*Every state A has n incoming edges (self edges included). (3)*
Here's an example: Let A be 0 1 2 1 0 0.

We can reach this state from 3 0 1 0 0 0 applying a move on cup 1, 2 0 1 1 0 0 applying a move on cup 1, 1 0 2 1 0 0 applying a move on cup 1, 0 1 2 1 0 0 applying a move on cup 5 (self edge), 0 1 2 1 0 0 applying a move on cup 6 (self edge)

We can reach this state from 0 1 2 1 0 0 applying a move on cup 1 (self edge), 3 0 1 0 0 0 applying a move on cup 1, 2 0 1 1 0 0 applying a move on cup 1, 1 0 2 1 0 0 applying a move on cup 1, 0 1 2 1 0 0 applying a move on cup 5 (self edge), 0 1 2 1 0 0 applying a move on cup 6 (self edge)

From (2), (3) the graph is Eulerian, from (1) the graph is also connected. It follows that we have an *Eulerian cycle* that goes through all edges and nodes. Which means we can reach any state from any other state.