  By introducing the busy beaver competition of Turing machines, in 1962, Rado
defined noncomputable functions on positive integers. The study of these
functions and variants leads to many mathematical challenges. This article
takes up the following one: How can a small Turing machine manage to produce
very big numbers? It provides the following answer: mostly by simulating
Collatz-like functions, that are generalizations of the famous 3x+1 function.
These functions, like the 3x+1 function, lead to new unsolved problems in
number theory.
