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 had hunted for five dice that could be thrown to decide who moves first in a board game. Crucially, each dice 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 have also spent a lot of time reporting on the growing power of artificial intelligence to solve mathematical problems, including this week鈥檚 shock Fermat鈥檚 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 dice that decide who goes first?
Brute force wasn鈥檛 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 鈥渢hought鈥� for around 13 minutes. Then it delivered.
Advertisement
I promptly fired the solution off to 聽at Auburn University, Alabama, one of the researchers working on the five-dice problem. 鈥淚 checked your numbers,鈥� he says. They were correct.
I accept my solution wasn鈥檛 a particularly elegant or aesthetically pleasing one. Five of my dice had 720 sides; another had 20. Try rolling a 720-sided dice in the real world, or convincing players that a lopsided 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 dice have 360 sides, they told me. This isn鈥檛 good enough for a public announcement and still isn鈥檛 very practical, although it is 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, such as GPT-6, are emerging all the time with . We live in interesting times.
When I passed on the news, Paul Meyer, a software engineer at Google and another dice researcher who collaborates with Harshbarger, wondered whether AI had truly arrived at the problem or whether it had 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-dice solution is , as is a description of a method they have long known can 鈥渋nduct鈥� a solution for more dice from one with fewer. Perhaps the AI model found these parts, filled in the gaps and did its own work.
鈥淚 think it鈥檚 a big deal,鈥� says Harshbarger, who claims he has never knowingly interacted with an AI model in any way. 鈥淎 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鈥檚 intelligence there, but it may have figured something out. It鈥檚 amazing 鈥� a little scary, in some ways 鈥� but it is amazing.鈥�
Meyer, who uses AI during his day job, is even more optimistic. 鈥淚 do believe there鈥檚 some intelligence there, personally. I鈥檓 almost constantly impressed 鈥� surprised and impressed 鈥� by what it can do on a regular basis, because it鈥檚 come so far just in this last year, or six months.鈥�
But in a strange sort of way, I am 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 had 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 materials scientists will feel when their paper on groundbreaking AI-derived research is published?
Possibly not. 鈥淚 would be thrilled,鈥� says Harshbarger. 鈥淚 don鈥檛 care how a solution is found. I would certainly like to know, if we could, how the AI did it.鈥�
But how much more advanced will these models get? We have seen extraordinary progress in the past six months when it comes to mathematical ability 鈥� is this a plateau or part of a long-term trend?
at OpenAI is, as you might expect, pretty positive about the future of AI and, when I explain what I had just done with one of his models, is seemingly neither surprised nor impressed.
鈥淢y impression is that the progress we鈥檝e seen in the last six months, I fully expect we鈥檒l see the same amount of progress in the next six months,鈥� says Bubeck. 鈥淚 feel we鈥檙e really going to be able to raise our ambition, specifically in mathematics, but in science in general.鈥�
Currently, AI models are proving adept at finding counterexamples or solutions to existing problems, but have yet to demonstrate an ability to create new concepts, ideas or fields of mathematics 鈥� and that鈥檚 where dramatic progress comes from.
鈥淲e don鈥檛 have models that can do that yet. I don鈥檛 think anyone has, but it is a natural evolution,鈥� says Bubeck. 鈥淲ill that happen at the end of this year, or next year? That I don鈥檛 know, but it鈥檚 clearly going to happen.鈥�
Bubeck concedes things will change and this 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 the public sphere. Bubeck says human mathematicians may be forgiven for a period of grieving.
In the meantime, he is enjoying his work, and says that in some sense, working on new AI models is like pushing at an open door. 鈥淚 have spent years, decades, being stuck on math problems where it鈥檚 very, very hard to make any progress,鈥� says Bubeck. 鈥淎I research is the polar opposite of that.鈥�
