between a Circle and Polygon
Eric Sweet: Shooting free throws with changing success probabilities

2004–05 TOPICS

The 2004 Honduran National Mathematics Olympiad
Don Hooley — September 7
We will view pictures of the Association of Bilingual Schools of Honduras 2004 National Mathematics Olympiad and the National Autonomous University of Honduras. I will report briefly on the invited lectures at the Olympiad (one of which I gave) and present several sample items from the exams.

How to Win(?) at Quiddler
Darryl Nester — September 14
The rules to Quiddler solitaire are simple: You have 16 cards, in eight stacks of two cards each. Each card has a letter (or pair of letters). From the eight top cards, make a word, and remove those cards. Repeat until all cards are gone, or no words are possible. Your score is based on the letters used, minus the letters remaining, plus bonus points for creating words with 5 or more letters. When you are finished, you'll see how your score ranks among others who have played that day.
A few names show up consistently in the list of top scores, most of them impressive or intriguing "handles" such as sardonimous, Vinster, cc, and UK Lady. And in the last several months, the slightly-less-intriguing Darryl from Ohio. What skills do these Quiddler wizards possess? A colossal vocabulary? Prodigious lexicographical intuition? Or, you know, just being, like, really good with words and stuff? Perhaps all of these things ... but almost certainly, some of them have a computer doing all the grunt work while they take all of the glory. We'll take a look at that grunt work, and the mathematics behind this word game.

How to Play for a Billion
Darryl Nester — September 21
Pepsi recently concluded its "Play for a Billion" contest with a primetime special to award a $1 million prize, and reveal whether the prizewinner had won the titular BILLION DOLLAR prize (he did not). The seven finalists were offered the opportunity to take home various cash prizes if they would give up their chance at $1 million. No one took that option—so six went home with nothing. Was that a good decision? We'll look at how to answer that question with probability and utility theory.

Least Squares Regression, Geometry, and Symmetry
Darryl Nester — September 28
(Preview of talk to be given at Miami University Math and Statistics Conference)
Given a set of points {(x1,y1), (x2,y2), ... (xn,yn)}, the usual least-squares regression line y = a + b x is the one which minimizes the sum of the squared vertical distances from the points to the line. Why do we use (a) squared and (b) vertical distance? How does the equation change when we seek to minimize the sum of squared horizontal distances? Perpendicular distances? What about other ways of measuring distance? Visit
this page to explore this for yourself.

Speed traps, 205 mph motorcycles and the unknown number in the Mean Value Theorem
Don Hooley — October 5
We will consider the recent motorcycle speeding ticket for going 205 mph measured over a half mile in Minnesota and the mathematics justifying it. We also explore the question of precisely when this speed may have been achieved and its relationship to the unknown number in the Mean Value Theorem.

A Gambler's Fortune
Darryl Nester — October 19
A gambler starts with $M. Each time he plays a certain game, he specifies $k, the amount of his bet. He then wins $k with probability p; otherwise, he loses that amount. The gambler decides to choose k to be half of his current capital each time he plays. For example, after the first round, he will have either $1.5M or $0.5M, and his second-round bet will therefore be either $0.75M or $0.25M.

(This is problem 782 in the September 2004 issue of the College Mathematics Journal.)

Squares through four points
Don Hooley — October 26
Is it possible to construct a square with sides going through four arbitrary points in the plane? How might one do this? How many such squares exist? We will use Paul Kunkel’s
wonderful website and applets to help consider answers to these questions and more.

A Mathematical and Statistical View of the Election
Darryl Nester and Don Hooley — November 2
We will take a look at a few of the web sites attempting to project the outcome of the election based on interpretation of available poll data. What methods are used? How can the same data (more or less) be used to arrive at contradictory conclusions? And who is going to win?
We’ll try to answer the first two questions. For the answer to the last question, you’ll have to wait until Tuesday night. Or perhaps Wednesday. Or … well, who knows?
For a head start, try these pages (there are MANY more):

Low Down Triple Dealing
Darryl Nester and Don Hooley — November 9
We’ll examine a trio of mathematical card tricks, published in the most recent issue of the Mathematical Association of America’s newsletter Focus. As reading a description of a card trick is about as entertaining as listening to a tape-recording of a mime, those interested will have to come to Seminar to see the tricks live.

Fun (?) with Mathematica
Darryl Nester — November 16
The computer program Mathematica can do complex computations, produce beautiful and intricate graphs, and greatly simplify complicated algebraic expressions. It can also be an incredible pain in the neck: Until one becomes accustomed to its notation and syntax, it will stubbornly refuse to obey one's commands. We'll take a look at basic Mathematica commands, and explore some of its more complicated features, too.

Earning Points on the Putnam Exam
and The 2004 Putnam Exam
Darryl Nester and Don Hooley — November 30 and December 7
(part 1) We will look at a few problems about algebra, number sequences, area and geometry, and probability in an attempt to identify strategies for earning points on the annual William Lowell Putnam Mathematical Competition exam.
(part 2) Last Saturday, thousands of college students spent up to six hours competing on the annual William Lowell Putnam exam. We'll see what we can make of it in half an hour.

A "Fish-Eye" view of the Plane
Darryl Nester — February 10
Can you squeeze the whole Cartesian (xy) plane inside a circle? (And if the answer is "yes"... how do you do it, and why would you want to?) We’ll examine a view of geometry where "objects in the mirror may be closer (or farther) than they appear."

Tossing Hammers, Rabid Raccoons, and Florida Panthers: A Report from the Atlanta Joint Mathematics Meetings
Don Hooley — February 17

Estimating Probabilities with Logistic Regression
Darryl Nester — February 24
Regression is a commonly-used tool to estimate some unknown quantity from one or more known quantities—for example, you know the weight of a car, and want to guess its fuel efficiency, or you know today’s predicted high temperature and humidity, and want to estimate the demand for lemonade at your concession stand.
Suppose the quantity you want to estimate is a probability; e.g.:

For problems like these, we need to modify the usual regression procedures a bit. We’ll take a look at some examples of this process.

The “Fish-Eye” Plane (Revisited)
Darryl Nester — March 17
We’ll look again at different ways of squeezing the entire Cartesian (xy) plane into a circle, including the mathematics involved with choosing the “best” function to squeeze the plane, and the related idea of stereographic projection (see examples
here and here).

2003–04 TOPICS

Using Mathematics to Solve a Word Puzzle
Darryl Nester — September 4
(preview of Miami conference talk, Oct. 3)
On the April 27, 2003, puzzle segment of National Public Radio’s Weekend Edition Sunday, Will Shortz asked listeners to rearrange the 16 letters in the phrase “The Conversations” into a 4-by-4 grid to form eight words—four across and four down. Anagram generators are widely available on the Internet; one yields from these letters 254 four-letter words, which can be combined into 15,220 four-word sets! How can we weed through this daunting list to find the solution (and confirm that there is only one)? One possibility would be to write a computer program to eliminate unusable combinations, but it can be done more quickly and with less work (by the user) with some moderately clever mathematics and a spreadsheet such as Excel.

Exploring Interactivate
Steve Harnish — September 11 and 18
Interactivate provides over 100 lessons, activities and Java-based computational tools to assist math and science instruction. Some are designed for elementary school classrooms while others could augment introductory college courses. We will explore the features of several key tools including the Function Flyer (a grapher), several statistical tools, and simulation packages for predator-prey models and Conway’s Game of Life. For more information, see the Interactivate web-site:

New Tricks for Old Dogs:
Symbolic Computation on the TI-8x, where 2 ≤ x ≤ 6

Darryl Nester — September 25
(preview of ICTCM talk, Nov. 1)
Students (or teachers) who use TI-82, -83, -85 or -86 calculators may envy their friends (or colleagues) who have a TI-89 or -92. While it may be impossible (or at least, prohibitively difficult) to get such calculators to perform all of the functions of their "big siblings," all is not lost! With no or minimal programming, one can use them to find, for example, the expansion of (x–4)(x+2)(3x+5), or to determine the asymptotic behavior of a rational function (that is, find the quotient of a polynomial division), or to write cos(4x) in terms of cos(x). And in the process of learning to do these tricks, one just might learn a bit of mathematics along the way.

GIMPS: The Great Internet Mersenne Prime Search
Mike Bumbaugh — October 2
GIMPS uses the power of many computers linked through the Internet to do the massive computing required to check if very large numbers are prime. Specifically, GIMPS attempts to identify Mersenne primes, primes of the form 2P–1, where P is a prime. The first seven of these are 3, 7, 31, 127, 8191, 131071, and 524287. To date, the effort has identified five—the five largest primes known. Two years ago, a GIMPS contributor was credited with identifying the 39th Mersenne prime, with over 4 million digits. For more information about GIMPS, visit More Mersenne primes can be seen here.

Physics, Functions, and Flying Discs:
The Mathematics of Frisbee Flight

Brian Lichtle & Erin Rae Stopher — October 9
While surfing the web, we found that often people who claim to enjoy math, also enjoy Frisbee. Wondering what the connection was, we searched a little deeper, to find this quote:

We didn't doubt this at all, but wondered if there still was a deeper cause. Possibly it is calculating the angles one needs to run to receive a pass, estimating the probability that the disc will land right-side up in order to win pulling-privileges, or maybe one simply needs fresh air?

A Report from a Mathematics Alumnus (and Bluffton College Lifetime Service Award Recipient)
Dr. Isaac Riak — October 16
Wondering what you can do with a mathematics major? Dr. Isaac Riak graduated from Bluffton in 1971 with a mathematics degree, and went on to teach mathematics in Jamaica for several years. Since then, his career path has turned towards economics, specifically the area of international development. At the Alumni Awards Banquet last weekend, he and his wife, Dr. Pauline Riak, received the Lifetime Service Award for their work. At this week's Math Seminar, he will tell us a bit about his journey from Bluffton College to his current home in Nairobi, Kenya.

Amusing Math Problems
Darryl Nester — October 23
We will explore a number of math problems, from relatively simple to quite complicated ones. In particular, we will discuss the annual William Lowell Putnam Mathematical Competition, and talk about strategies for scoring points on this exam. Beginning soon, we hope to schedule weekly problem sessions, during which we will discuss solutions to problems and introduce new ones for consideration.

Hard Cubes, Slippery Hexagons, and Interstellar Soccer Balls; One Element's Disguises
Ron Rich — October 30
Which element arranges its atoms with cubic symmetry (in diamonds from the deep), hexagonal symmetry (in graphite in pencils), or soccer-ball symmetry (in truly astronomical amounts among the stars)? I'll show models and analyze a wee bit.
   A student asked how to predict boiling points, and for several decades I worked on this uncool topic without competitors breathing down my hot neck, at Bluffton College, on down through Bethel College, Harvard, Japan, Stanford, etc. Now, in articles in Japan, the U.S. and Europe, you can see how to predict a great variety, using statistics and a surprising power law. Some predictions for soccer-ball molecules have already been verified, in spite of a persistent myth.

ICTCM Report
Darryl Nester — November 6
At the 16th Annual International Conference on Technology in Collegiate Mathematics, held last weekend in Chicago, I:

In this presentation, I'll share some of these things (sorry, the candy is all gone).

Modelling with STELLA and Madonna
Steve Harnish — November 13
If this title isn't enough to pique your interest, then let's try simulations of dynamical systems:

Recursion Excursion
Mike Bumbaugh — November 20 and December 4
We will look at Italian rabbits that never die and hop down the bunny trail to an efficient method for finding Mersenne primes. Also, we will examine a very simple problem that elementary kids could have fun with, but the proof eluded mathematicians for many years: Ulam's Conjecture. Much of the discussion should be accessible to a broad audience.

A Game Show Strategy
Darryl Nester — January 13
The game show host has chosen a (secret) random number M between 0 and 1. You are presented with a random number X (between 0 and 1/M). If you accept X, you win M*X dollars. Otherwise, you are presented with Y (also between 0 and 1/M). If you accept Y, you win M*Y dollars. Otherwise, you are presented with Z, and win M*Z dollars. How do you decide whether you should accept X, Y, or Z? We'll answer this question using a simulation on Excel, as well as a bit of formal mathematical analysis. (This is problem #11051 from the December 2003 issue of American Mathematical Monthly.)

Recursion Excursion (part 3)
Mike Bumbaugh — January 20
We will look at Italian rabbits that never die and hop down the bunny trail to an efficient method for finding Mersenne primes. Also, we will examine a very simple problem that elementary kids could have fun with, but the proof eluded mathematicians for many years: Ulam's Conjecture. Much of the discussion should be accessible to a broad audience.

Three (of the many) Faces of Gamma
Steve Harnish — January 27 & February 10
What should be the next entries in these lists?

The Gamma function has been studied and "rediscovered" many times during the past three centuries. Euler first defined it in 1729, Gauss later named it, and many mathematicians since have used it in areas as diverse as generalized factorials, differential equations, probability and statistics, and theoretical physics (e.g., Veneziano applied it in 1968 and inspired the development of string theory). We will look at three connections to factorials, normal distributions, and differentiation. Along the way I will give a personal account of my initial encounter with Gamma in 1982.

Three sticks
Darryl Nester — February 3
You have three sticks, of length x, y, and z, all attached at one end to a point P. The sticks can rotate freely about point P. You want to arrange them so that the other (unattached) ends lie at the vertices of a right triangle. (1) When is this possible? (2) Of those triangles formed in this way, which one has the largest possible area? You can explore this further with the applet
here. (This is a restatement of problem #11057 from the January 2004 issue of American Mathematical Monthly.)

Recursion Excursion (conclusion)
Mike Bumbaugh — February 17
We'll explore visual demonstrations of recursion with the
Koch snowflake (and variations on that idea). You can see another interesting view of recursion here (Shockwave animation).

Typesetting (and doing) Mathematics with TEX
Darryl Nester — February 24
Long before Microsoft Word's Equation Editor, computer scientist Donald Knuth developed TEX (pronounced "tech"), a system to simplify the creation of documents containing mathematical formulas. We'll see how TEX works, both for typesetting mathematics, and also for "doing" math (generating lists of primes, performing synthetic division, etc.).

Coin Flips, Random Walks, and Standing Waves ... Matrices to the Rescue
Steve Harnish — March 9, 18, & 23
Several talks on a mathematical model for solid-state physics (both in Math Seminar and in the Thursday Math Physics Discussion).

Permutation Inversion Excursion
Darryl Nester — March 16
A permutation simply means a rearrangement of a set; for example, "real nerds try" is a permutation of "darryl nester," and "13524" is a permutation of "12345." Within a permutation, an inversion is an instance of two items having their order reversed relative to the original ordering. For example, "13524" has three inversions: the pairs (3 and 2), (5 and 2) and (5 and 4).
We'll explore the number of inversions in a randomly selected permutation of the set 1,2,3,..., n.

Slide Rules
Duane Bollenbacher — March 30 & April 6
Slide rules will be provided for your use. We will discuss its history, its role in the classroom, and its overnight demise. Old-timers (like Professor Harnish) will wax nostalgic; youngsters (like Professor Nester) will be in solemn wonder. We even have music for you!!