= Specific values of the Busy beaver function
The following things come to mind when you look into research in this area, especially the search for <BB(5)> which was hard but doable:
* it is largely <recreational mathematics>, i.e. done by non-professionals, a bit like the <aperiodic tiling>. Humbly, they tend to call their results <lemma (mathematics)>[lemmas]
* complex structure emerges from simple rules, leading to a complex <classification (mathematics)> with a few edge cases, much like the <classification of finite simple groups>