## Sunday, February 19, 2017

### The benefits of Recreational Mathematics

Why study Recreational Mathematics?

Why do recreational Mathematics?

1)  The line between recreational and serious mathematics is thin. Some of the problems in so-called recreational math are harder than they look.

2) Inspiring. Both Lance and I were inspired by books by Martin Gardner, Ray Smullyan, Brian Hayes, and others.

3) Pedagogical: Understanding Godel's Inc. theorem via the Liar's paradox (Ray S has popularized that approach) is a nice way to teach the theorem to the layperson (and even to non-laypeople).

4) Rec math can be the starting point for so-called serious math. The Konigsberg bridge problem was the starting point for graph theory,  The fault diagnosis problem is a generalization of the Truth Tellers and Normals Problem. See here for a nice paper by Blecher on the recreational'' problem of given N people of which over half are truth tellers and the rest are normals, asking questions of the type is that guy a normal'' to determine whose who. See here for my writeup of  the algorithm for a slightly more general problem. See William Hurwoods Thesis: here for a review of the Fault Diagnosis Literature which includes Blecher's paper.

I am sure there are many other examples and I invite the readers to write of them in the comments.

5) Rec math can be used to inspire HS students who don't quite have enough background to do so-called serious mathematics.

This post is a bit odd since I cannot imagine a serious counter-argument; however, if you disagree, leave an intelligent thoughtful comment with a contrary point of view.

1. I don't disagree but a valid counter-argument is that research costs money (even if we sometimes don't feel it), so the question is whether that money spent on rec math could have been spent better.

2. I originally thought he prior “anonymous” commenter (who no doubt inspired this post) was just trolling for attention, they seemed so over-the-top, but perhaps I’m wrong... the value of recreational math just seems obvious. One might just as well argue against doing “pure” math versus applied, or hey, there’s a lot of theoretical physics (versus applied physics) out there these days that some view as junk science and a waste of  if one wants to so argue. The value comes later and is often unforeseeable.
Also, the number of professional mathematicians who trace their interest/careers right back to old Martin Gardner columns is remarkable.

3. Among the many benefits of recreational math mentioned above...

A more profound question might be: What is the point of life? Does everyone have to work on P vs. NP or solve the Riemann Hypothesis? Is it okay to enjoy participating in sports, or to enjoy watching sports? Is it okay to enjoy recreational mathematics?

-Clyde Kruskal

4. I didn't see your reply in the Smullyan thread until today, so this is my argument against recreational mathematics versus more focused research:
Recreational mathematics is TOO recreational/motivating with respect to fundamental progress in mathematics methodology and tools.
The "overall landscape" is so vast and intricate that you could burn several lifetimes (thousands maybe) joyfully trundling around (and actually solving) many very nice and very interesting problems but...
What are the chances that among those solved problems there were some leading to decisive KEY methods and insights, improving on the whole progress pace of mathematics mastery?
Not that much IMHO, it is akin to exploring a beautiful garden caring only about the appearance of the plants and a few hints about some of their preferences sun/shade, soils, more or less water etc... this lack of purpose and method hinders your progress as a gardener despite still giving you some skills.
This metaphor is the best I can come up with to explain my rejection of mostly recreational mathematics.
I am not a mathematician but I read quite a lot of mathematics texts and I am soooo often irked and pissed off by those, to the point, to the point, damn it!
Many texts are only a glose on some intricate proof details, fail to give the proper pointers to rebuild the base ideas and do not allow to pin down the crux of the matter: is there a fruitful core idea buried in the jargon and cryptic notations or is it just a fancy decorative elaboration of a fine technical point without much import?
I think it's not just a matter of not being familiar with this of that domain but that it is very difficult to figure out if it is really worth the trouble to dig deeper in any field.
For instance I wasted a lot of time on bounded arithmetic and theory interpretations, Buss, Visser and als to finally decide that it was hopelessly "bounded" (pun intended) in that it will not lead to improvements in proof search and management.
I think professionals mathematicians are also prone to this but since they enjoy the riddles as such they don't care about the "waste"...

5. It is difficult to foresee the value of a particular research topic but still we should make the best of our effort to invest in the more likely "useful" topics. "Usefulness" here depends on our preferences (philosophical, practical, elegant, etc.) I think for most people recreational math is low in these measures.

I know of no one getting stuck into math by recreation math.. Most of us enjoyed playing with it in our teen days but it seemed far from being the push. Btw even if you think it's helpful to get students interested in math, does it justify *more* rec math research being done? I would think that existing results are more than enough to play this role.

My take is that you should do rec math only if you truly enjoy it. The other "collateral" benefits are too weak to justify so.