> What's the point of spending "time" (brain cycles) on such problems?
What's the point of anything?
> Is it just a mathematician's version of brain teaser?
All of mathematics can be characterised as "brain teasers". Make of that what you will.
> What useful development comes out of such?
I'm not qualified for this particular example. However, it seems quite number-theoretic, and number-theory has given us such "practical" things as asymmetric encryption and universal computation (Turing's famous paper from 1936 was titled "On Computable Numbers", after all).
What's the point of anything?
> Is it just a mathematician's version of brain teaser?
All of mathematics can be characterised as "brain teasers". Make of that what you will.
> What useful development comes out of such?
I'm not qualified for this particular example. However, it seems quite number-theoretic, and number-theory has given us such "practical" things as asymmetric encryption and universal computation (Turing's famous paper from 1936 was titled "On Computable Numbers", after all).