I don't doubt the claim, but this story is so specific ("the cube root of 1729.03") that it seems plausible there may be at least some truth to it. And it start

I don't doubt the claim, but this story is so specific ("the cube root of 1729.03") that it seems plausible there may be at least some truth to it. And it starts off with him being soundly beaten in adding and multiplying, so it's not totally egotistical. (I get that it's still pretty egotistical, since the parable is that he understands the actual numbers while the salesman is just using a rote, mechanical process without fully understanding the underlying theory.) I could definitely see him totally making this up by working in reverse (decide to teach a lesson about mechanical vs. fundamental understanding, start with a tough operation, pick a mentally tractable number like a bit over 1728, tack on the simpler arithmetic contests before it to contrast and build tension), so I'm not saying it's a good argument for it being real. Just that it's tough to say one way or another.