But think of this trick we use about turning a tricky research paper in maths/science into something more tangible by making an LLM whip up an interactive version. That works at every level of education, and it means that you can completely tailor a piece of educational material to the kid

Tiny example: one kid was introduced to fractions and found it abstract that it was both about partitioning stuff and about numbers on a number line. So while we were practicing, I had an LLM make https://fuglede.codeberg.page/broeklegeplads/ to make it more hands-on.

Obviously for the small kids, this has to be an experience guided by teachers and parents, but for bright older kids with sufficient discipline that ought to be a useful trick for enhancing education.

Of course when we were kids, we would just write such educational programs ourselves and get the same effect /and/ learn to program (before getting banned from the computer room for putting spooky /binaries/ on the computers anyway), so maybe that's better for older kids. And maybe these kids will never have to do any maths or programming because the AI overlords have taken over when they grow up.

So in the game you're taking turns moving coins down a line until no moves are possible. If you've taken a course on combinatorial game theory, you'll have come across the game of nim and you can think of this one as a variant of nim where there can only be a single coin in every square. The theory of how to play it isn't quite as well-established as for nim though; play with enough coins and I certainly find it quite tricky to mentally reason about whether a position is good or not.

One fun fact about Welter's game is that its winning positions form the codewords of a particular error-correcting code called the constant weight binary lexicode (Conway and Sloane, 1986). What that means is that being good at the game is the same as being good at decoding messages in that code. Probably more fun to just play the game though.

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It will, at least, have to be acknowledged by making GDPR Art. 33 notifications.

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What I don't like is that all the resulting pages look the same. It's so easy to tell when a page is authored by Claude, and by now, I get the same “ugh another one” feeling as Dall-E generated images would give when they were completely overdone.

Maybe it's just be, but if not, maybe HTML generated outputs will also end up being dialed down from where they're at now.

But maybe not, since it's obviously a useful thing to be able to do. I wonder if there's a way out. To be able to introduce some natural entropy so not everything ends up looking the same. I guess not, since we're using machines whose natural mode of behavior is regression toward the mean. And maybe having it be harder to tell the author wouldn't even be desirable anyway.

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If the reason it was able to output the proof is that it happened to be included in an in-house university report written in Georgian, then that would make it less useful for research than if it's new entirely.

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One extension that I ran into, and which I think forms a nice problem is the following:

Just like the XOR swap trick can be used to swap to variables (and let's just say that they're bools), it can be extended to implement any permutation of the variables: suppose that the permutation is written as a composition of n transpositions (i.e., swaps of pairs), and that is the minimal number of transpositions that let's you do that. Each transposition can be implemented by 3 XORs, by the XOR swap trick for pairs, and so the full permutation can be implemented by 3n XORs. Now here's the question: Is it possible to come up with a way of doing it with less than 3n, or can we find a permutation that has a shortcut through XOR-land (not allowing any other kinds of instructions)? In other words, is XOR-swapping XOR-optimal?

I'm not going to spoil it, but only last year a paper was published in the quantum information literature that contains an answer [0]. I ended up making a little game where you get to play around with XOR-optimizing not only permutations, but general linear reversible circuits. [1]

[0] https://link.springer.com/article/10.1007/s11128-025-04831-5

[1] https://swapple.fuglede.dk

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