The maximum length of the shortest string accepted by an n-state two-way finite automaton is currently known to be at least of the order Ω(1.626n) and at most (Formula presented). In this paper, a family of n-state automata with shortest accepted strings of length (Formula presented) is constructed, thus improving the lower bound. Also a modest improvement to the upper bound is made: the length of the shortest accepted string is at most (Formula presented). For the special case of direction-determinate automata (those that always remember in the current state whether the last move was to the left or to the right), the maximum length of the shortest accepted string is determined precisely as (Formula presented) .