@Lopsy I got stuck with the same observation you made, that the warden can ensure that groups of people have identical histories

and this ruins attempts to assign everyone a unique ID like the randomized strategy

maybe you learn something about the number of twins of a person by having them broadcast their history, then having everyone with that history broadcast, and counting the number of steps it takes to spread to everyone

which is between lg(n/k) and n-k where k is the cluster size and n is the total number of people

yeah, the problem with the simple bounds searching is it only discovers a zone in which the warden has control to accelerate or decelerate signal propagation to distort any answer you're seeking

discovering the upper bound allows you to set a cycle length that he cannot exceed even with maximum acceleration

but at that point you need a different technique to identify a precise number