I'm somehow fascinated by the Velcro invention story because, as far as I can guess, Velcro would hardly have been invented if someone was looking for an elegant solution of this kind in per-meditated, logical manner.
Let's assume we had the explicit task of discovering a method for connecting two fabric surfaces in a way that they would hold tight while fastened and would be extremely easy to unfasten:
a) sticking by the "theoretical" approach, my personal starting points of thinking would be glue, magnets, vacuum, knobs, etc;
b) the approach of exploring existing solutions in the fabric field or analogical fields - the "innovation transfer" approach, so to say - would lead us to the Velcro solution sooner or later, just the way it has actually happened - the inventor had loaned an idea seen in nature. But we must note he had not been explicitly looking for this solution then.
What bothers me, is that the first approach, while looking quite exhaustive, scientific and reliable, would hardly bring up such abstract, non-evident solution as the Velcro. Pure logic and reasoning are usually stem from more or less familiar concepts.
If we are to base our reasoning on some "synthetic" concepts (or so called "constructs") - that is, concepts which have not been observed in reality, but are supposed to possibly exist - we would spend a huge amount of time and energy describing those constructs, that we would hardly be able to actually proceed to thinking how they would be applied to our problem. It might be something like trying to do a series of very complex mathematical calculations in your mind alone.
Therefore, I'm wondering, is there is any special theoretical problem-solving approach has the potential to bring solution of the Velcro type?