... whenever there is a 0 in the binaryrepresentation of n − k, there is a 1 in the corresponding digit of k.Then the corresponding(n−k)ikiequals 0, and by Lucas’s Theoremn−kkis even.Therefore, ... Ramsey’s Theorem, either three of them(call them X, Y, Z) are all adjacent or all not adjacent. But then in the group {V, X, Y, Z}, none of the players draws exactly once with the other players, a ... from the corner, prove that the remaining part of the floor cannot be tiled with the availabletiles.Solution: Choose a coordinate system so that the corners of the square floor lie along the lattice...