My first class of the day was with JM. According to the schedule, he was supposed to teach us generating functions, but instead, he taught us about alternating sums.
We’ve been getting JM a lot recently. Apparently, the reason we’re sort of overloading him so much is that many of the other teachers aren’t actually cleared yet. CMU requires very in-depth clearing of staff to make sure they’re safe. Prospective staff members need to get their fingerprint scanned and sent to the FBI in order to be cleared. JM seemed kinda salty: he was like, “The process takes 2 months, which of course meant the MOP staff thought it would only take 2 weeks.”
A lot of the TAs aren’t cleared either, so the first few days the only TAs we had were Turtle and MH (lol, I just realized that their initials are reversed). All the other non-cleared TAs are staying at an Airbnb nearby. We’re not allowed to interact with them under the purview of MAA or MOP because they’re not cleared, but we can actually interact with them not under the purview of MAA or MOP. That means that, if we interact with them under the guise that they’re strangers, then it’s fine. Technically. It’s all a bit silly.
JM is a very interesting character. I mentioned him to my brother when I called him. I also mentioned JM to JX, a TA who just got cleared, when I ran into her outside my dorm room and she asked me about my classes (she lives in the room next to me). Both of them had the same question: was he chugging his huge 2-liter bottle of coke? I find it very amusing that Jason and JX both remember this about JM, particularly considering Jason went to MOP an eternity ago, in 2016 and 2017. Sadly, JM wasn’t chugging 2-liters: he just had a bunch of little cans.
The class was fun. We were learning about “DIE” (because we already all know PIE; he mentioned that AoPS and summer camps and stuff are totally changing the landscape of math competitions. Like, ten years ago, walking into Red, no one knew PIE, yet now we all do). When JM mentioned to Po he wanted to teach us DIE, Po was like, “Do you need to see someone?”
No, JM was not encouraging death. Rather, DIE is an acronym for Description Involution Exception. Essentially, you’re working with a set of things, right? So you toggle something in the elements of the set to pair each element with another one (involution). Then, the hope is that with an alternating sum, the pairs of elements will sort of cancel out, and you just need to count those not paired (exceptions) to get your alternating sum.
For instance, if you’re doing the alternating sum of binomial coefficients from 1 to m of nCk, you can consider this as counting the subsets of [n]. Then, you toggle the 1 so that if a subset contains 1, it’s paired with a subset not containing 1, and vice versa. The subsets of size m containing no 1 is not paired with anything (in the sum), so those are the exceptions you have to count (which gives you (n-1)C(m-1)).
IT’S SO COOOL!!!!!!
The next class was Error-correcting codes, taught by MG (on the schedule, the class was written as “Rrr qrect kowd” :P). Error-correcting codes is MG and RZ‘s research topic. It was quite interesting.
Finally, the third class was SR‘s Combinatorics Philosophy. We spent the first thirty minutes not even talking about combo. Instead, he was like, “Appreciate this time when you sleep at a decent hour. In a few weeks, you’ll be up at 2 am and keep insisting ‘one more game’ and look back on now as the good, responsible times.”
He insisted we should think of our “long-term happiness,” except he was also telling us to figure out what our sleep limit is so that we can go to that limit and maybe a bit beyond. After all, MOP is a once-in-a-lifetime chance that we should take advantage of. We can always just go home and divebomb on our bed for two weeks later, but now, we have to take full advantage.
After talking about sleep and sending VERY mixed messages, we started talking about how we solve math problems.
A math problem is like a many-dimensional landscape. You start at the problem and take many exploratory paths out from the problem, and maybe you start at the solution and take many exploratory paths from there as well, until maybe both ends meet up and you’re happy. He was saying that you’re not good, you’re just lucky if you solve IMO 3 in five minutes. You’re lucky that you managed to get that right path so quickly. Unless you can tell him every single nook and cranny of the landscape, he would say you’re lucky.
He also commented on the ineffectualness of trying problems for a while and then going directly to the solution. He said this is ineffective because a solution is like a one-dimensional path. You follow the solution, and you understand it, but you don’t understand any of the rest of the landscape, and that’s bad.
Also, he talked about depth-first versus breadth-first searching in solution-finding. He said the best way would be a combination of the two, by sort of figuring out the different ways, maybe write them down in a tree, consider the probability distribution of that method working, along with the time required, and then go depth-first after that.
We described our problem-solving method as being sort of going at it and playing with the problem until we get something, which he compared first to a random walk, then an intuition walk, and then finally we settled on calling it a drunken walk.
The drunken walk idea was then expanded upon: he said we need a combination of both sober and drunken because the drunken subconscious sometimes knows things that your sober self doesn’t. So while sober, maybe you plan how long you want to be drunk and decide you’ll take two shots and then you go on your drunken walk. Then, once you’re sober again, maybe you decide to go to the solution and start taking two shots and doing another drunken walk over there. Maybe after that, your two ends are somewhat close and you just need something in the middle to connect the two, and so you throw a few stones into the landscape, take some shots at some lemmas in the middle that might connect the two ends, and that’s where those crazy weird lemmas at the start of solutions come about.
He also mentioned how sometimes, when you’re so familiar with a problem and its landscape, it can be hard to see how to solve it. Thus, if you have someone else you can talk to, they can just spitball ideas about the problem. A lot of the ideas are absolutely off the mark and nonsense, but maybe there’s something that’s actually perfect.
SR mentioned that in Ukraine, they have a lot of team math competitions. The best team he knows of had the captain literally solving no problems. Instead, his role was to assign problems and go around talking to his team members and spitballing ideas.
As we can’t do that on the USAMO or IMO, we can write some spitball, crazy ideas beforehand. Then, after we go on our drunken walk, we can look back on those crazy ideas and maybe there will be some that work.
I’m honestly still not sure why that class was called “Combinatorics Philosophy,” but it was fun, anyway.
At 4 pm, we had the 2σ talk. They brought us all into the Doherty 2210 auditorium (the test room) so that 2σ could Zoom in on the big screen. They were talking about using satellite images to count cars outside shops. That way, they can figure out which businesses are doing well and which ones aren’t.
During dinner at Resnik, the boys (JC as the main information-collector, with DD, MLu, AB, JLin, KWu watching on) are attempting to get everyone’s location so that they can make a map of all the MOPpers. I think someone already did a map of MOPpers for last year’s MOP, too. JC plans to go a step further this year and try to get everyone’s addresses, as well (apparently, for the sake of sending Christmas cards).
We were singing by the piano today. Turtle was playing, and he’s a truly excellent sheet reader. He can literally play any song we mentioned from just looking at a piece of sheet music. I think LR dropped in to sing a few songs. RZ and Evan Chen were singing basically the whole time. Apparently, EB joined them around the time I left for bed at around 11 pm.
I also did my laundry today because I’m responsible like that!
Po did his first office hours today, too. I initially planned to go. However, I couldn’t actually find him, so I sang with Turtle instead. I thought that Po‘s office hours would actually be, you know, indoors. I looked for them inside the lounge, but apparently, they were out in the backyard. Apparently, people were climbing out of the window and shoving chairs through the window to get to the office hours 😝
“Turtle was playing, and he’s a truly excellent sheet reader. He can literally play any song we mentioned from just looking at a piece of sheet music.”
I’m honored :O I’m glad you enjoyed!