4 problems

Let $S(n)$ denote the maximum number of steps executed by any halting $n$-state Turing machine started on a blank tape. The Marxen–Buntrock machine is a 5-state machine achieving…

Fifth busy beaver number conjecture. Although there is no general algorithm to compute $\operatorname{BB}(n)$ in general, researchers have long conjectured that

Let $\operatorname{BB}(n)$ denote the maximum number of steps made by any halting deterministic Turing machine with $n$ states and two symbols, starting on a blank tape. A machine…

Let $S(n,m)$ denote the maximum number of steps before halting, and let $\mathnormal{\Sigma}(n,m)$ denote the maximum number of symbols written, among halting $n$-state, $m$-symbol…