For the ring, once again we are looking at the area between two concentric circles. Therefore the areas are the same for every plane intersecting the solids. Therefore their volumes are the same. Observe that triangles AOB and DOC are similar. L1 and L2 are given, and so L is a constant.
Every square tile covers an a-multiple of black squares.
Every square tile covers an a-multiple of black squares. This statement of how to position the colouring isn't good enough, I'm afraid. Suppose N columns are black with this positioning. The difference between these two numbers of black cells is H, which is not a multiple of a.