I would simply have called Bruno and asked if he remembered the solution that Paul gave him in the taxi. And if the cell phone signal not could penetrate the cave I would just have turned on the light on my phone! Why make it so complicated? (Traders versus Quants I guess)
"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
the taxi number being 1729?.
Cubing the Cube? Unfortunately then we need to call someone else than Bruno. Or did he tell about it on another cab ride?