I actually started by saying "reasoning" is a strong statement because as you note, it's really not reasoning. It's about intuition. And yes, if you've seen it before then intuition doesn't play a role. But if you haven't had sufficient experience with it, even if you know "double each square", and even if I walk you through the first ten squares on the chess board "OK, now we're at $10.24 cents... and we're at $10 million dollars with the other method -- it's not looking too good for the doubling method is it!?".

But a philosophical question is, with sufficiently large memory -- does almost everything just reduce to a measure of memory?