This year, a number of LRMs, which try to solve a problem step by step rather than spit out the first result that comes to them, have achieved high scores on the American Invitational Mathematics Examination (AIME), a test given to the top 5% of US high school math students.
At the same time, a handful of new hybrid models that combine LLMs with some kind of fact-checking system have also made breakthroughs. Emily de Oliveira Santos, a mathematician at the University of São Paulo, Brazil, points to Google DeepMind’s AlphaProof, a system that combines an LLM with DeepMind’s game-playing model AlphaZero, as one key milestone. Last year AlphaProof became the first computer program to match the performance of a silver medallist at the International Math Olympiad, one of the most prestigious mathematics competitions in the world.
And in May, a Google DeepMind model called AlphaEvolve discovered better results than anything humans had yet come up with for more than 50 unsolved mathematics puzzles and several real-world computer science problems.
The uptick in progress is clear. “GPT-4 couldn’t do math much beyond undergraduate level,” says de Oliveira Santos. “I remember testing it at the time of its release with a problem in topology, and it just couldn’t write more than a few lines without getting completely lost.” But when she gave the same problem to OpenAI’s o1, an LRM released in January, it nailed it.
Does this mean such models are all set to become the kind of coauthor DARPA hopes for? Not necessarily, she says: “Math Olympiad problems often involve being able to carry out clever tricks, whereas research problems are much more explorative and often have many, many more moving pieces.” Success at one type of problem-solving may not carry over to another.
Others agree. Martin Bridson, a mathematician at the University of Oxford, thinks the Math Olympiad result is a great achievement. “On the other hand, I don’t find it mind-blowing,” he says. “It’s not a change of paradigm in the sense that ‘Wow, I thought machines would never be able to do that.’ I expected machines to be able to do that.”
That’s because even though the problems in the Math Olympiad—and similar high school or undergraduate tests like AIME—are hard, there’s a pattern to a lot of them. “We have training camps to train high school kids to do them,” says Bridson. “And if you can train a large number of people to do those problems, why shouldn’t you be able to train a machine to do them?”
Sergei Gukov, a mathematician at the California Institute of Technology who coaches Math Olympiad teams, points out that the style of question does not change too much between competitions. New problems are set each year, but they can be solved with the same old tricks.
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