I’ve determined computationally that for an n×n grid the number of tilings is 2n/2 if n is even and 0 if n is odd, but I don’t have a proof of this fact yet. I’ll be looking for one—the answer looks quite nice so there ought to be a neat argument for seeing why it’s like this.
More precisely, it looks like the possible tilings correspond to the ways of tiling the first row of the grid with 2x1 white or black dominoes. It’s easy to see that the first row has to be of this form and that any such first row can’t correspond to more than one tiling of the grid, but showing that there is always a valid extension of the coloring to the whole grid seems to be the hard part.
About the second problem:
I’ve determined computationally that for an n×n grid the number of tilings is 2n/2 if n is even and 0 if n is odd, but I don’t have a proof of this fact yet. I’ll be looking for one—the answer looks quite nice so there ought to be a neat argument for seeing why it’s like this.
More precisely, it looks like the possible tilings correspond to the ways of tiling the first row of the grid with 2x1 white or black dominoes. It’s easy to see that the first row has to be of this form and that any such first row can’t correspond to more than one tiling of the grid, but showing that there is always a valid extension of the coloring to the whole grid seems to be the hard part.