A study of the output of a heuristic computer program reveals two four-state binary Turing machines which yield the highest known score for four states in Rado's co-called "Busy Beaver" logical game. There is evidence which supports the conjecture that this score of 13 is the particular value of Σ(4), where Σ is a noncomputable integer function associated with this game. It is also conjectured that S(4) = 106, where S is another noncomputable function, the maximum shift number, of interest in Rado's study. Complete solution of the problem for four states has been reduced to a relatively small set of machines.