[7373737373]

The https://en.wikipedia.org/wiki/Busy_beaver value for machines with 2 symbols and 5 states (the maximum number of steps that such a machine can make on an initially blank tape before eventually halting) was recently proved to be 47,176,870: https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-prov...

Finding the value for machines with 2 symbols and 6 (or more) states will require solving, among other things, the problem of whether the following Collatz-like program ("Antihydra": https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html, https://wiki.bbchallenge.org/wiki/Cryptids, https://wiki.bbchallenge.org/wiki/Antihydra) halts or not, because there exists such a machine that exhibits this behavior:

    a = 8
    b = 3
    while b:
        a += a>>1
        b += 2-(a%2)*3

This looks simple, but no one expects it to be solved in the near future. So Collatz(-like) problems are intimately connected to the halting problem and in this sense represent the "simplest" type of problem (statement) in the Cryptid/complexity hierarchy we cannot (yet?) solve. Other ((closer to the) real world) problems are higher in this hierarchy (require more states to describe: https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids), and solutions/increased understanding of the Collatz problems could assist in their solution.

[zekchelovek]

Thank you for giving me an answer that nobody else seems to know. I have no idea if you're correct, since i haven't studied the halting problem, although i understand the general concept.