In this paper, we present a 4-point compact finite difference scheme for the solution of the Helmholtz equation that gives O(h^3) accuracy despite a local truncation error that is O(h). We will present a proof for why this happens and provide a number of computational tests to verify the third-order behavior. Copies of this and a related paper are at http://www.cs.uky.edu/~jzhang/pub/PAPER/hexgrid3.pdf and http://www.cs.uky.edu/~jzhang/pub/PAPER/hexgrid2.pdf