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Last month, I interviewed mathematicians who were seeking a very peculiar set of dice. For no reason other than they found the problem fun, they’d hunted for five die that could be thrown to decide who moves first in a board game. Crucially, each die must have the same odds of winning and throws must never tie. Their search took more than a decade, but they finally cracked it.
I’ve also spent a lot of time reporting on the growing power of AI to solve mathematical problems, including this week’s shock Fermat’s last theorem formalisation. The two ideas began to commingle in my head after I got off the phone with the dice hunters. Could AI, I wondered, go one better and find a solution for six players, with six die that decide who goes first?
AI’s solution to 87-year-old riddle takes mathematicians by surprise
Brute force was not the answer: there are more possible designs of five dice than there are atoms in the universe. So, I fired up ChatGPT, described the problem, and asked for a solution. I nudged and clarified twice, and in total it “thought” for around 13 minutes. Then it delivered.
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I promptly fired the solution off to Eric Harshbarger at Auburn University, Alabama, one of the researchers working on the five-die problem. “I checked your numbers,” he says. They were correct.
I accept my solution was not a particularly elegant or aesthetically pleasing one. Five of my die had 720 sides; another had 20. Try rolling a 720-sided dice in the real world, or convincing players a lop-sided set is fair.
Nor was my AI-derived solution even the best we know of. Harshbarger and his colleagues have already been able to adapt their five-dice solution to create a six-dice one, where all die have 360 sides, they told me. This is not good enough for a public announcement and still isn’t very practical, although undeniably much better than mine.
However, a layman like myself, with at best dusty mathematics, found a solution to a thorny, if unserious, problem in minutes by doing nothing more than describing it in plain English to an AI model. Something fundamental in mathematics has shifted, and new models like GPT-6 are emerging all the time with ever better benchmark scores. We live in interesting times.
The next generation of AI models are meant to be trained by people paid to have conversations with them, but several of these workers have admitted to New Scientist that they simply get chatbots to do it instead. This “AI inbreeding” may reduce the power and usefulness of future models, warn experts
Paul Meyer, a software engineer at Google and another die researcher who collaborates with Harshbarger, wondered when I passed on the news whether AI had truly arrived at the problem, or simply found it in some obscure part of the internet. But a search revealed no trace.
The best explanation that Harshbarger, Meyer and I came up with on a call is that the five die solution is published on the website used by the researchers to track their work, as is a description of a method they have long known can “induct” a solution for more dice from one with less. Perhaps the AI model found these parts, filled in the gaps and did its own work.
“I think it’s a big deal,” says Harshbarger, who claims he has never knowingly interacted with an AI model in any way. “A non-math person, at least, if they read those two components, might not be able to do [that] on their own. I hate to say there’s intelligence there, but it may have figured something out. It’s amazing – a little scary, in some ways – but it is amazing.”
Meyer, who uses AI during his day job, is even more optimistic. “I do believe there’s some intelligence there, personally. I’m almost constantly impressed – surprised and impressed – by what it can do on a regular basis, because it’s come so far just in this last year, or six months.”
But in a strange sort of way I’m almost embarrassed to have done it. It took no skill on my part. I also feel it somehow undermines the work of the human mathematicians I’d reported on and I fear it may be the thin end of the wedge for diminished human input in the field. What happens to pet mathematical ideas when a free online chatbot can chip away at them? Is this how future mathematicians, physicists and material scientists will feel when their paper on groundbreaking AI-derived research is published?
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Possibly not. “I would be thrilled,” says Harshbarger. “I don’t care how a solution is found. I would certainly like to know, if we could, how the AI did it.”
But how much more advanced will these models get? We’ve seen extraordinary progress in the last six months when it comes to mathematical ability – is this a plateau or part of a long-term trend?
Sébastien Bubeck at OpenAI is, as you may expect, pretty positive about the future of AI and, when I explain what I’d just done with one of his models, seemingly neither surprised nor impressed.
“My impression is that the progress we’ve seen in the last six months, I fully expect we’ll see the same amount of progress in the next six months,” says Bubeck. “I feel we’re really going to be able to raise our ambition, specifically in mathematics, but in science in general.”
Currently, AI models are proving adept at finding counterexamples or solutions to existing problems, but have demonstrated no ability to create new concepts, ideas or fields of mathematics yet – and that’s where dramatic progress comes from.
Ten conundrums that stumped human mathematicians for years have been cracked by OpenAI’s Astra model, continuing a hot streak of AI-driven breakthroughs that are shaking up the field
“We don’t have models that can do that yet. I don’t think anyone has, but it is a natural evolution,” says Bubeck. “Will that happen at the end of this year, or next year? That I don’t know, but it’s clearly going to happen.”
Brubeck concedes things will change, and it may cause some upset. The enjoyment derived from manually picking away at a problem and finding a solution may be a thing of the past. The community will also have to grapple with thorny problems like whether or not they can take credit for work AI does, or whether they are just facilitators who usher these findings into public. Brubeck says human mathematicians may be forgiven for a period of grieving.
In the meantime, he’s enjoying his work, and says that in some sense, working on new AI models is like pushing at an open door. “I have spent years, decades, being stuck on math problems where it’s very, very hard to make any progress,” says Brubeck. “AI research is the polar opposite of that.”
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