Optimal Packings of 13 and 46 Unit Squares in a Square
نویسندگان
چکیده
منابع مشابه
Optimal Packings of 13 and 46 Unit Squares in a Square
Let s(n) be the side length of the smallest square into which n non-overlapping unit squares can be packed. We show that s(m2 − 3) = m for m = 4, 7, implying that the most efficient packings of 13 and 46 squares are the trivial ones. The study of packing unit squares into a square goes back to Erdős and Graham [2], who showed that large numbers of unit squares can be packed in a way that is sur...
متن کاملMore Optimal Packings of Equal Circles in a Square
The problem of nding the maximum radius of n non-overlapping equal circles in a unit square is considered. A computer-aided method for proving global optimality of such packings is presented. This method is based on recent results by De Groot, Monagan, Peik-ert, and WWrtz. As an example, it is shown how the method can be used to get an optimality proof for the case n = 7, which has not earlier ...
متن کاملOptimal Packings of Two Ellipses in a Square
For each real number E in ]0, 1], we describe the densest packing PE of two non-overlapping congruent ellipses of aspect ratio E in a square. We find three different patterns as E belongs to ]0, 1/2], [1/2, E0] where E0 = √ (6 √ 3− 3)/11, and [E0, 1]. The technique of unavoidable sets – used by Friedman for proving the optimality of square packings – allows to prove the optimality of each packi...
متن کاملEfficient Packing of Unit Squares in a Square
Let s(N) denote the edge length of the smallest square in which one can pack N unit squares. A duality method is introduced to prove that s(6) = s(7) = 3. Let nr be the smallest integer n such that s(n + 1) ≤ n + 1/r. We use an explicit construction to show that nr ≤ 27r/2+O(r), and also that n2 ≤ 43.
متن کاملPacking 10 or 11 Unit Squares in a Square
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 + √ 1 2 ≈ 3.707 and that s(11) ≥ 2 + 2 √ 4 5 ≈ 3.789. We also show that an optimal packing of 11 unit squares with orientations limited to 0◦ or 45◦ has side 2+2 √ 8 9 ≈ 3.886. These results prove Martin Gardner’s conjecture that n = 11 is the first case in which an optimal result r...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2010
ISSN: 1077-8926
DOI: 10.37236/398