Can the Sierpinski graph be embedded in the Hamming graph?

The (generalized & expanded) Sierpinski graph, S(n,m), and the Hamming graph have the same set of vertices (n-tuples from the set {0,1,...,m-1}. The edges of both are (unordered) pairs of vertices. Each set of edges is defined by a different property so that neither is contained in the other. We ask if there is a subgraph of the Hamming graph isomorphic to the Sierpinski graph and show that the answer is yes. The embedding map leads to number of variations and ramifications. Among them is a simple algebraic formula for the solution of the Tower of Hanoi puzzle.