I am in a problem solving mood. Termination can be proved for finite graph with sink as: For every grain, compute its distance to the sink along the shortest route. For grains that entered the sink, use 0. Sort the values and put them in a vector. Note that there are finitely many possible vectors for a given number of grains and a given graph. Compare vectors lexicographically. With every topple at least one grain comes closer to the sink. This means that the vector decreases for every topple.