I can't find any description of the search algorithm that found it. Is Timothy Browning keeping it a secret? Obviously, he didn't simply try every pair of 16-digit numbers to see if their difference, plus 33, was a cube. There isn't enough computer power in the world for such a search. Maybe something to do with modular arithmetic? It looks like, modulo any prime in the form 6k+1, cubes have only about a third of the possible residues (e.g. cubes have 35 residues mod 103). And they're of course all independent, so by combining a bunch of them you can shrink the search space by as much as desired. Unless that just triggers a different combinatorial explosion. It's cool that the next unknown is 42. I already presented the problem to a Douglas Adams fan as the 42 problem years ago, skipping all mention that 33 was also, at the time, unknown.