Untying The Gordian Knot via Experimental Mathematics

In this methodological article on experimental-yet-rigorous enumerative combinatorics, we use two instructive case studies, to show that often, just like Alexander the Great before us, the simple, "cheating" solution to a hard problem is the best. So before you spend days (and possibly years) trying to answer a mathematical question by analyzing and trying to 'understand' its structure, let your computer generate enough data, and then let it guess the answer. Often its guess can be proved by a quick 'hand-waving' (yet fully rigorous) 'meta-argument'. Since our purpose is to illustrate a methodology, we include many details, as well as Maple source-code.