OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 1, 2006 Vol. 5, No. 5
http://www.math.hope.edu/newsletter.html


Upcoming Colloquia

A number of Hope mathematics students typically do mathematics research in the summer here at Hope College.  Last summer was no exception.  I our colloquium tomorrow, three students will present their research findings. 

Martha Precup and Dan Emmendorfer worked with Prof. Aaron Cinzori to study piecewise linear spirals.  Their research categorized all geometric sprials and their qualities.  Using these characteristics, they were able to divine an analytic expression for the length of  these spirals.

Brett Jager worked with Prof. Tim Pennings and Prof. Tom Bultman on a mathematical biology project.  Brett looked at the relationship between wasps parasitizing on army worms as they feed on grass infected with a fungus as well as those that are unaffected with a fungus.

The students will not only present their research findings, but will also answer questions over what it is like to do summer research at Hope College.  If you are interested in doing research this summer or next, this is a "must attend" colloquium!
In a continuation of this week's colloquium, next week two more students will give results of their research experiences from this past summer.  Dan Lithio will present "The Dynamics of a Volleyball Serve."  Dan worked with Prof. Tim Pennings to model the serve of a volleyball in order to find the serve that would give the receiving team the minimal time to react.

Megan Patnott will present "Graph Pebbling," based on work done with Prof. Airat Bekmetjev.  She will describe the game where pebbles are placed on vertices of a connected graph and moved according to specific rules.   She will also discuss when a pebble can be moved to any vertex under different configurations of pebbles.

As with this week's colloquium, Dan and Megan will also be available to discuss what it is like to do research at Hope College during the summer.


  Join us for Tea Time on Thursdays before colloquia

As part of our colloquium series this year, the mathematics department will host a "tea time" in the Reading Room (VWF 222) at 3:45 pm.  If tea isn't really your cup of tea, have no fear -- we'll provide some other beverages and snacks, too.  So please join us for a little food and fellowship before you go to the colloquia.  It'll be a great time to chat with the speaker, your professors and other students.


Recent Events at Hope 

MUMC

On Saturday, October 21, Hope hosted the 9th annual Michigan Undergraduate Mathematics Conference (MUMC).  Students from across Michigan presented their research.  Professor Bob Devaney from Boston University delivered the keynote address, entitled The Fractal Geometry of the Mandelbrot Set."

The Hope students' talks were: "Spiraling out of Control," by juniors Dan Emmendorfer and Martha Precup, based on work done with Dr. Aaron Cinzori; "The Dynamics of a Volleyball Serve," by sophomore Dan Lithio based on work with Dr. Tim Pennings; "Mathematical Modeling of a Grass-Army Worm-Wasp Ecosystem," by senior Brett Jager, based on work with Pennings and Dr. Tom Bultman; and "Graph Pebbling" by senior Megan Patnott, based on work done with Dr. Airat Bekmetjev.

The conference concluded with the game Induction Seduction.  Eight teams of students competed to determine the rule that was behind a sequence of numbers.  Different sequences were determined acceptable or not acceptable until a rule was finally discovered.  All students competing received some great prizes that were donated by sponsors of the conference
Megan Patnott, Dan Emmendorfer, and
Sarah Dix cast their vote while playing Induction Seduction.

Dan Lithio describes how to model the
flight of a volleyball serve.

This past Saturday morning at 9:30 a.m. eleven Hope College students gathered in VanderWerf Hall to participate in the thirteenth annual Michigan Autumn Take Home Challenge. Fortified by donuts, the students worked in teams of two or three for three hours to solve ten challenging mathematics problems. Their solutions were sent off to be judged in competition with those of students from 23 different colleges in the Midwest.

The Hope College participants were Jackie Lewis, Samantha Dunmire, Brian McLellan, Christopher Hall, Josh Kinder, Kimberly Klask, Mandy Ferguson, Jeffrey Meyers, Dan Lithio, Forrest Gordon, and Martha Precup.

After the competition, the students gathered with faculty advisor Aaron Cinzori for lunch at 84 East. Results of the competition will be available in a few weeks.

Here's a sample problem from the competition: The three points (4,14,8,14), (6,6,10,8), and (2,4,6,8) are vertices of a 4-dimensional cube in 4-space. Find the center of the cube.


Book Sale!

There is currently a mathematics book sale going on in the Reading Room (VWF 222).
The books are located in boxes by the windows.  The cost of each book is just 50 cents. 
You may pay for books in the main office.


Problem Solvers of the Fortnight 
Jeffrey Meyers, Rachel Hashimoto, Laura Smallegan, Jill Immink and the Anonymous Mathematician with no class all flexed their algebra muscles to crack the last Problem of the Fortnight.  Congratulations on a job well done!  The question was:

What is the product of the real roots of the equation x2 + 18x + 30 = 2 (x2 + 18x + 45)1/2 ?

Laura and Jill provided an insightful solution, which we'll paraphrase here:  Let u = (x2 + 18x + 45)1/2 so that the equation becomes u2 - 15 = 2u, which of course can be rearranged to u2 - 2u - 15 = (u -5)(u + 3) = 0, in which case it's easy to see that u = 5 or u = -3.  Since u = (x2 + 18x + 45)1/2 is the positive square root, we can disregard the solution u = -3.  Hence u = 5 = (x2 + 18x + 45)1/2  and squaring both sides gives x2 + 18x + 20 = 0.  The solutions to this equation can be obtained from the quadratic formula, of course, but since the question asked for the product of the roots, a more revealing approach is available.  Factor the expression as follows: x2 + 18x + 20 = (x - a)(x - b) = x2 - (a+b)x + ab.  Then the roots are clearly a and b, and the product ab = 20 can be read directly from the expression.  Thus, the product of the real roots of the equation is 20 (and the sum of the roots is -18).

Sadly we received no World Series tickets.  Had the editors of Off on a Tangent been in attendance to cheer on the Tigers, the Tigers just might have won game one!


Problem of the Fortnight 

In the figure below angle AOB has a measure of 15 degrees and the length of segment A1B1 is 4.  Segment AiBi is perpendicular to OB for each i = 1, 2, 3, ...  The lengths of segment AiBi is the same as Ai+1Bi for each i = 1, 2, 3, ...  Find the total length of the zigzag path A1B2A2B2A3B3A4B4 ...  Give your answer in closed form.



Write your solution on a candy wrapper (with the candy still included) and drop it in the Problem of the Fortnight slot outside Professor Pearson's office (VWF 212) by 3 pm on Friday, November 10.  Please be sure to include your name, your math class(es) and the name(s) of your professor(s) -- e.g.  Kal Q. Lus, Math 131, Professor Leibniz -- on your solution.

Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

Dear Elvis,

How many zeros are at the end of the whole number that is the product of the first 300 positive integers?

Al

Dear Al,

Another way to phrase your question is how many consecutive zeros are at the end of 300!.  We have Maple available to us on campus and it can find the value of 300! quite easily.  When I typed this in the answer was the following.

3060575122164406360353704612972686293885888041735769994167767412594765331
7671686746551529142247757334993914788870172636886426390775900315422684292
7906974559841225476930271954604008012215776252176854255965356903506788725
2643218962642993652045764488303889097539434896254360532259807765212708224
3763944912012867867536830571229368194364995646049816645022771650018517654
6469340112226034729724066333258583506870150169794168850353752137554910289
1264071571548302822849379526365801452352331569364822334367992545940952768
2060806223281238738388081704960000000000000000000000000000000000000000000
0000000000000000000000000000000

As you can see, there are 74 zeros at the end of this number.

You can also do this quite easily without Maple.  To get a zero at the end of a number, it must have a factor of 10.  Since 10s are made up of factors of 2 and 5 and we know there are lots and lots of factors of 2 in 300! we need only count the number of factors of 5.  Since 300/5 = 60, there are 60 numbers that contain at least one factor of 5.  Since 300/25 = 12, there are 12 numbers that have at least two factors of 5.  Since 300/125 = 2.4, there are 2 numbers that have three factors of 5.  Adding these up, we get 60 + 12 + 2 = 74.  This means there are 74 factors of 5 contained in 300! and hence there will be 74 consecutive zeros at the end of 300!.

Dear Elvis,

I hear that some mathematicians are still trying to discover whether or not there's any repeating pattern in the infinite decimal of pi. I've a few questions in this connection: first, is it true that in 1768 Johann Lambert proved that there cannot be a repeating pattern in pi? And if such a thing was indeed proven, why are there folks still trying to find a pattern? Finally, does it matter whether there's a pattern? That is, what difference would it make in the world of mathematics, and in the wider world?

Thanks for any insight you can offer, and please know that I always appreciate your dogged pursuit of the truth.

Best wishes,
Danny Hoffman


Dear Danny,

Yes, Johann Lambert was the first to prove that pi is an irrational number and hence cannot have a repeating pattern of digits.  That has not, however, stopped people from trying to find some sort of pattern.  Since pi is irrational, they know that they are not looking for a repeating pattern of digits, but there might be some more complex pattern among clusters of digits flopping around after the decimal place.  Some might view their search as irrational, but then the world is full of people devoted to crazy notions -- or at least what you and I would call crazy notions! 

Life is like that, though.  When some people say that something cannot be done, that just makes some people want to try to do it all the more.  Things like running a four-minute mile or climbing to the top of Mt. Everest were once thought impossible to do.  Now even humans regularly do these things.  Would it matter if there were a pattern in pi?  Well, would it matter if Sir Edmund Hillary had never climbed Mt. Everest?  In some sense, probably not.  Mt. Everest would not have changed if Sir Hillary had not scaled it; but people's perceptions of the great mountain changed once he did.   It was no longer crazy to attempt to climb it.  If you or someone else were to find a pattern in the digits of pi, it wouldn't change the number -- only our perception of it.  The search for a pattern in pi might be a Sisyphean task, but some people are obsessed with such things. 

Pattern or not, that crazy number pi will always be irrational, thanks to Lambert's proof.   Keep up your own dogged pursuit of the truth (rational and/or otherwise), Danny!


... the next best thing to playing and winning is playing and losing.
~ R.L. Castleman