Janet, my original strategy was apparently trying NOT to find the magic fish. The idea of finding the magic fish seemed too daunting, so I just wanted to match the fins to their appropriate bodies to end up with a nice tank full of pretty fish. Seemed simple enough.

But that's how I ended up with a few of them. In fact, I was a teeny bit disappointed when they turned out to be magic fish, because all those trailing sparkles were messin' up my serene little aquariums!

Turned out that the pursuit of matching fins to bodies was almost as difficult, as some of the fins seem to work for sevral bodies!
But over all, keeping track of what crossed with what produces what (via spreadsheet) was the only way to progress, no matter what direction I was going or what my goal was.