Consider Turing machines that read and write the symbols 1 and 0 on a one-dimensional tape that is infinite in both directions, and halt when started on a tape containing all O's. Rado'sbusy beaver function ones(n) is the maximum number of 1's such a machine, withn states, may leave on its tape when it halts. The function ones(n) is noncomputable; in fact, it grows faster than any computable function.

Other functions with a similar nature can also be defined. The function time(n) is the maximum number of moves such a machine may make before halting. The function num(n) is the largest number of 1's such a machine may leave on its tape in the form of a single run; and the function space(n) is the maximum number of tape squares such a machine may scan before it halts.

This paper establishes a variety of bounds on these functions in terms of each other; for example, time(n) ≤ (2n − 1) × ones(3n + 3). In general, we compare the growth rates of such functions, and discuss the problem of characterizing their growth behavior in a more precise way than that given by Rado.