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Color the interior squares of R black and white in checkerboard fashion, and check if it passes the color test. If so, pick an arbitrary vertex on the boundary and label it zero. Trace clockwise around the boundary. As you go around, label each vertex with the previous label plus one when you pass a white square, and with the previous label minus one when you pass a black square. Once all the border vertices are labeled, find a vertex v with a minimal label; this will not occur on a corner. Place a domino so v is on the midpoint of its long side. What remains is a smaller region (or regions, if the added domino disconnects the region), so this procedure can be recursively applied to the remainder (really, you don’t need to recompute the border labels, only local adjustments need to be made near the added domino).

]]>Best,

Colin.

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