OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
February 25, 2004 Vol. 2, No. 10


Making pop-up Valentine's Day cards proved to be a very popular colloquium

The large crowd in attendance at a recent colloquium enjoyed making fractal pop-up Valentine's Day cards.  Not only did students make their cards, but also got to make an interesting "Go Hope - Beat Calvin" card.  Shown at the left are students busy doing mathematics with paper and scissors.

A mathematician takes a look a Minesweeper

Professor Mike Catalano will present a colloquium about some of the mathematics behind the popular computer game Minesweeper tomorrow.  The title of his talk is "Minesweeper Musings: Some specific strategies for sequentially summing safe sites."  The idea for this talk originated with one of his students.  This student was playing minesweeper on her computer while avoiding some homework.  She started thinking about the numbers that appear on the "safe sites" on the board.  One of the questions that arose was, "What is the largest sum possible of all of the numbers on the safe sites?"  This talk will look at their investigation into this question.  Along the way he will discuss some more famous problems and will even tell us how you could win $1,000,000 "playing minesweeper."  Dr. Catalano is an NSF/AIRE Fellow in the mathematics department this year and is on sabbatical from South Dakota Wesleyan University.  His colloquium is scheduled for 4:00 p.m. on Thursday, February 26 in 297 VZN.


Math in the News:  Packing M&M's

In probably the biggest news in mathematics since the discovery that dogs know calculus, a group of scientists based at Princeton recently discovered, through experimentation, that randomly packed ellipsoids pack more tightly than spheres.  To put this in terms the general public can understand, randomly packed M&M's pack more tightly than randomly packed gumballs.  For example, a pyramid of neatly stacked oranges (or spheres) in a grocery store, occupy 74 percent of the available volume. Arranged randomly, however, the spheres fill only 64 percent of the space.  Neatly stacked ellipsoids, like the M&M's shapes, take up 74 percent of the space, just like spheres. But in random arrangements, computer simulations and experiments with M&M's showed that ellipsoids could be packed much more densely, filling up to 71 percent of the space.  For more information about this, visit http://www.sciencenews.org/20040214/fob7.asp.


Problem of the Fortnight

Although it is the dead of winter, the Black River is frozen over, and the annual competition is months away, ‘The Pull’ is back in the news again.  Four students have been called into the Dean’s Office each suspected of having made off with the official ‘Pull’ rope.  Each is known to own an SUV or pick-up capable of carrying the massive rope, and all have been seen at different times in the vicinity of the storage site for the rope.  The Dean questions each student separately.  He knows that none of them are being completely truthful.  He also knows that no two of the students make the same number of true statements.  The student statements are as follows:

A.    1.  D was near the rope on several occasions.
        2.  None of us took the rope.
        3.  B was near the rope at least once.
        4.  All of my statements are false.

B.    1.  A is not the guilty one.
        2.  D owns an SUV.
        3.  I have never been near the rope storage site.
        4.  D was seen near the rope several times.

C.    1.  A’s first and third statements are false.
        2. D does not own anything that could carry the rope.
        3.  D has only been near the rope once.
        4. B’s statements are all true.

D.    1.  I do not have an SUV.
        2.  B is the guilty one.
        3.  I was only near the rope once.
        4.  B’s third statement is true.

Identify which of these 16 statements is true, which is false, and who the guilty party is.  Tie your solution up in a bowline knot and pull it over to Dr. Catalano’s office (209 VWF) by 3:00 p.m. on Friday, March 5.


Problem Solvers of the Fortnight

Our most recent problem received an overwhelming response with 19 student and one faculty entry.  The problem was to write at least 15 of the numbers from 0 to 50 using the digits in 2004 and any operations.  Several entries were especially noteworthy. 

Vishnu Desaraju successfully represented all 51 numbers by clever use of the floor and ceiling functions.  Mike Cortez, Kevin Butterfield, Nick Sumner, Carrie Thomason, Utsab Khadka, and Heidi Libner all had more than 40 numbers on their lists.  Heidi, noting that we did say there might be several ways to write a given number, takes the cake for giving more different expressions than any other student with 178. 

Our faculty entry, from Mike Misovich, comes in as the most weighty entry at more than 40 pages.  Mike programmed Maple to produce expressions, and found more than 1000 giving 30 different numbers from 0 to 50.  Jennica Skoug and Megan Vivian submitted the most creative entries, following the suggested Valentine theme.  Jennica submitted her solution on a pop-up Valentine as done in Professor DeYoung’s recent colloquium.   Successful entries were also received from Joe Ellis, Alyssa Johnson, Stefan Coltisor, Robert Dody, Brett Jager, Lindy Babcock, Dave Girardot, Jeff Mulder, Andrew Wells, and Megan Patnott.

Everyone who submitted an entry wins a prize.  Under a change in policy, blue ribbon prizes (pizza!) will now be determined by random drawing.  This week, we are awarding three prizes and they go to Stefan Coltisor, Carrie Thomason, and Utsab Khadka.  All other solvers may claim two coupons redeemable for chocolate chip cookies from Java Joe’s.  Congratulations to all!!  We would love to see more from all of you in the future.


Surfing the Web

The 7 UP company has an intriguing ad campaign going on and one of their ads tickled our mathematical funny bones.  Go to http://digicc.com/fido/ and play their game.  Then see if you can figure out why it works as you sip a refreshing bottle of 9 UP.  (That's a hint!)   


Famous Curves: Catenary

Conic sections had been studied for over 2000 years when, in the early 1600s, scientists found two new applications: Kepler's discovery that the planets move around the sun in elliptical orbits and Galileo's finding that the path of a projectile, disregarding air resistance, is a parabola.  Galileo also thought that the curve that results when a chain, wire, or rope hangs under its own weight is in the shape of a parabola.  This time Galileo was mistaken.  The curve formed by a hanging chain is actually a catenary.  This shape is defined symbolically as a combination of exponential functions:

f(x) = (a/2)(ex/a + e-x/a),

where a > 0 and a depends on the tension and physical properties of the chain.  It can also be defined using the hyperbolic cosine function, 

f(x) = a cosh(x/a). 

The shape of a catenary can be seen in common items such as electric and telephone lines, necklaces, and jump ropes.  In fact, the method of supporting trolley wire in a horizontal position using other wires is referred to as a catenary system. One of the more notable catenaries in the United States is the Gateway Arch  in St. Louis.  It is in the shape of an inverted catenary (as shown in the accompanying figure).  See http://www-gap.dcs.st-and.ac.uk/~history/Curves/Catenary.html or http://mathworld.wolfram.com/Catenary.html for more information about catenaries.


Mathography: Georg Cantor

"The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity."  Those are the words the esteemed mathematician David Hilbert used to describe the work of Georg Cantor (whose birthday is one week from today).  Cantor's work shook the mathematics world of his time and his ideas changed mathematics forever.

Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845. As a child, he and his family moved to Germany where he would spend the rest of his life.  Cantor excelled in mathematics and wanted to study this subject.  His father, however, wanted him to become an engineer.  The young Georg, not wanting to displease his father, started down this path.  After a number of years of training in engineering, he became so fed up with his studies that he finally wrote a letter to his father asking his permission to study mathematics.  His father agreed and the grateful son thanked his father and said he would not "displease him."

Cantor received his doctorate in 1867 and two years later he received an appointment at the University of Halle.  He soon began writing papers involving set theory.  Shortly after his appointment, he showed that the rational and algebraic numbers are countable.  (Or they can be placed in one-to-one correspondence with the natural numbers.)  He then showed that the real numbers were not countable.  In 1877 he showed that there was a one-to-one correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote, "I see it, but I don't believe it!"

Before Cantor, infinity had been a taboo subject (some even thought that negative numbers, fractions, irrational numbers, and imaginary numbers had no place in mathematics)  Gauss had stated that infinity should only be used as "a way of speaking" and not as a mathematical value.  Most mathematicians followed his advice and stayed away.  Cantor, however, did not.  This caused him to have a number of enemies.  He had trouble getting his work published, there were written and verbal attacks against him, and he was blocked from getting a position at the prestigious University of Berlin.  While other mathematicians supported him, Cantor could not handle the criticism.  He suffered the first of many nervous breakdowns in 1884 and then spent the rest of his life in and out of mental institutions. 

Much too late for him to really enjoy it, his ideas finally began to gain recognition by the turn of the century. In 1904, he was awarded a medal by the Royal Society of London and was made a member of both the London Mathematical Society and the Society of Sciences in Gottingen. He died in a mental institution on January 6, 1918.  For more information and Gregor Cantor visit http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cantor.html.


"Here's another math problem I can't figure out.  What's 9 + 4?"
"Ooh.  That's a tricky one.  You have to use calculus and imaginary numbers for this."
"IMAGINARY NUMBERS?!"
"You know, eleventeen, thirty-twelve and all those.  It's a little confusing at first."
"How did you learn all this?  You've never even gone to school!"
"Instinct.  Tigers are born with it."
Calvin and Hobbs