Suppose we have a board of 121 squares ($11\times11$) but with a black square in the tile in the middle (6,6). What is the maximum number of $3\times4$ tiles we can tile the checkerboard with?? (without putting anything on the black one). We can rotate the $3\times4$ pieces.
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Consider the rows 3, 6 and 9.
Find such a construction with 8 tiles, which you should already have done.