Tiling

Probability Level 5

A 2×22 \times 2 square is cut into two pieces, A,A, which has area 3, and B,B, which is a square of area 1,1, as shown in the figure below.

A 2×82 \times 8 grid is to be tiled with various pieces having shape AA or B.B. Pieces of shape AA can be rotated by 0,π2,π, or 3π20, \frac{\pi}{2}, \pi, \mbox{ or } \frac{3\pi}{2} before being placed. Subject to the condition that three pieces of shape BB cannot occur on the grid in such a way that they could be replaced by a piece of shape AA, how many different tilings are there of the grid?

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