OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
October 6, 2004 Vol. 3, No. 3


Students spend part of last summer helping Prof. DeYoung update Math 205 Curriculum

Megan Scholten, Tara Baase and Erica Pagorek worked with Professor Mary DeYoung to explore some curricular ideas in elementary mathematics. (The four of them are shown to the right.)  The summer of 2004 will be remembered for their lively mathematical discussions that led to the writing of new curricular materials for the Hope classroom.

The central focus was effectiveness. Which classroom activities motivate the students to learn mathematics and which ones promote long-term understanding that will be useful to classroom teachers? The discussions of background differences eventually pushed the group toward a “flexible syllabus,” allowing students to complete different out-of-class assignments that match their individual preferences.

Since the three students are not mathematics majors (or even minors), they provided an important diversity of perspectives that might be seen in a typical classroom of future elementary teachers. The three of them helped to create some new practice activities that will enable similar students to deepen their own understanding before they embark on their teaching careers.

The Michigan Undergraduate Mathematics Conference will be held this month

The Department of Mathematics at Central Michigan University will be hosting the seventh annual Michigan Undergraduate Mathematics Conference (MUMC) on Saturday, October 30, 2004 from 9:00 a.m. to 5:00 p.m.  Hope College will be taking a group of students and faculty.  They will leave early in the day and return in the evening.

Students are invited to give 20-minute oral presentations on any area of mathematics, statistics or related discipline. Such areas include undergraduate research projects, interesting class projects, history of mathematics, or expository talks on interesting mathematics.  Students are also encourage just to attend as there will be presentations on careers in mathematics, information about mathematics graduate programs and REU programs. All students are invited to participate in a mathematical game show called Mathematical Fights.

For students interested in speaking, the registration deadline is October 14, 2004.  For those interested in just attending, the deadline is October 20.  Contact Prof. Darin Stephenson if you are interested in presenting or attending (he has a sign-up sheet outside his office door, VWF 210).  Visit the MUMC web page at http://calcnet.cst.cmich.edu/org/mumc/  for more information about the conference.


Colloquium: Tales from the Crypt---The Mathematics of Secret Messages

Due to the growing popularity of online shopping, ATM and credit card transactions, and e-mail, the need for secure communication has become more apparent.  However, the desire to transmit sensitive information from place to place privately dates back many centuries.  This need for security has resulted in the development of cryptography, a field that relies heavily on computational abstract algebra and number theory.

In next week's colloquium, "Tales from the Crypt: The Mathematics of Secret Messages," Professor Darin Stephenson will discuss some of the mathematics behind cryptography.  The talk is scheduled for Thursday, October 7 at 3:30 p.m. in VWF 104.

In this talk, he will discuss the basic goals involved in cryptography, as well as the mathematics involved in creating simple cryptosystems.  Primary examples include affine cryptosystems, block ciphers and rotating key ciphers.  He will give historical information relating to the development of cryptography and indications as to what new directions this field has taken since the invention of public-key cryptography nearly 30 years ago.  This talk will be accessible to all students.
 

Help!

If you are experiencing a mid-term crisis in your mathematics class or if you just want some help on a problem, don't forget that the Mathematics Lab is there for you.  The Academic Support Center sponsors these help sessions five nights a week in VZN 274.  The hours are Sunday 6:00 to 8:00 p.m. and Monday through Thursday at 7:30 to 9:30 p.m.  The lab is staffed by some of the best mathematics majors we have and they are eager to answer your questions.


There is still time to sign up for the Michigan Autumn Take Home Challenge

The 2004 Michigan Autumn Take Home Challenge (or MATH Challenge) will take place on the morning of Saturday, November 6 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.  Each year 20-30 teams compete in this competition with teams from Hope regularly placing in the top three.  Last year, the team of  Daniela Banu, Stefan Coltisor, and Heidi Libner from Hope College won the event. 

For more information about this competition visit http://www.mcs.alma.edu/mathchallenge/.  If you are interested in competing, you need to sign up with Prof. John Stoughton before October 15.

Problem Solvers of the Fortnight

Your enthusiasm for the second coin weighing problem nearly balanced the zeal with which you weighed in on the first problem, and we again received over 40 entries!  Most of you proposed a solution something like this: First, weigh 17 coins on each pan of the balance; if they balance, the bogus one is in the remaining 16.  Let's consider this case first.  For a second weighing, weigh two piles of 5 from among the 16 against each other; if they balance, the bogus coin is one of the six remaining, and if they don't, it's one of the five in the heavier pan.  If the bogus coin is in the pile of six, weigh two against two for the third weighing; if the pans balance, then a fourth weighing of the two remaining coins will determine which is heavier and thus counterfeit; if the pans do not balance, a fourth weighing of the two coins in the heavier pan will determine the counterfeit.  So, a minimum of four weighings is needed in the case the two original piles of 17 balance each other. 

On the other hand, if the two original piles of 17 do not balance, put piles of 6 coins from among the 17 on each side of the balance for a second weighing.  If they balance, the bogus coin is among the 5 remaining, and a third weighing of two against two will either identify the heavier coin as the one left over or identify a heavier pan, in which case a fourth weighing will determine which of the two coins in that pan is heavier.  If the two pans of 6 do not balance, then weighing two against two for the third weighing will identify a heavier side, in which case a fourth weighing determines the heavier coin, or will determine that the counterfeit is among the two remaining, and again a fourth weighing reveals the counterfeit.  In any case, four weighings will determine with certainty which of the coins is heavier and thus counterfeit.  

Congratulations to Daniela Banu, Jim Boerkoel, Luke Boote, Carrie Brandis, Andre Brau, Stephen Christensen, James Daly, Sarah Dix, Marti Ebert, Kristin Ellsworth, Lindsay Ellsworth, Adam Fitchpatrick, Nick Hinkle, Chris Johnson, Brian Lajiness, Jamie Lajiness, Heidi Libner, Jane Louwsma, Josh Morse, Jeff Mulder, Nicole Mulder, Allison Pautler, Zak Rohde, Amanda Runge, Troy Schrock, Justin Shaler, Vicki Speyer, Sara Stevenson, Nick Sumner, Kevin Vanden Bosch, David Visser, Aimin Walsh, Amanda Zoratti, and to the Phantom Problem Solver who submitted a nameless paper.  Please drop by Dr. Pearson's office to claim your sweet reward!


Problem of the Fortnight

Now that you've mastered the fine art of determining which coin in a pile is heavier, we invite you to weigh the following problem in your minds. . . .

You have twelve coins, numbered 1 through 12, say, and you know one is counterfeit, but you do not know whether it is heavier or lighter than the other eleven, which are of equal weight.  Using a balance scale, what is the minimum number of weighings needed to determine with certainty (1) which of the twelve coins is bogus and (2) whether the counterfeit coin is heavier or lighter than the other eleven . . . and how do you do it?

Inscribe your solution on an "Omega" counterfeit of a $20 U.S. gold piece (see http://rg.ancients.info/bogos/ for details on how a coin dealer bought such a counterfeit for $3500 from a fellow who looked like Newman from "Seinfeld"), or write your solution on the back of a $12 bill and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office by 3:00 p.m. on Friday, October 15.  (Over fall break, you might try using a rusty balance scale to convince your folks that they have a bunch of bogus currency in the house and offer to "get rid of" it for them.)  Please include your math course (number and professor) on your solutions.



 Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu
Before I answer this week's letter, I want to yelp for your help!  I've sat very patiently at my computer these past weeks, waiting for letters from my readers.  When an email with a subject of "Spam" came into my mailbox the other day, I drooled a bit on my keyboard, thinking of those special Friday dinners of Spamburgers and milkbones.  I LOVE Spam!  However, somebody explained to me that spam email is different from the canned meat product, so I won't expect anything great from spam emails anymore.  But, I'd really like to hear from my readers.  So this is my yelp for help: if you have a math question, please email me at elvis@hope.edu, and I'll work like a dog to try to answer one or two questions in each issue of "Off on a Tangent." 
                           
Thanks, Elvis

Dear Elvis,
What's cryptography?  Is that anything like a graph that's really hard to understand?  I've seen a few of those in my calculus book!  Or maybe it's a picture on the wall of a mausoleum?  Or is the meaning more cryptic than that?

                            Cryptically, Un Signed

Dear U.,
I didn't learn about cryptography from my studies of calculus (though I have encountered some graphs that took me a while to decipher!), so I asked around the department.  Cryptography is the name given to encoding and decoding secret messages.  We dogs use a kind of cryptography around humans, communicating with each other "in code" so that our owners don't know what we're saying to each other.  Unlike public key cryptography, though, I'm not allowed to tell you the secret to the dog code.  All you really need to know, though, is that we dogs love having our tummies rubbed, our ears scratched, and, of course, we love FOOD!  (Especially Spam!)
                           
       Hrtmvw,
       Voerh
  (That's "Signed, Elvis" in code)                              
                                                                  

Mathography

In anticipation of Dr. Stephenson's colloquium talk about cryptography on Thursday, we highlight Len Adleman -- the "A" in the RSA public key encryption scheme, which, along with its analogues, has been used widely in web servers and browsers, email programs and electronic financial transactions.  Joe Gallian offers the following synopsis of the RSA cryptosystem in his book "Contemporary Abstract Algebra": The RSA public encryption scheme "permits each person who is to receive a secret message to publicly tell how to scramble messages sent to him or her.  And even though the method used to scramble the message is known publicly, only the person for whom it is intended will be able to unscramble the message" (158).  Adleman was an assistant professor of mathematics at MIT in 1976, when Ron Rivest (the "R") and Adi Shamir (the "S") approached him excitedly about an idea they had for an unbreakable public encryption scheme.  Adleman was less thrilled about the project than Rivest or Shamir but agreed to try to break their codes for them.  After Adleman broke their first 42 codes (!), they hit upon the now-famous RSA scheme in their 43rd attempt.  It is based on some nifty modular arithmetic and group theory.  Adleman, whose path into mathematics was highly nonlinear, sees great aesthetic appeal in mathematics.  "People think of mathematics as some kind of practical art.  [But] the point when you become a mathematician is where you somehow see through this and see the beauty of power of mathematics" (Gallian 167).

To learn more about the RSA scheme, please visit http://www.math.nmsu.edu/crypto/public_html/BegRSA.html or hit Adleman's homepage at http://www.usc.edu/dept/molecular-science/fm-adleman.htm.  If you'd like further reading about Adleman himself, please see http://encyclopedia.thefreedictionary.com/Leonard%20Adleman.  Also, Joe Gallian's "Contemporary Abstract Algebra" contains an interesting biographical essay on Adleman and a lucid presentation of the RSA scheme.         
 

Black holes are where God divided by zero.

                              – Steven Wright