I found a great interactive version of Kirkman's schoolgirl problem in Quanta. While I haven't done research on designs in a few years, this stuff is still near and dear to my heart. The way the demo is organized leads to a really natural solution, I think. Some of the others might be tougher to come up with...
Bamboo and products of small primes
Carl Zimmer has a really interesting article at National Geographic on bamboo flowering cycles (which can be on the order of tens of years!) and products of small primes. I won't give away the punch line, but it's definitely worth the read!
True for form, Quanta has a great article on Persi Diaconis's current research pursuit: smooshing.
Cryptography in Context
Last spring, Babson's Teaching Innovation Fund was kind enough to sponsor my writing a course pack, including lecture notes, problems, solutions, and teaching notes, for our cryptography course. I've polished up these notes and finally published them here on the site.
Some notable features of the course include:
This document is offered free of charge under the Creative Commons license. Feel free to edit, redistribute, remix, etc. as you see fit. If you end up using part or all of these notes in a course, I'd appreciate your letting me know.
It may be a little too whiz-bang at the beginning, but this video is still very cool! Back to grading finals...
Information Theory and Ecology
Quanta has a really interesting article on a new application of information theory to ecology. The basic idea is to use information theoretic techniques to estimate biodiversity of large regions from relatively small samples. Super cool!
The new MacArthur fellows have been named, and it seems that mathematics is especially well represented, with Craig Gentry (cryptography), Danielle Bassett (complex systems), Yitang Zhang (number theory), and Jacob Lurie (algebraic geometry). Tons of good stuff going on!
I saw this short but interesting article on Digg about new data on how sharks hunt for food. It turns out that the seemingly random way they move around resembles a Levy flight. From the article:
He explained that the technique's movements "can be advantageous when searching for randomly distributed resources because they reduce 'over sampling' without the need for cognitive maps and sophisticated navigational abilities."
It seems like there are at least two awesome things here: 1) an idea developed in a pure mathematical context proves "useful" well after its invention; 2) evolution drives behaviors towards some sort of optimal solution. Very cool stuff!
I heard of this fun little combinatorics problem from David Dralle, who (I think) heard about it from Sean Rule.
Imagine a group of $n$ people is sitting in a circle. Let's label the participants from 1 to $n$ counterclockwise. A person on the outside of the circle begins by eliminating person 1. She then skips one remaining person and eliminates the next remaining person. She repeats this process until only one person remains. For a given value of $n$, where is the last person to be eliminated sitting? Let's denote the position of this person $f(n)$.
Let's do an example to get our bearings. Here's one with $n = 8$ people. If the first person chosen is in position 1 and we number people counterclockwise, then we have $f(8) = 7$.
Here's another with $n = 20$ people. Here we have $f(20) = 8$.
Notice that the last remaining person is located in very different parts of the circle in the two examples.
One thing you might have noticed is during the first pass around the circle, we simply eliminate alternating people without much fuss. Once we're done with this first pass, we're left with $\lceil n/2 \rceil$ remaining people in the circle. We could also confirm that if $n$ is odd, then we begin the next revolution around the circle by skipping person 2, and if $n$ is even, we begin by eliminating person 2.
Let's continue with the $n$ even case first. There are now $n/2$ remaining people in the circle, namely those indexed 2,4,$\ldots$,n. Since all eliminated people are ignored in the choosing process, this setup is nearly identical to starting a new game of duck-duck-goose with $n/2$ people. The only difference is the indexing; the fresh game as people indexed $1,2,\ldots, n~/~2$ and the original game has participants index $2,4,\ldots,n$. If we imagine starting a fresh game with $n/2$ people and finishing with a goose at position $f(n/2)$, then the goose will have been sitting at position $2 f(n/2)$ in the original game. So for $n$ even, we have the recursive definition $f(n) = 2 f(n/2)$.
What if $n$ is odd? Here, we skip person 2 at the beginning of our second pass around the circle. Again, since eliminated participants are ignored when choosing the next person to eliminate, we can consider starting a new game with $n/2$ with participants index $1,2,\ldots,\lfloor n/2 \rfloor -1, \lfloor n/2 \rfloor$. Note that these new indices correspond to indices $4,6,\ldots,n,2$ in the original game. The conversion between new indices and old indices is $i \mapsto 2(i + 1 \bmod \lfloor n/2 \rfloor).$ So, if we finish the fresh game on a goose at position $f(\lfloor n/2 \rfloor)$, then the goose will have been sitting in position $2(f(\lfloor n/2 \rfloor) + 1 \bmod \lfloor n/2 \rfloor)$.
To see the strange floor functionality at work, consider the case of $n = 9$. We could easily verify that $f(9) = 2$. Our result claims that $f(9) = 2(f(4) + 1 \bmod 4)$. A quick execution of the algorithm shows that $f(4) = 4$, and so $f(9) = 2(4 + 1\bmod 4) = 2.$
These past couple of weeks I've been busy developing content for a new masters-level data analytics course I'll be teaching in the fall. One of the big challenges has been finding compelling, manageable, and well documented data sets for the students to investigate. This is probably a challenge in the most general circumstances, but aiming content at Babson students makes management, economics, sustainability, marketing, and similar data sets really valuable. I've found a bunch I like on Kaggle, a site that runs data analytics competitions. Even if a competition is closed, you can submit your predictions and see how your model would've stacked up. Could make for a fun open-ended midterm or final.