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If f (m, n) is the (vertex) independence number of the m n grid graph, then we show that the double limit ₌, ₍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 and prove that 1. 503047782. . . 1. 5035148. Numerical computations suggest that the true value of (the "hard square constant") is around 1. 5030480824753323. . . .
Calkin et al. (Sun,) studied this question.