Problem 1883

Six children are evenly spaced around a circular table. Initially, one has a pile of $n > 0$ sweets in front of them, and the others have nothing. If a child has at least four sweets in front of them, they may perform the following move: eat one sweet and give one sweet to each of their immediate neighbours and to the child directly opposite them. An arrangement is called perfect if there is a sequence of moves which results in each child having the same number of sweets in front of them. For which values of $n$ is the initial arrangement perfect?