In each vertex of a regular n-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The result of the shooting is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let P(n) be the number of possible results of the shooting. Prove that for every positive integer k⩾3, P(k) and P(k+1) are relatively prime.(4th Middle European Mathematical Olympiad, Team Competition, Problem 3) combinatorics proposedcombinatorics