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Octants are Cover Decomposable

We prove that octants are cover-decomposable, i.e., any 12-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into two coverings. As a corollary, we obtain that any 12-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into two coverings. We also show that any 12-fold covering of the whole plane with open triangles can be decomposed into two coverings. However, we exhibit an indecomposable 3-fold covering.