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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?

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  • $\begingroup$ This is a bit tough because I think there are different categories of very different recreational mathematics. For example, many would suggest that the entire field of Combinatorial Game Theory falls under that umbrella (see this conference or Wikipedia), even though it often involves things that wouldn't be easily accessible to people without prior knowledge of the field. $\endgroup$
    – Mark S.
    Commented Oct 8, 2020 at 13:23

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Recreational math tends to comprise topics that are fun, and easily accessible to people without a lot of specialized training, but, for one reason or another, are not usually topics of serious mathematical research. Factors in that lack of serious mathematical research may include lack of applications of the topic, lack of mathematical tools to apply to the topic, a feeling that the topic itself is "silly" or arbitrary, or simply outside the mathematical fashion of the day.

You might look at "What is recreational mathematics?" by Charles Trigg, Mathematics Magazine 51(1), 18-21. doi:10.2307/2689642

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The history of mathematics is replete with examples of something novel being done to address a problem at the time. It might have been "applied" in the sense of helping with a technological innovation, or it might have been "pure", e.g. mathematicians trying to make sense of something unrigorous or paradoxical in their present formulations of mathematics. Recreational mathematics is different again. Let's contrast three examples:

Applied: how can we encrypt data so eavesdroppers can't decrypt it fast enough to crack our security before we update it? At least one main approach relies on number theory.

Pure: asymptotically, how many prime numbers are there below $n$? (Answer here.) More to the point, what kinds of mathematics do we need to prove the answer? Complex numbers came up in early proofs, but we eventually found we didn't need them. There are all sorts of interesting stories there.

Recreational: when is a perfect power plus one also a perfect power? (We know now.)

Recreational mathematics involves solving a lot of Diophantine equations that don't enrich our understanding of mathematics as a whole.

Now, the applied/pure distinction gets foggy at times, especially when something that seemed pure at first becomes "useful" later, or when something invented for applications has "pure" spin-offs. But distinguishing these two from recreational is easier, as recreational problems aren't expected to enrich even our mathematical insight in general.

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I don't think there is a definitive answer. Some characteristics I have noted that set recreational problems apart are

  • Particular instead of general problems
  • Depend on the digit representation of a number
  • Involve finding a trick of logic to find the answer
  • Worded in a way to make finding the right approach difficult
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