math riddle: King and His Prisoners (no content, long)

[SumZero]

Take the answer for k=0 (which is the typical problem). Now have each prisoner do what they have do once in the k=0 solution k more times. And have the "special" prisoner change his counting scheme to account for the (k+1)*n chalices. I think that works. There might need to be a tweak since the king can turn the chalice each direction or can not do anything, but I think it is just a wrinkle of what I've (intentionally vaguely) described.