Abstract
Let T be a tile made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be tilable by finitely many Q -weighted tiles with the same shape as T, and necessary and sufficient conditions for a square to be tilable by finitely many Z -weighted tiles with the same shape as T. The main tool we use is a variant of F.W. Barnes's algebraic theory of brick packing, which converts tiling problems into problems in commutative algebra.
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