Yes, these are theorems! I don't think they're too hard, either, though it's been a while since I've thought about this sort of thing, and maybe there's some r

Yes, these are theorems! I don't think they're too hard, either, though it's been a while since I've thought about this sort of thing, and maybe there's some reason they're harder than I expect. (You'll probably want to look up the diamond lemma for the second.) The Wikipedia article is surprisingly bare. There's a lot more known about sandpiles than that. So, this seems to be an appropriate time to infodump on some basics. :) For instance, you can consider sandpiles on finite graphs (a vertex topples when the number of grains on it is at least its degree). Of course, this will never terminate by itself, but if we designate beforehand one of the vertices as a sink, then (assuming the graph is connected) it will. We can then ask which configurations are reachable (by adding sand anywhere) from any other configuration. (For instance, the configuration where all vertices are just about to topple is necessarily reachable from any other configuration, by just adding the appropriate amount of sand to each vertex). This is often phrased in terms of Markov chains -- we consider a random process where you choose randomly which vertex to add sand at, and these are the recurrent states of this process. Then in fact the number of such recurrent states doesn't depend on the chosen sink, and is equal to the number of spanning trees of the graph. Not only that, you can actually make the recurrent states into a (commutative) group -- the group operation is to just add up two states vertex by vertex, and then topple to get the result. Note that the identity in this group is not the all-zero state, because that isn't recurrent! (Well, unless your graph is a star graph and you chose the center to be the sink.) Again, which vertex you pick as the sink doesn't matter; the groups you get this way are all isomorphic. They can also be defined more intrinsically, without picking a sink; look up the "Jacobian group" or "Picard group" of a graph. Sandpiles are also studied under the name "chip-firing games". Another term to search on. There unfortunately seems to be a proliferation of names for the same thing in this field...