Williams saw that a simulation based on squishy pebbles would make the new algorithm’s space usage much smaller—roughly equal to the square root of the original algorithm’s time budget. That new space-efficient algorithm would also be much slower, so the simulation was not likely to have practical applications. The weird part is how Williams got there, by first proving a result that applies to all algorithms, no matter what problems they solve. That repeated improvement doesn’t work with the current version of the algorithm, but researchers don’t know whether that’s a fundamental limitation. “But often, the thing I prove is way better than what I wanted.”Editor’s note: Scott Aaronson is a member of Quanta Magazine’s advisory board.