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We prove that asymptotically almost surely $N\\ge\\left(1-o\\left(1\\right)\\right)\\,n^{2}/4$, and that $\\mathbb{E}N\\le\\left(1+o\\left(1\\right)\\right)\\,n^{2}/2$ (therefore asymptotically almost surely $N\\le fn^{2}$ for any $f\\to\\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. 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