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In the past two weeks, I’ve taken a mini-course taught by Yakov Pesin at the Penn State REU. The topic is “Fractal Geometry and Dynamics”. Instead of handwriting my notes, I tried to live-TeX notes during the lectures. This was inspired by Akhil Mathew; his thoughts on live-TeXing exist here.
The live-TeX experiment turned out much better that I thought it would. My first reaction to the idea of live-TeXing was that it would be infeasible and horribly inefficient. It seemed indisputable that I could write faster than I could type. Magically, however, I realized that my live-TeXed notes were more comprehensive and more legible than my typical handwritten notes. I thought that I would regularly fall behind while live-TeXing, but this did not happen; I rarely fell behind — certainly no more often than when I used paper and pencil. By this point, I think I’m going to live-TeX most if not all of my classes from now on.
It takes some effort to get used to Live-TeXing. The first few lectures were difficult; I struggled to keep up with all of the formulas. As the class continued, I became faster at typing and learned to think less about trivial matters such as formatting. I also used a number of LaTeX shortcuts for lazy people that sped up the typing process significantly. By the end of the class, I could keep up with even the most complicated formulae.
There are a number of benefits to live-TeXing. For example, having all of my notes on my computer means that it is less likely for me to lose my notes and (hopefully) less likely for me to spill water on them. Furthermore, they’re easier to use (because pdfs are searchable) and they save trees. With respect to content, my live-TeXed notes include all of the historical notes and heuristic remarks that I would have missed while writing manually. Furthermore, they can be read without a magnifying glass.
The main problem with live-TeXing is that it is difficult to copy down pictures during lectures; this was a particularly big problem in the fractal geometry class. I tried to describe each picture in words, though I’m sure that this was not an optimal solution. In the future, I might try to draw pictures by hand and scan them, or draw pictures in Paint. In addition, live-TeXing ended up taking up a bit more time; I spent a few minutes every evening cleaning up the day’s notes (fixing typos, dealing with formulae that run off the page, adding subsection headings). That’s probably a good thing, however; it’s good that I was forced to look back and think about the lecture.
Here are the results on my two weeks of live-TeXing. At two hours per day, this constitutes approximately 20 hours of lectures. If anyone took good handwritten notes with pictures from the class, I’d be happy to scan them and insert them into my file.
Here are the technical details of what I did to make live-TeXing happen.
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There exists a new website for mathematics:
Users can post and answer questions about mathematics. Questions from all levels of mathematics are permitted and encouraged. The site is not a discussion forum; instead, through an efficient voting system, the site aims to obtain definitive answers.
A related site is MathOverflow. This site is an amazing tool used for mathematical research. Mathematicians ask and answer questions on the site, making communication in research much easier. The new Math.SE site is a more open relative of MathOverflow.
Math.SE was created through an open democratic process at Stack Exchange’s Area 51. The site entered private beta a week ago, and as of today, it is in public beta and everyone can use it. If you’re interested, go make an account. Spread the word and help this site succeed!
Here are some fun logic (math?) puzzles involving hats:
1. Fifteen prisoners sit in a line, and hats are placed on their heads. Each hat can be one of two colors: white or black. They can see the colors of the people in front of them but not behind them, and they can’t see their own hat colors. Starting from the back of the line (with the person who can see every hat except his own), each prisoner must try to guess the color of his own hat. If he guesses correctly, he escapes. Otherwise, he is fed to cannibals (because that’s the canonical punishment for failing at hat problems). Each prisoner can hear the guess of each person behind him. By listening for painful screaming and the cheering of cannibals, he can also deduce if each of those guesses was accurate. Of course, this takes place in some magical mathematical universe where people don’t cheat. Assuming that they do not want to be eaten, find the optimal guessing strategy for the prisoners. (Hint: The cannibals should eat no more than one prisoner.)
2. In the year 3141, Earth’s population has exploded. A countably infinite number of prisoners sit in a line (there exists a back of the line, but the other end extends forever). As in the previous problem, white and black hats are placed on their heads. Due to modern technology, each person can see the hat colors of all infinitely many people in front of them. However, they cannot hear what the people behind them say, and they do not know if those people survive. Assuming that they do not want to be eaten, find the optimal guessing strategy for the prisoners. Assume that there are enough cannibals to eat everyone who fails. (Hint: The cannibals should eat no more than finitely many prisoners. Assume the Axiom of Choice.)
3. There are seven prisoners, and colored hats will be placed on their heads. The hats have seven possible colors (red, orange, yellow, green, blue, indigo, violet), and may be placed in any arrangement: all the same color, all different colors, or some other arrangement. Each person can see everyone else’s hat color but cannot see his own hat color. They may not communicate after getting their hats, and as in the previous problems, they remain in a magical universe where no one cheats. They must guess their hat colors all at the same time. If at least one person guesses correctly, they are all released. If no one guesses correctly, however, the entire group is fed to cannibals. Assuming that they don’t want to be eaten, find the optimal guessing strategy for the prisoners. (Hint: By this point, the cannibals have probably eaten far too much. It would be cruel to force them to eat any more, so spare the cannibals and find a way to guarantee that the seven prisoners survive.)