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In [BNRR], it was shown that tiling of general regions with two rectangles is NP-complete, except for few trivial special cases. In a different direction, Rémila [Rém2] showed that for simply connected regions and two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10 6 rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in thedoi:10.1016/j.jcta.2013.06.008 fatcat:3ihbnp43drgojopcevxojzfzgi