K light-emitting points are placed on a three-dimensional integer lattice (N*N*N cube). Three spectators observe these lights from three different perspectives. Each of them watches a projection on two of three coordinates: XY; YZ; and ZX. For example, the first observer sees the point (2,7,3) as (2,7); the second sees it as (7,3); and the third as (3,2).

A set of light points is defined as revealing if given any point (u,v) in the N*N square, there is an observer who sees a light at that point in his projection and one can uniquely reveal both the exact source of light and the identity of the observer: each point of light (u,v) is seen by exactly one spectator and he or she sees it exactly once (no light hides behind another light).

For which N is there a revealing set? Show how to construct such a set for all feasible N and prove the nonexistence of them for all other N.