Mathematicians Puzzled Over a Famous Problem for 80 Years. Now, They've Used A.I. to Identify a Clever Solution

Ellen Wexler | Writer and Special Projects Editor Add as preferred source For 80 years, most mathematicians assumed Paul Erdős strategy was correct. OpenAI Place any number of dots on a two-dimensional plane—say, a piece of paper—and measure the distance between each pair. If you rearrange the dots, how many pairs could be positioned exactly the same distance apart? The Hungarian mathematician Paul Erdős first posed this question in 1946. Even as the number of dots grew, he argued that the best way to maximize the total pairs was to draw a grid-like arrangement. Erdős also speculated that the number of pairs could only be slightly higher than the number of dots. For the next 80 years, most mathematicians assumed he was correct. But now, an artificial intelligence model has challenged this approach. OpenAI recently announced that one of its internal models had identified a strategy to produce more pairs than the arrangements proposed by Erdős. “This is the unique, interesting result produced autonomously by A.I. so far,” Daniel Litt, a mathematician at the University of Toronto, tells Scientific American’s Joseph Howlett. This particular puzzle, known as the planar unit distance problem, was “one of Erdős’ favorite problems,” Noga Alon, a mathematician at Princeton University, writes in a companion paper to OpenAI’s announcement. Alon had heard him mention the problem “multiple times” in his lectures. In 1982, after decades with few breakthroughs, Erdős offered $300 for a proof or disproof of his proposed upper bound. He increased that amount to $500 around 1995. Born in Budapest in 1913, Erdős was the son of two math teachers. Both of his older sisters died of scarlet fever while his mother was in the hospital for his birth. Soon after World War I broke out in 1914, his father was detained in a Siberian prisoner-of-war camp. He wouldn’t return for six years. In the midst of this anguish, Erdős’ mother became very protective of her surviving child, keeping him at home most of the time. “I fell in love with numbers,” he later recalled. “They were my friends. I could depend on them to always be there and always behave in the same way.” After graduating from a Hungarian university with a PhD in mathematics, Erdős lived an itinerant lifestyle, staying with mathematicians around the world who he wanted to collaborate with. He would turn up, often without notice, uttering his signature greeting: “My brain is open!” “He lived and breathed mathematics and could fall asleep at a dinner table if mathematics wasn’t the topic of conversation,” the Los Angeles Times’ Hector Tobar wrote in Erdős’ 1996 obituary. He co-authored more than 1,500 papers, making him one of the 20th century’s most prolific mathematicians. In addition to his published papers, Erdős posed hundreds of problems for fellow mathematicians to solve, often offering monetary rewards. He priced some as low as $10 or $25. Others came with bounties in the thousands. (His largest was $10,000.) In practice, this reward system was rather informal. “I solved a $250 problem, but I only got $50 because Erdős didn’t like the proof,” the late András Hajnal, then a mathematician at Rutgers University, told Science magazine’s Charles Seife in 2002. Because Erdős had posed so many problems, he didn’t always know when someone had solved one. “Some years ago, he was visiting Athens, Georgia,” mathematician Carl Pomerance recalled in 2002. “He was going back into town, and my colleague Helmut Maier was giving him a ride.” When Maier mentioned a theorem he’d recently proven, Erdős wondered whether he’d offered a prize for it. The two men rerouted to the library, where they discovered that Erdős had, indeed, promised $100. “I said to Erdős, ‘That’s a pretty expensive taxi ride,’ and he found that hilarious,” Pomerance told Science. When OpenAI announced its recent breakthrough, “everyone in math lost their minds,” writes the Wall Street Journal’s Ben Cohen. The idea of such a discovery would have been laughable even a year ago—or, as OpenAI’s Sébastien Bubeck tells the publication, even “a month ago.” OpenAI’s models had attempted to tackle Erdős problems before. Last year, Kevin Weil, then a vice president at the company, announced that “GPT-5 just found solutions to ten (!) previously unsolved Erdős problems.” In reality, the model had identified existing solutions that were buried in the literature. The Erdős Problems website listed these problems as “open,” but “the ‘open’ status only means I personally am unaware of a paper which solves it,” Thomas Bloom, a mathematician at the University of Manchester who maintains the site, wrote on social media. He called OpenAI’s claim a “dramatic misrepresentation.” This time, OpenAI appears to have been more careful. In the companion paper, a group of mathematicians—including Bloom—reviewed the breakthrough and reflected on A.I.’s role in the process. OpenAI’s strategy involves arranging the dots using techniques

Mathematicians Puzzled Over a Famous Problem for 80 Years. Now, They've Used A.I. to Identify a Clever Solution

Key takeaways

More in publications