Studying recreational math professionally?
There are a lot of folks who do this, I think. I also think that a lot of them are members of the Mathematical Association of America (MAA). That isn't their main focus, of course, but it would be a good source of contacts and information. 

The Wikipedia Article points to some other resources and maybe ways to contact people. 

Whether you can make a career out of it depends on some things. At an R1 university, probably not, but maybe at a liberal arts college in the US. But it can be fun as a break from more serious(???) pursuits. And some aspects of it are certainly valid research topics. Game Theory, for example.