After 65 Years, Supercomputers Finally Solve This Unsolvable Maths Problem

Mathematician Andrew Booker was searching for topics to present at his children’s maths club when he stumbled on a YouTube video about the sum-of-three-cubes puzzle. ‘It was a case of serendipity,’ the University of Bristol mathematics professor says. The video – produced by the popular YouTube channel Numberphile – led him to solve a curious equation that has stumped mathematicians for decades.

- David Grossman

In 1825, mathematician S Ryley proved in the magazine Ladies’ Diary that any fraction can be represented as the sum of three cubes of fractions. Mathematician Louis Mordell took the puzzle a step further in 1953, when he questioned whether the same type of solution could be found for x3 + y3 + z3 = k, a Diophantine equation, which involves only positive and negative integers. He started with the number 3, and the hunt was on to find solutions for all integers between 1 and 100.

In some cases, finding the solution to x3 + y3 + z3 = k is easy enough. To reach 53, for example, all that’s needed is 33 + 33 + (–1)3. Other numbers, such as 51, grow increasingly complicated: (–796)3 + 6593 + 6023. For others – 4, 5, 13, 14, 22, 23, 31, 32 – it is mathematically impossible. Then there are the numbers that verged on unsolvable but seemed hypothetically possible, such as 33 and 42.