Today I woke up at 8:55. That’s not good news since I have analysis at 9:00 on Mondays. Shortly after I left I realized that my left foot was full of painful blisters, and thus running was very painful. Still, I was only two or three minutes late. He started talking about Cantor sets, and that was good since I actually understand it. He used it to show that the cardinality of the measurable sets is 2^continuum, which is blatantly obvious but still neat in my opinion. Then he showed us a weird set that wasn’t measurable. It’s a shame they exist, but he said it’s of no consequence to analysts, and he pointed out that Hardy had said “It’s a gentleman’s agreement that all sets and functions are measurable.” Yay. I approve. Anyway, it was hard to think about analysis when my left foot was in as much pain as it was, and after class I managed with great difficulty to get back to my room and try to ease the pain. It didn’t work very well. Anyway, after that, I headed off to my music discussion group. We dealt with Italian and English madrigals and opera, and then we had a quiz consisting of material that we had been told no more than three minutes before the beginning of the quiz.
After discussion and on the way to lunch, a man who was clearly Jewish stopped me and asked me if I was Jewish, so I said yes. He had a lulav and etrog and asked me if I would like to shake them around, so I say yes and did. That was amusing. He is a member of the campus Chabad, and there seems to be quite a strong Jewish community on campus. I don’t know if I really want to become a Chabad member. I think I’ll stick to my conservative services at UCSB Hillel.
Putnam practice was fun. I got everyone interested in a problem that someone on AoPS had brought up yesterday as a generalization of a class problem in Intermediate Counting: “How many ways are there to get from one corner of an m*n grid given that you can’t pass through any tile more than once?” Prof. Ryavec thought it would be a good idea to look at only the square grids to begin with, but even that seems very difficult. It starts 1, 2, 12, ???. Yes, we couldn’t even count the next case. I think Jeff is going to write a computer program for it tomorrow, so I should help him with that. Or we could try to be not so lost in algebra. That would be a good idea too. But we’re all more interested in the grid problem. Even an asymptotic expression would be nice. I conjecture it’s not O(n!) but not much more than that for square ones. Does anyone have any ideas? Jon wanted to set up mazes, but I don’t see that working since it seems more complicated than the original problem.
Then Jon, Jeff, and I went to Jeff’s suite to learn a bit of algebra. Now I’m less lost in that class. Yay.
In orchestra we only played Dvorak. We aren’t doing Corigliano this concert. Yay.
Shrenik asked me to guess the USAMO problems. So I’ll share my guesses with the world: one normal geometry question, one geometric inequality, one number theory question, a game, a combinatorics question, and an algebra/polynomial question. Watch my guesses fail miserably now.