  A pebbling move on a graph G consists of the removal of two pebbles from one
vertex and the placement of one pebble on an adjacent vertex. Rubbling is a
version of pebbling where an additional move is allowed, which is also called
the strict rubbling move. In this new move, one pebble each is removed from u
and v adjacent to a vertex w, and one pebble is added on w. The optimal
rubbling number of a graph G is the smallest number m, such that one pebble can
be moved to every given vertex from some pebble distribution of m pebbles by a
sequence of rubbling moves. In this paper, we give short proofs to determine
the rubbling number of cycles and the optimal rubbling number of paths, cycles,
ladders, prisms and Mobius-ladders.
