OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
March 10, 2004 Vol. 2, No. 11


Hope seeks a threepeat in the Lower Michigan Mathematics Competition
It is time again to defend Hope's first place title in the Lower Michigan Mathematics Competition (LMMC) and keep the Klein bottle trophy (shown to the left) here at Hope. Teams from Hope have won this competition for the last two years.  Last year's winning team consisted of Daniela Banu, Stefan Coltisor and Caleb Gleason.

This year's competition will be held on Saturday, April 3, 2004 at Kalamazoo College.  Undergraduate students from colleges and universities from around the state of Michigan will gather to challenge themselves on some interesting mathematics problems. The students will compete in teams of up to three without the use of calculators, computers, or books.  Transportation to Kalamazoo will be provided. If you are interested (and plan to go) sign up with Prof. Cinzori by Wednesday, March 24.

The exam will be held from 9:30 a.m. to 12:30 p.m.  Lunch will follow and a solution session will be held from 1:15 to 2:15 p.m.



Problem Solvers of the Fortnight

As you may recall, two weeks ago the problem was to determine which of four students made off with the gigantic rope used for the annual Pull.  Fourteen people successfully determined that B was the guilty party.  B was guilty even though he or she was the most truthful of the bunch, making 3 true statements out of four.  Student C was the pathological liar, making no true statements, and D made only one true statement (which was the statement that B was the guilty one.)

‘Knotted solutions’ (i.e. tied up in one way or another) were received from Sara Jongekrug, Kristen Hanna, and Adam Witt.  Adam gave the best rendition of an actual bowline knot.  The most thorough solutions were submitted by Justin Shaler, and Nick Sumner, who gave complete arguments that B was guilty and that the problem did indeed have only one solution.  Katie Wandell gave the most colorful solution, coding true statements blue and false statements red.  The solution which traveled the greatest distance came from Dr. Pearson’s brother Paul, and was pulled all the way across Lake Michigan from Northwestern University using a 24-guage steel cable.  Jeff Mulder, Carrie Thomason, Scott Ibbotson, Sarah Story, Katie Hinkle, Kyle Cox, Ryan Terlouw, and Jim Boerkoel complete the list of solvers. 

Congratulations to all solvers!!  This week's blue ribbon prize winner (pizza), determined by random drawing, is Kyle Cox.  All other solvers may claim two coupons redeemable for chocolate chip cookies from Java Joe’s.   


Problem of the Fortnight

 Those who were at the last colloquium, which presented a minesweeper related problem, were also introduced to rectangular tiling problems.  This weeks problem also features rectangular tiling, and is related to a two-person combinatorial game called Chomp.  Here’s how it works.

Start with a rectangular grid, say the 4 by 5 grid shown below.

                             
A move consists of picking one of the squares and ‘chomping’ off that square and all the squares above and to the right of it  (imagine each square contains a yummy chocolate if you wish).  Here is a sample move,

               
and here is a sample response by the second player.
    
               
Play continues until one of the players chomps the bottom left square, and this player loses (perhaps this square has a yucky piece of broccoli instead of a chocolate).

Your challenge this week is to consider each of the following cases for grids and decide if there is a winning strategy for one of the players and which player this is (first or second).  In each case, be sure to either
  1. explain what the winning strategy is OR
  2. prove that a winning strategy exists.
Here are the cases.
  1. Any square grid.
  2. Any 2 by N grid, where N is any natural number.
  3. A ‘half-infinite’  grid with two rows and infinitely many columns going off to the right.
  4. A 4 by 5 grid.
  5. A 3 by N grid
  6. And finally, a general M by N grid!
Those attempting cases V and VI should feel free to submit solutions for specific grids with N specified in case V, or both M and N specified in case VI.  Any student submitting three correctly solved cases will be eligible for a prize.  Solvers are encouraged to attempt as many cases as they like.

Write your submission on the inside of some Hershey wrappers and turn in (preferably re-wrapped around the chocolate bars) to Dr. Pearson by Friday afternoon, March 26th. 

Treat Yourself to a Slice of Pi Day!

What do movie star Billy Crystal, baseball Hall of Famer Kirby Puckett and theoretical physicist Albert Einstein have in common?  They were all born on Pi Day, March 14 (get it?) -- the transcendental date mathematicians prefer to other dates by a ratio of about 22:7.  Here are a number of irrational suggestions -- or perhaps just an irrational number of suggestions -- for celebrating Pi Day:  (1) Send 3.14 Pi Day e-cards to all your friends by visiting http://www.123greetings.com/events/pi_day/ -- or (2) Gather your friends in a circle and sing some of the Pi Day songs at http://www.winternet.com/~mchristi/piday.html -- or (3.14159...)  Just have a sector of your favorite fruit- or custard-filled pastry (http://www.pierecipe.com)!

To read more about Einstein see http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html. To find out more about Pi see http://www.verbose.net/Pi.html. Editor's Note: Billy Crystal toured with an improv group called "3's Company," and Kirby Puckett batted .314 in his final season (1995) with the Minnesota Twins. Einstein, of course, knew a bit about Pi, too.


Famous Curves: Trifolium

When an arachnologist hears the word trifolium, he or she probably thinks it is a species name for a certain spider. When a botanist hears the word trifolium the genus name for clover probably comes to mind.  When a mathematician hears the word trifolium, he or she will think it is a curve. We thought since St. Patrick's Day was right around the corner, we could celebrate that by taking a look at the mathematicians version of a trifolium. The equation for this curve in polar coordinates is   An enhanced version of the graph of a trifolium is shown at the right.  For more information about the trifolium visit http://www-gap.dcs.st-and.ac.uk/~history/Curves/Trifolium.html.


Mathography:  Daniel Bernoulli (February 8, 1700 - March 17, 1782)

Daniel Bernoulli is our mathematician of the fortnight this issue. Son of Johann and nephew of Jacob (the Bernoulli brothers who brought you the catenary highlighted in the last issue), Daniel was an accomplished mathematician in his own right, winning the Grand Prize of the Paris Academy ten times. Daniel Bernoulli died on St. Patrick's Day in 1782, 1321 years after the death of St. Patrick himself in the year 461 (see http://wilstar.com/holidays/patrick.htm for more on the patron saint of Ireland). To read more about Daniel Bernoulli see http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html.


Why do truncated Maclaurin series fit the original function so well?
Because they are Taylor made!