# Tiling

Discrete Mathematics Level 5

A $$2 \times 2$$ square is cut into two pieces, $$A,$$ which has area 3, and $$B,$$ which is a square of area $$1,$$ as shown in the figure below.

A $$2 \times 8$$ grid is to be tiled with various pieces having shape $$A$$ or $$B.$$ Pieces of shape $$A$$ can be rotated by $$0, \frac{\pi}{2}, \pi, \mbox{ or } \frac{3\pi}{2}$$ before being placed. Subject to the condition that three pieces of shape $$B$$ cannot occur on the grid in such a way that they could be replaced by a piece of shape $$A$$, how many different tilings are there of the grid?

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