Answering physics FAQs without preparation

Experts don’t know everything. Often, they only know how to look things up, and how to understand what they find. If you’ve ever seen physicists answering a physics FAQ, those answers took a lot of effort to get right. Some common questions are in fact really complicated, or hard, or maybe they just aren’t about the things that physicists normally think about.

With humorous intent, I’m going to answer a bunch of frequently asked questions, sampled from this physics FAQ by John Baez. And I’m doing it without preparation, so the answers will be bad.

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Sexual identity and topology

One of the consequences of having a great deal of math and physics education, is that whenever I learn about something, I internally encode it as math, even if nobody else is thinking of it that way. Today I’m going to share one of the more ridiculous examples, the analogy between identity labels and topology.

I’m mainly thinking about sexual identity labels, and especially arguments over boundaries of those labels. I’m thinking of how people claim “everyone is a little bisexual”; or they argue about the validity of bisexual lesbians; or they ask “isn’t demisexuality just normal?”; or they draw sharp distinctions between asexual, gray-asexual, and allosexual.

In all these arguments, there is the essentialist viewpoint, which says that everyone has an underlying sexuality, and each word covers (or should cover) a specific space of sexualities. If your underlying sexuality falls within the domain of the identity label you use, then your label is “correct”, and if it doesn’t, then your label is “incorrect”.

I disagree with the essentialist viewpoint, and I frequently point to prototype theory, family resemblance theory, and Wittgenstein as alternatives. But I also feel that if you’re going to take the essentialist viewpoint, obviously you should take it all the way, and learn about the math that you’re implicitly using. I am not going to “prove” that essentialism is wrong, and if you summarize my essay as “Mathematics disproves essentialism” then so help me, you did not read the fourth paragraph. The goal is to explore the implicit mathematical framework of essentialism, and point out its unaesthetic aspects.

Of course, I don’t recommend actually using this in an argument, since it relies on teaching people math.

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Unbiased estimators in a Monty Hall problem

In my previous post, I talked about the German Tank Problem. And while discussing the frequentist approach, I defined the “unbiased” estimator. But seriously, unbiased estimators are really weird. Let me show you an example, in the form of a Monty-Hall-like problem.

Suppose that I’ve set up three closed doors A, B, and C, each with a prize behind it. Two of them have $1000, and one has $2000. Doors A and B don’t really matter, your prize is behind door C. How much is this prize worth to you? But before you answer, please, look behind one of the other doors, A or B.

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The German Tank Problem

The German Tank Problem supposes that there are N tanks with ID numbers from 1 to N. We don’t know what N is, but have looked at a single tank’s ID number, which we’ll denote m. How would you estimate N?

This is a well-known problem in statistics, and you’re welcome to go over to Wikipedia and decide that Wikipedia is a better resource than I am and, you know, fair. But, the particular angle I would like to take, is using this problem to understand the difference between Bayesian and frequentist approaches to statistics.

I’m aware of the popular framing of Bayesian and frequentist approaches as being in an adversarial relationship. I’ve heard some people say they believe that one approach is the correct one and the other doesn’t make any sense. I’m not going to go there. My stance is that even if you’re on Team Bayes or whatever, it’s probably good to understand both approaches at least on a technical level, and that’s what I seek to explain.

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Logic puzzles, overexplained

By “logic puzzle”, I don’t just mean puzzles involving logic, but rather a specific genre of puzzles, whose most famous types are Sudoku and Picross. There are many other types of such puzzles, and creators of logic puzzles can create entirely new types, if they are so inclined. If you’re not sure what I’m talking about, or if you’re just interested in finding logic puzzles, at the bottom of this post I’ve included a list of places you can find them.

I’m fairly good at logic puzzles. I’ve done the US Puzzle Championship for over a decade, and I placed in the top 25 once? So not like top-of-the-world good, but decent. And I’m a generalist, which is to say that relatively speaking I’m not very good with Sudoku, and I do better with other types of puzzles, including entirely new types.

My goal here is to overexplain my understanding of logic puzzles, and solving strategy. I am not confident that this is actually helpful to someone trying to get better at solving logic puzzles, but that’s not really the point. The point is to explicitly describe what would otherwise only be understood intuitively.

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The physics of jigsaw puzzles

Wholesome jigsaw puzzle youtuber Karen Kavett recently did a challenge where she assembled a 1000 piece puzzle by selecting 100 pieces at random at a time. For a while, this just looked like a bunch of scattered pieces with only a few connections. But once she had 700 or 800 pieces, the whole puzzle started to come together, despite the gaps.

I found this fascinating, because it is a live demonstration of a concept in physics/math: the percolative transition. This is something I often think about when assembling puzzles.

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Computational complexity of jigsaw puzzles

During the pandemic, I started doing more jigsaw puzzles. Not real puzzles mind you—I found a jigsaw simulator on Steam that was fairly authentic to the real experience. And since I was doing jigsaw puzzles through the medium of video games, I couldn’t help but think about them in the context of puzzle video games. I realized, jigsaw puzzles are kind of weird! In your typical puzzle video game, the ideal is to have a set of levels, each of which require some crucial insight. In contrast, a jigsaw puzzle is more like a large task that you chip away at.

One way of thinking about this is through the lens of computational complexity. Take Sokoban, the classic block pushing puzzle upon which many puzzle video games are founded. In general, a Sokoban puzzle of size N requires exp(N) time to solve, in the worst case. However, the typical Sokoban puzzle does not present the worst case, it presents a curated selection of puzzles that can be solved more quickly. This gives the solver an opportunity to feel clever, rather than just performing a computation.

Jigsaw puzzles, on the other hand, are about performing a computation. And, if you wish to do a large jigsaw puzzle in a reasonable amount of time, you look for ways to perform that computation efficiently. This raises the question: what is the computational complexity of a jigsaw puzzle?

According to the open access paper, “No easy puzzles: Hardness results for jigsaw puzzles” by Michael Brand, realistic jigsaw puzzles require Θ(N2) steps both in the worst case and on average. On the other hand, this is not born out by my own statistics, which seem to fit a straight line.

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