Geodesic
On a Two-Dimensional Search Problem
Serdica Mathematical Journal, Tome 21 (1995) no. 3, pp. 219-230.

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In this article we explore the so-called two-dimensional tree− search problem. We prove that for integers m of the form m = (2^(st) − 1)/(2^s − 1) the rectangles A(m, n) are all tight, no matter what n is. On the other hand, we prove that there exist infinitely many integers m for which there is an infinite number of n’s such that A(m, n) is loose. Furthermore, we determine the smallest loose rectangle as well as the smallest loose square (A(181, 181)). It is still undecided whether there exist infinitely many loose squares.
Keywords: Two-Dimensional Search Problem 
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Kolev, Emil; Landgev, Ivan. On a Two-Dimensional Search Problem. Serdica Mathematical Journal, Tome 21 (1995) no. 3, pp. 219-230. http://geodesic.mathdoc.fr/item/SMJ2_1995_21_3_a3/