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Báo cáo toán học: "Packing 10 or 11 Unit Squares in a Square"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Packing 10 or 11 Unit Squares in a Square | Packing 10 or 11 Unit Squares in a Square Walter Stromquist Department of Mathematics Bryn Mawr College Bryn Mawr Pennsylvania USA walters@chesco.com Submitted Nov 26 2002 Accepted Feb 26 2003 Published Mar 18 2003 MR Subject Classifications 05B40 52C15 Abstract Let s n be the side of the smallest square into which it is possible pack n unit squares. We show that s 10 3 ỵj 2 K 3.707 and that s 11 2 2 4 K 3.789. We also show that an optimal packing of 11 unit squares with orientations limited to 0 or 45 has side 2 2 9 3.886. These results prove Martin Gardner s conjecture that n 11 is the first case in which an optimal result requires a non-45 packing. Let s n be the side of the smallest square into which it is possible to pack n unit squares. It is known that s 1 1 s 2 s 3 s 4 2 s 5 2 y1 and that s 6 s 7 s 8 s 9 3. For larger n proofs of exact values of s n have been published only for n 14 15 24 35 and when n is a square. The first published proof that s 6 3 is by Kearney and Shiu 3 and the other results are reported in Erich Friedman s dynamic survey 1 . We prove here that s 10 3 ự2 3.707 Theorem 1 and that s 11 2 2 4 K 3.789 Theorem 2 . The 10-square packings in Figure 1 are optimal. The most efficient known packing of 11 squares shown in Figure 2 and due to Walter Trump has side about 3.8772 and includes unit squares tilted at about 40.182 . THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R8 1 s 3.8772 Figure 2 Best known packing of 11 squares tilt 40.182 s 3.886 Figure 3 Optimal 45 packing for n 11 In the case of n 11 we also show that any 45 packing that is one in which the unit squares are tilted only at 0 or 45 with respect to the bounding square must have side at least 2 2 9 tt 3.886 Theorem 3 . This bound is realized by the packing by Hamalainen 2 in Figure 3. Together these results establish the truth of Martin Gardner s conjecture in 7 that n 11 is the first case in which non-45 packings are required. These results were first reported in 4 5 6 . We take