Mathematics is not my primary discipline, but I know enough about both it and academics in general to know that many to most mathematical researchers do what they do because they enjoy doing it. This would seem to make "recreational mathematics" a rhetorical tautology, yet the term is used as if it were a subdiscipline in its own right. Universities offer courses on recreational mathematics. There are academic journals on recreational mathematics. There's a tag right here on SE Mathematics that reads
recreational-mathematics. It's defined as "Mathematics done just for fun, often disjoint from typical school mathematics curriculum." Yet it does seem to be applied to a certain specific, albeit rather eclectic category of mathematical problems.
A lot of math that is called recreational falls quite neatly under some other category of math, often logic or combinatorics. In some cases, a kind of recreational math seems to be characterized as such only because some other formulation of the same ideas "got there first": it's relaxing to draw shapes with a compass and straightedge in much the same way that folding paper is relaxing, and the mathematics behind origami apprehend most if not all of the same mathematical truths that Euclidean geometry does, but mathematical origami is considered recreational while Euclidean geometry generally isn't.
With that in mind, what is recreational math? "Math that's done for fun" doesn't seem to make sense, because, again, that applies in some capacity to all math. Is it math that's done casually, with less concern for rigorous proof? Is it math that has not (yet) found an application in engineering or the empirical sciences? Is it the mathematical counterpart to popular science? What is it?
It's good math, but I don't understand this label attached to it. What makes a particular bit of math recreational?