A round table is surrounded by $n\geqslant 2$ people, each assigned one of the integers $0, 1,\ldots , n-1$ such that no two people have the same number. In each round, everyone adds their number to their right neighbour’s number, and their new number becomes the remainder of the sum when divided by $n{}.$ We call an initial configuration of integers glorious if everyone’s number remains the same after some finite number of rounds, never changing again. For which integers $n\geqslant 2$ is every initial configuration glorious? For which integers $n\geqslant 2$ is there no glorious initial configuration at all?