one of these things...

I remember, in high school, when a teacher was talking about series and sequences of numbers. She presented a series, such as 3, 6, 9* and asked what the next number is. Of course, she was expecting us to answer 12. I noted that, if you say that the sequence is defined by a polynomial expression, you can justify anything as the next number by picking the right coefficients. 3, 6, 9, 75? There's a third-degree polynomial that fits it. And, by the way, there are infinitely man fourth degree polynomials that fit as well.** 

I used to have a friend named Angry Bob. On more than one occasion he called me, annoyed about a sample Mensa test (or somesuch) he saw in a magazine. He didn't score in the genius range, and wanted to plead his case. Specifically, there would be four pictures, and a question of which one doesn't belong. It was easy enough to figure out what the quiz-writers intended, but Angry saw it differently. He pleaded his case. I explained that the quiz was looking for a different answer. He'd plead again, arguing that his choice was just as legitimate***.This went on until I said that, yeah, he's right, he deserves credit. We would go through several questions this way until he had enough questions right to score in the "genius" range.

I was reminded of the above vignettes as we listened to Christopher Danielson at the National Museum of Mathematics (Mo Math). Blair, Sharon and I (along with one of Sharon's friends and her family) went to see Danielson's presentation, "Which One Doesn't Belong?" which was part of the museum's "Family Fridays" program.

Danielson is an educator of educators. That is, his "thing" is helping people figure out how to teach math to kids. His websites are talkingmathwithkids.com and christopherdanielson.wordpress.com.

The main part of the program consisted of Danielson showing slides with pictures of four objects, and asking for the particpants' thoughts about which object didn't belong. But instead of focusing on what is the one right answer, he focused on the fact that any answer could be right. Consider, for example, the picture above.

• The shape in the lower left doesn't belong because it's the only one that's convex.
• The shape in the lower right doesn't belong because it's the only one with multiple lines of symmetry.
• The shape in the upper left doesn't belong because it's the only shape whose boundary has no straight lines.
• The shape in the upper right doesn't belong because it's the only shape whose boundary consists entirely of straight lines.

This wasn't as advanced a topic as Francis Su's cake-cutting talk (which I write about here). And to a degree we were out of place; Sharon and her friend were older than most of the kids there. But it was interesting nonetheless. Danielson was trying to introduce the youngsters to flexible mathematical thinking. And it was vindication for that former high schooler in me who bristled at an inflexible teacher. I only hope that Angry Bob, wherever he is, would feel vindicated as well.

*I don't remember what the actual numbers were, so I'll use these for the purposes of the narrative.

**Yeah, I was a smart-ass in high school.

***In all fairness, he was right.

splitting cakes with francis su

The classical question "How can we cut a cake fairly?" has been around since antiquity, but what happens when mathematicians and economists focus on the situation? Is there always a way to divide things fairly? Join Harvey Mudd Professor of Mathematics Francis Su as he shares envy-free (and tasty!) solutions to this very human problem.

That was the message on the poster, and it gave as good an introduction to Professor Su's talk as you're going to find. It sounds frivolous, but anyone who has been to a children's birthday party knows that splitting a cake fairly -- or at least in a way that is perceived as fair by a roomful of children -- is not an easy task. "He got a bigger piece!" "She got the flower!" "I didn't get enough icing!" You get the idea...

It's an interesting subject at the intersection of economics and mathematics. So when Sharon asked me to go with her to Mo Math (technically, the National Museum of Mathematics) for a presentation on the topic -- well, great!

Su started with the well known "You cut, I choose" method of splitting a cake (actually, pretty much anything) between two people. One of the people splits the object in two, and the other chooses which piece she wants. From there, he went on to the situation with three people, and then higher numbers. Of course, along the way he had to define fairness. In order to adress the topic with mathematical rigor, it's necessary to have precision in one's definitions. He specified several definitions of a "fair" division:

• Envy free: Each person believes he got the biggest piece
• Proportional: Each person believes he got 1/N (where N is the number of people)*
• Equitable: Each person thinks he got exactly the same percentage of the total as each other person thinks he got.
• Efficient: A division that is not dominated by another. For these purposes, a division is said to dominate another if it is at least as good for each player.**

Su hit that sweet spot, balancing mathematical rigor (or, at least the hint of it) and accessibility for the educated layman. The children in the audience seemed to stay interested, as did the adults, many of whom, had mathematical backgrounds.  For the simpler splits, he explained the methodology in detail. So he continued from the simple "You cut, I choose." For more than two people, he described the "Dubins-Spanier Moving Knife Procedure." But at some point he stopped trying to describe methods in detail, and just explained that solutions exist, but may involve very large numbers of steps.***

It was also interesting getting a flavor of the thinking that leads to the solutions. Su explained and demonstrated Sperner's Lemma, which is at the intersection of topology and combinatorics. And at another point he talked about how he got interested in the cake-splitting problem -- trying to figure out how much rent each person should pay in a shared house where rooms differ in desirability.

Now, if only I weren't dieting -- there was free cake. Su used it to demonstrate the Dubins-Spanier moving knife procedure.

*I describe this by saying that each person thinks he got his fair share.
**We weren't precise to the point of indicating whether the exact definition of "dominate" is written so that all divisions dominate themselves or none do.
***This year, Aziz and McKenzie proved the existence of a method for an envy-free division for n people, but it can take up to n^(n^(n^(n^(n^n)))) steps.

a trip to the gallery

Today we visited the Adelson Galleries in midtown Manhattan to see the exhibit, "Animals and Friends in Pencils, Books and Bullets." The exhibit, which featured the works of Colombian artist Federico Uribe, featured works created from broken pencils, torn books and (most impressive) spent bullet casings and shotgun shells.

The stuff made from the books and pencils was OK, but I didn't find it particularly interesting. The bullet shells and casings, though, were amazing. Uribe somehow managed to imbue the pieces with a lifelike look. Staring in the taxidermy eyes of the leopards and wolves, they seemed like living animals.In some pieces, Uribe was able to get different colors through controlled corrosion. He used those effects to create such things as spots on leopards. Sadly, the price tags mean that none of these works will be gracing my home anytime soon.

We did this as an event for our chapter of Stack Up, a charity that Ethan got us involved with. Stack Up supports our country's armed forces (both active and veteran) through a passion for video games.

If anyone in the New York area has any interest, go to the gallery's website. It says that this closes on Saturday (6/11), but we were told that it's been extended another couple of weeks.