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binary number binary triangle obtained binary triangles inverse binary triangles satisfy column containing CORNELL UNIVERSITY critical point denote the j-th denotes the binary denotes the greatest determined distribution easy to check ending run equality holds exactly exist at least Figure follows from Lemma four possibilities fourth row ft ft greatest integer Harborth hence i-th row implies impossible induction hypothesis integer less intersection j-th digit l+l n left columns form lemma follows listed binary triangles n-i+l otherwise possible binary triangles possible values present paper Proof prove Q.E.D. Lemma r+s+2 r+s+l right columns form right parallelogram row of Tn run of zeros second row Similarly smallest possibility Suppose it holds Suppose the theorem T(IoT T(llOOOl T(OrllOS T(OrlOS T(oT theorem holds theorem is complete third row three consecutive zeros three rows trapezoid marked triangle of order