I really like this puzzle because it does such a good job of illustrating the essential approach to all problem-solving; identifying principles working together, then using them to narrow down the possibilities, ideally to just one result.
Sherlock Holmes’ most famous line teaches the same lesson: “once you have eliminated the impossible, whatever remains, however unlikely, must be the truth.” (His line, “Elementary, My Dear Watson” came from an American movie – it appears nowhere in A. C. Doyle’s books about Sherlock Holmes.)
We can begin such puzzles by taking an inventory of principles, which means writing down or thinking of everything we know that must be true in our situation. We really have little else to work with in an open-ended situation, and indeed, the entire history of science can be considered the (re)discovery and use of such relationships.
It may have been a while, but for this puzzle, if you write down everything which is true for all circles and all rectangles, you’ll be able to identify the two which work together to force X to be a particular size, and not some other. You’ll know when you’ve found the right ones, because you’ll immediately know what the answer MUST be, based on the logic and that is the entire point of working that way; so that you can make your own judgement about it, and can trust it, and will know that you can trust it.
