I read this with pleasure, right up until the bit about the ants. Then I saw the note from myself at the end, which I had totally forgot writing seven years ago. I probably first encountered the article via HN back then as well. Thanks for publishing my thoughts!

The ants argument feels rather like a retelling of Zeno's Paradoxes.

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

Thanks! I'm glad to see I'm not the only one who went down this rabbit hole. :)

I considered parallelizing my solution as well, but the problem is that it only gives a linear speedup, while the problem space increases exponentially. I decided to focus on pruning the search tree instead, and that seemed to work pretty well (after much thinking).

It's getting crowded down here in the rabbithole... One more to peek at: https://github.com/prb/pips-solver/blob/main/README.md

  Last updated 2025-10-27.
  [...]

  The puzzles with the most solutions are:


  • 2025-09-15 hard: 2,764,800 solutions
  • 2025-10-05 hard: 344 solutions
  • 2025-09-30 hard: 110 solutions
  • 2025-09-04 hard: 86 solutions
  • 2025-08-23 hard: 80 solutions

Hah...it's like the NYT was just waiting for you to update so they could immediately release a puzzle that makes your list out of date. 2025-10-28 hard has 166 724 solutions.

That's great! Your experience with the 2025-09-15 and 2025-10-14 puzzles was very similar to mine, I think. I'm impressed that you were able to get AI models to solve this game effectively. I coded it the old-fashioned way myself, mostly, with occasional help from Gemini Pro.

I did write the spec first — data model, algorithm, etc. That may have helped the agents get traction.

Particle Lenia is awesome. I did an online version that you can interact with here: https://brianberns.github.io/ParticleLenia/

I made a game out of creating proofs without words: https://brianberns.github.io/Tactix/

Yes, I use F# with TorchSharp to do machine learning. My most recent project is a Hearts AI that is quite good. Here are some links:

* https://github.com/brianberns/Hearts

* https://github.com/brianberns/MinGptSharp

* https://github.com/brianberns/ModestGpt

* https://github.com/brianberns/DeepKuhnPoker

This idea comes from a functional pearl called "Power Series, Power Serious" [0], which is well worth reading.

I implemented the same thing myself in F#. [1]

[0]: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&d...

[1]: https://github.com/brianberns/PowerSeries

Huh, there must have been something in the water leading up to this. Also from 1998 is this paper, "Calculus in coinductive form" and neither of these cites the other. https://ieeexplore.ieee.org/document/705675

These are indeed very similar. Thanks for the link!

The math is a bit over my head, but this formulation seems more difficult than the one I'm familiar with. For example, x^2 is represented as 0::0::2 instead of 0::0::1 (because 2! = 2) and x^3 is represented as 0::0::0::6 instead of 0::0::0::1 (because 3! = 6). Is there a benefit to that?

I was introduced to the notion of power series two weeks ago, and now it's seemingly everywhere...

Power series are possibly one of the most powerful tools in analysis.

Power series have been everywhere for 200 years!

I am learning that indeed.

I translated this to F# for my own edification. It's more verbose, but perhaps easier to understand for non-Haskellers.

https://github.com/brianberns/AnnotatedStack

FWIW, you might like this: https://github.com/brianberns/FYampaSynth

That looks right up my alley, thanks for the link!

Did you read the linked paper? They consider this possibility and reject it.

Sure. Did you realize that epistemically they don't have the power to reject explanations? It would certainly be news to Stephan Guyenet if they did have in this case.

Using media statements to sway public opinion is… a totally normal thing for organizations to do.

Exactly. And yet people in this thread are arguing against the idea that the CIA would do it.