Way back in 2007, I got my first taste of mathematical research. I was in an REU at the University of West Georgia run by Professors Bruce Landman and Abdollah Khodkar. They taught us some basics of graph theory and Ramsey theory, and then presented a broad range of open problems. One which caught my attention that summer was about the existence of so-called Silver Cubes. These are vertex colorings of the graph which have certain properties, and the open question was whether these exist for all . The paper which introduced these graphs showed that they can be constructed multiplicatively: that is, given two cubes of orders and , one can construct a cube of order . The paper also gave explicit constructions for , and asked whether there existed one of order 7. I became kind of obsessed with the problem that summer, and after trying many different approaches, both computational brute force and testing various symmetry ansätze, I found success! I was able to construct a cube of order 7. Professor Khodkar and I wrote up the paper and this became my first publication. The original code is lost to time, but I do remember it was written in Java and involved a truly grotesque tower of nested for-loops.

The natural next steps were of course to attempt to construct cubes of higher order. I spent a fair amount of time that summer and the following academic year trying to construct cubes of order 11 or 13, to no avail. The computational complexity just grew too quickly, and there was no obvious emergent pattern in the cubes I was able to construct. As of today, as far I’m aware, no progress had been made on the Silver Cube problem since the result.

So why write about this now? As the reader may be aware, there’s been a lot of hype lately about LLM’s making progress on open questions in mathematics, especially those with a combinatorial or graph theoretic flavor (see e.g. Tao’s wiki on AI for Erdos problems). That brings us to last Monday, May 25. After I got home from work, I thought hey I’ve got a Claude Code subscription, why not let it take a crack at this? I spent some, not all, of that evening on a vibey research session, just presenting the problem without any other context and let it run overnight. It didn’t make much progress, not even able to reconstruct the cube for . The next evening, I dredged up some vague recollections of how I’d approached it twenty years prior, and this was apparently enough for Claude to construct the cube in about 20 seconds. After that, I tasked it with finding a cube for , and let it run overnight. On Wednesday morning, I woke up to discover that it had constructed a cube of order 11 (!!!).

As I am still paranoid of hallucinations, I did independently verify the cube with a python script written by hand. I spent Wednesday and Thursday evening prompting Claude to solve or even try to come up with a general construction to no avail. I’ll probably put the problem down for now and pick it back up in a year when the models have gotten an order of magnitude better, but this was a fun and demonstrative sign of the times. The code and more technical details can be found in the Github repo here.

Special thanks to Professor Khodkar for introducing me to this problem nearly twenty years ago. Sadly, Professor Khodkar passed away last year, but I’d like to think he would have been excited to see this result.