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Shameless self-promotion time: if you enjoyed this thread and/or are interested in these sorts of aspects of Conway's Game of Life, have a look at my (free) book "Conway's Game of Life: Mathematics and Construction", co-authored with Dave Greene: www.conwaylife.com/book/

The cover of the book "Conway's Game of Life: Mathematics and Construction"

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#3 (1/4): Oscillator density. In the early days of Life, it was conjectured that the maximum density (i.e., maximum ratio of alive cells to dead cells) of an infinitely large still life is 0.5. This density is easily attained by alternating rows of dead and alive cells.

An infinite still life in Conway's Game of Life, made up of alternating rows of alive and dead cells.

#2 (1/4): Still life glider synthesis. A glider synthesis is a way of crashing together 2 or more gliders so as to create another object. For example, in the image below three gliders collide so as to create a lightweight spaceship.

Three gliders colliding so as to create a lightweight spaceship in Conway's Game of Life.

#1 (1/4): Phoenices. A phoenix is an oscillator in which every cell dies in every generation (and thus every alive cell was dead in the previous generation, hence the name). The first known phoenix has period 2: conwaylife.com/wiki/Phoenix_1

The "phoenix 1" oscillator in Conway's Game of Life.

But there's a problem: the answer was found by computing the thing for n = 1, n = 2, and n = 3, spotting a pattern, and hoping that the pattern continues. Seem familiar?

A screenshot of ChatGPT's method of finding the solution to problem A6: computing small values.

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Wow, ChatGPT got the correct answer of 10^(n(n-1)/2)! People proclaim this would be scored 10/10 and that the AI apocalypse is here.

A screenshot of ChatGPT finding the correct solution to problem A6.

To see why this line of reasoning falls flat to me, let's look at its answer to this year's question A6. If you don't want to read the technical details of the question, all that really matters here is that the question asks you to compute a formula for something.

Problem A6 from the 2024 Putnam math competition, which asks for a formula for the determinant of a certain matrix.

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And the people reading the "proof" don't notice that it's garbage since they're exhausted from the previous five paragraphs which were almost entirely content-free and irrelevant (but technical and impressive-sounding!).

A screenshot of ChatGPT writing several paragraphs of introductory text, most of which is irrelevant or unhelpful.

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This wouldn't be so annoying if ChatGPT were honest about how it came up with and verified (or didn't verify) the solution. But instead, it acts like a grade-grubbing student and throws some crap against the wall, hoping that something will stick.

A screenshot of ChatGPT writing a gibberish proof.

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ChatGPT gets the right answer, which on the surface seems super impressive. But then you read its "proof" and see that it didn't actually prove anything: it just did a ton of computations, noticed a pattern, and hoped against hope that the pattern continues to hold forever.

A screenshot showing ChatGPT solving a problem via "experimentation with small cases".

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To illustrate what I mean by this, let's look at its answer to question B1, which asks for which sizes of grids it's possible to pick squares in a certain way.

Problem B1 from the 2024 Putnam math competition, which says "Let n and k be positive integers. The square in the i-th row and j-th column of an n-by-n grid contains the number i+j-k. For which n and k is it possible to select n squares from the grid, no two in the same row and column, such that the numbers contained in the selected squares are exactly 1, 2, ..., n?"

ChatGPT, most of the time, simply does not prove its answer is correct. This isn't a matter of "lack of rigor" or however else people try to deflect this point. It's that ChatGPT quite simply ignores the entire crux of the given problem.

A screenshot of a Tweet that says someone instructed ChatGPT to act as though it's writing the Putnam competition, causing ChatGPT to write more "rigorous" solutions.

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The problem is that Putnam questions all require you to prove something. Not just compute it or get the right answer, but prove that it's right. Heck, lots of Putnam questions *tell* you the answer, and just want you to prove it correct.

Problem B3 from the 2024 Putnam math competition, which says "Let r(n) be the n-th smallest positive solution to tan(x)=x, where the argument of tangent is in radians. Prove that 0 < r(n+1) - r(n) - pi < 1/((n^2+n)*pi) for n >= 1."

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First, the claims are that ChatGPT o1 scored 80+ points on the Putnam, or 60+ points, or high enough to be in the top 2% of participants, or whatever other impressive-sounding claim someone can make up.

A screenshot of a Tweet claiming that OpenAI o1 scored between 80/120 and 90/120 on the Putnam math competition.

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There's been a bunch of claims (mostly on X) that ChatGPT did great on this year's Putnam math competition. Let's do a thread to talk about it! 🧵 #MathSky

A screenshot of a Tweet claiming that OpenAI o1 scored in the top 1%-2% of participants in the Putnam math competition.

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It looks like Overleaf's new AI writing tools don't like the notes that I make to myself while writing papers.

A screenshot of Overleaf recommending that I correct "How the hell to do this?" to "How else can we do this?"

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Spoiler: that sum equals...

A formula that says "0 if n is odd, 2^(2m-n) * (m choose n) * (n choose n/2) if n is even"

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Had a recent research project where we had to evaluate this hideous sum. Convolution to the rescue! Made a video to talk about how it works (or at least how some similar but simpler sums work): www.youtube.com/watch?v=aIj6...

An alternating double sum of a product of 4 binomial coefficients.

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