In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the Θ_2^p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates--although the number of voters may still be polynomial in the number of candidates--a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it ``knows'' that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.