  In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using
Cerf Theory, developed tools for comparing Heegaard splittings of irreducible,
non-Haken manifolds. As a corollary of their work they obtained a new proof of
Waldhausen's uniqueness of Heegaard splittings of S^3. In this note we use Cerf
Theory and develop the tools needed for comparing Heegaard splittings of S^3.
This allows us to use Rubinstein and Scharlemann's philosophy and obtain a
simpler proof of Waldhausen's Theorem. The combinatorics we use are very
similar to the game Hex and requires that Hex has a winner. The paper includes
a proof of that fact (Proposition 3.6).
