If f(m,n) is the (vertex) independence number of the m× n grid graph, then we show that the double limit ηlimm,n→∞f(m,n)^1 mn exists, thereby refining earlier results of Weber [Rostock. Math. Kolloq., 34 (1988), pp. 28--36] and Engel [Fibonacci Quart.,, 28 (1990), pp. 72--78]. We establish upper and lower bounds for η and { prove} that 1.503047782... ≤ η ≤ 1.5035148…. Numerical computations suggest that the true value of η (the "hard square constant") is around 1.5030480824753323... .
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Calkin et al. (1998) studied this question.
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