Interesting - if any computer program can be rewritten in the form of a Collatz-like function, is it at all surprising that the BB problems appear in this form?
The interesting part is how simple the resulting Collatz-like functions are to describe, suggesting that the Collatz-like form is the most 'natural' form for hard problems given the TM formalism. E.g., any program can also be rewritten in the form of a Diophantine equation, but most of those would be monstrously large.
The wikipedia article doesn't do full justice perhaps though the essence is there. The reason the halting problem is equivalent to solving collatz like problems is because you can rewrite any turing machine in terms of a generalized collatz function. The Fractran wikipedia page has more info.
(the original conway article itself is also very readable: FRACTRAN: A SIMPLE UNIVERSAL PROGRAMMING LANGUAGE FOR ARITHMETIC)
Given an input n, if n is of the form 3k then output 5k+6
if n is of the form 3k+1 then output 5k+9
if n is of the form 3k+2 then halt
So starting with 0, which is of the form 3x0, we go to 5x0+6=6, then to 5x2+6=16. Because 16 is 3x5+1 we then go to 25+9=34, etc.
The bb program is made such that one computation of the function takes an insane long time.
By the way Conway proved [0] that any computer program can be rewritten in the form of these collatz like functions. So they are pretty prowerful.
[0]: https://en.wikipedia.org/wiki/Collatz_conjecture#Undecidable...