Junior Mathematical Olympiad 2001 Q15

[JMO 2001 B6 edited] This question is about ways of placing square tiles on a square grid, all the squares being the same size. Each tile is divided by a diagonal into two regions, one black and one white. Such a tile can be placed on the grid in one of four different positions as shown below.

When two tiles meet along an edge (side by side or one below the other) the two regions which touch must be of different types (i.e. one black and one white). $(i)\quad$A 2 x 2 grid of four squares is to be covered by four tiles. $\quad(a)\quad$If the top-left square is covered by a tile in position A, how many different ways can the other three squares be covered? $\quad(b)\quad$In how many different ways can a 2 x 2 grid be covered by four tiles? $(ii)\quad$In how many different ways can a 3 x 3 grid be covered by nine tiles? $(iii)\quad$Explaining your reasoning, find a formula for the number of different ways in which a square grid measuring n x n can be covered by n 2 tiles.