Monthly Archives: February 2006

Appropriately frustrated? (updated)

The information age is sometimes as much a curse as it can be a blessing. I am not sure exactly how frustrated I should be when it comes to a situation that arose as I left one of my classes this afternoon. I’ll spare many of the details but suffice to say, a project is assigned for which a valid solution can be found online. The project is specific enough that there is little room for variation on the approach to the solution. Sure, I’ll agree with the students that they will still need to be able to understand the model to explain the solution, but surely something has been lost in not deriving the model for themselves. I don’t know a solution to the dilemma I have.

UPDATED (2/28/06): The students in the group involved are responsible students with character. I want to emphasize that I am not impugning or challenging their character in any way by my frustration. In fact, members of the group have already stepped forward to offer to change projects. I made the proposal that they may choose to switch projects or simply accept one additional exercise based on only a slight modification or direct application of the model. I also want say that Robert makes a very good point in the comments. In the real world of mathematical modeling, it is exactly the practice of mathematicians (as well as other problem solvers) to research and determine how has the problem been solved before and then tweak or revise the model to fit the context or improve its application. Can I get a witness?

Pass the napkins

The Wobbly Table Theorem

There’s a theorem I have recently come across that is much along the same lines as the Pancake Theorem or the Ham Sandwich Theorem. It’s called the Wobbly Table Theorem (by some). Basically it states that if you’re at a four legged table that does rest evenly on the floor due the fact that floor is slightly uneven, you can rotate the table around until eventually all four legs will rest on the ground. The proof of this fact depends on one of my favorite theorems. I used to have a series of Favorite Theorems, but it is faded into oblivion. This post will hopefully bring back those theorems. Perhaps tomorrow you can expect one on this: The Intermediate Value Theorem.

At any rate, here are a couple of references on the Wobbly Table Theorem. I’m not completely satisfied with the proofs I’ve found here. When I have time, I’ll come back and explain why.

HT: Gaussian Nodes

False advertising?

“Canadians eat enough Smarties each year to circle the Earth 350 times.”

So says the company Nestlé. They have also claimed that Canadians eat about 4 billion Smarties each year.

Now, if both those statements are true then, we have a problem. Considering that Earth is 24,901.55 miles about the equator, then for these statements to both be true, a smartie would have to be 11.5 feet in diameter. In case you are curious, Nestlé Smarties are a colourful sugar-coated chocolate confectionery popular in Australia, Austria, Canada, France, Germany, Ireland, South Africa and the United Kingdom. They are similar to M&M’s produced by Mars.

As a matter of fact, a 6th grade class was able to get Nestle to change their tune on that bit of fun trivia:

Then they took the size of a single Smartie and worked out how many it would take to form a 40,000-kilometre necklace around the planet.

That’s when they figured out that something was wrong.

Four billion Smarties would go around the world just once.

“We broke it down from centimetres to metres and then kilometres,” student Caitlin Henderson says. For the candies to make 350 laps of the planet its circumference would need to be much smaller, or each Smartie would have to be bigger.

“Three and a half metres,” says student Kaylie Rankin. “It’s about the size of our chalkboard. I don’t know if I’d be able to eat it.”

Teacher Tanja Coghill says it took three letters to get Nestlé officials to admit their mistake, but it was important for the students to get that acknowledgement.

“We can make a difference,” she says. “We’re a little class of 10 students in Grade 6 in Thunder Bay, Ont., and you can make a difference with a huge multinational company.”

HT: think again

Hairy Theorem

Hairy Ball

This one’s a new one for me. It is definitely intuitive but I would have never thought of name for it!

You cannot comb a hairy ball.

The Hairy Ball Theorem states that if you take a ball that is evenly covered with hairs, no matter how you comb the ball, there must be a part somewhere. In other words, the orientation of the hairs must be discontinuous. Compare this to a hairy cylinder which you could comb all the hairs in on direction around the outside of the cylinder with no part. Sorry, I have no picture of a hairy cylinder.

To think of the theorem in another way, let the hairs represent the velocity of the wind blowing across the surface of the Earth. Then, if the wind velocity is continuous, there must be a point where the wind speed is zero.

Koosh!

cf, Mathsnacks: Hairy Theorem

Fair Judging in Figure Skating

I’ll be honest. I don’t like to watch figure skating. Nevertheless, thanks to a blogger at Substandard Analysis, I have a newfound interest in watching or at least paying some attention to the results in Torino.

In an effort of establishing a “fair” judging system, the scoring system for figure skating has been changed. This is in response the the fiasco over biased judging at the last Olympics in Salt Lake City. John Emerson, Assistant Professor of Statistics of Yale University, claims that the new system is in fact less fair than before. Here is Dr. Emerson’s synopsis of the new scoring system:

In place since the 2004 World Championships and in use at the 2006 Olympic Games, the new system awards points for technical elements as well as five program components: skating skills, transition/linking footwork, performance/execution, choreography/composition, and interpretation.

The scores for the technical elements depend on a base value for the level of difficulty of the elements. The twelve judges add or deduct points from this base value, acknowledging the “grade of execution” of the performance of the elements. Program component scores range from 0 to 10, with increments of 0.25, reflecting the overall presentation of the program and quality of the figure skating.

Judging is now anonymous. Nine of twelve judges are selected at random for the Short Program and again for the Free Skate. Scores for each executed element or program component are calculated using a trimmed mean, as in the old system, dropping the maximum and minimum of the nine scores.

Dr. Emerson then proceeds to demonstrate that the random selection process, in the case of a tight competition, can result in more than one possible outcome. The sample data he cites is from the Ladies’ 2006 European Figure Skating Championships. Due to the trimming of the pool of judges by three, there is a possible 220 different outcomes. In the mentioned competition, Dr. Emerson demonstrates that although every one of the 220 possible random choices resulted in the same 1st place finish, there was a wide distribution of outcomes for the other places.

The one thing that Dr. Emerson lacks in his article is a clear interpretation of “fairness.” He claims that the random procedure is less fair due to the fact that he hopes to never hear “a 4th or 5th place finisher give the following interview: ‘I did my best, and I would have won Bronze if all twelve judges’ scores had been included. And if a different panel of 9 judges had been selected, I might have won Gold.'”

As a mathematician I’d like to know a bit more about this new system. For example, “What’s the probability that at least half of the 220 different possible random judge selections give different top 3 rankings than a given control set of selected judges?” If that probability is too high ([tex]alpha > 0.05 [/tex]), I might interpret that as “unfair”.

Olympic Puzzle

From think again!

(I’ll be passing this on to my College Algebra students who just completed their Rational Functions chapter.)

What’s the difference?

In the Olympic games in Munich in 1972 the gold medal on 400 m medley went to the Swedish swimmer Gunnar Larsson. Tim McKee got silver. Their respective times were 4 minutes 31.981 seconds and 4 minutes 31.983 seconds.
The gold medal should naturally go to the fastest swimmer. Let me ask two questions.
If Gunnar Larsson’s lane measured exactly 50 m, while Tim McKee’s lane was slighty longer, McKee may have been the fastest of the two. How much longer would the lane have to be for the two swimmers to be equally fast? Assume they swim with constant speed.
Assume sound travels exactly 344.4 m per second in air. How much closer would the start pistol have to be to Larsson for him to hear it 0.002 seconds earlier than McKee?

For the first question, I simply use the fact that we want the rates to be equal and know that rate is distance divided by time. Let [tex]x[/tex] be the extra length of Tim McKee’s lane, thus we have, because of 8 laps, [tex]displaystyle frac{8(50)}{271.981 mbox{sec}} = frac{8(50+x)}{271.983 mbox{sec}}[/tex] Thus, [tex]xapprox 3.67 times 10^{-4} mbox{m} = 0.367 mbox{mm} [/tex]. Holy Cow, that’s only 367 microns!

Secondly, the question about the gun is simply a question of how far does sound travel in 0.002 sec which is approximately 0.69 m or 69 cm.

That was certainly a close finish. I like one commenters statement, “Perhaps they should move to starting lights instead of starting guns.”