Results 11 to 20 of about 6,788,504 (325)

Packing Unit Squares in a Rectangle [PDF]

The Electronic Journal of Combinatorics, 2005
For a positive integer $N$, let $s(N)$ be the side length of the minimum square into which $N$ unit squares can be packed. This paper shows that, for given real numbers $a,b\geq 2$, no more than $ab -(a+1-\lceil a\rceil) -(b+1-\lceil b\rceil)$ unit ...
H. Nagamochi
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Packing Unit Squares in Squares: A Survey and New Results

The Electronic Journal of Combinatorics, 2009
Let $s(n)$ be the side of the smallest square into which we can pack n unit squares. We present a history of this problem, and give the best known upper and lower bounds for $s(n)$ for $n\le100$, including the best known packings. We also give relatively
Erich Friedman
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Translative covering by unit squares

Demonstratio Mathematica, 2013
Some results concerning translative coverings of squares and triangles by two, three and four unit squares are presented.
J. Januszewski
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Optimal Packings of 13 and 46 Unit Squares in a Square [PDF]

The Electronic Journal of Combinatorics, 2010
Let $s(n)$ be the side length of the smallest square into which $n$ non-overlapping unit squares can be packed. We show that $s(m^2-3)=m$ for $m=4,7$, implying that the most efficient packings of 13 and 46 squares are the trivial ones.
W. Bentz
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Efficient Packing of Unit Squares in a Square [PDF]

The Electronic Journal of Combinatorics, 2002
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 $n_r$ be the smallest integer $n$ such that $s(n^2+1)\le n+{1/r}$. We use an explicit construction to show that $n_r\le 27r^3/2+O(r^2)$, and also that $n_2\le43$.
M. J. Kearney, P. Shiu
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Inefficiency in Packing Squares with Unit Squares

Journal of Combinatorial Theory, Series A, 1978
AbstractIt is shown that, in packing a square of side n + 12 with unit squares, the wasted space always has area ⪢ n12. This answers a question of Erdös and Graham.
K. F. Roth, R. Vaughan
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Global unit squares and local unit squares

Journal of Number Theory, 2008
Let \(K\) be any Galois extension of \(\mathbb Q\), and \(U_K\) be the unit group of \(K\). For any place \(v\) of \(K\), let \(U_v\) be the group of local units of \(K_v\). Here the authors study the following problem: does there exist an odd prime \(p\) such that the map \(U_K/U_K^2\to \prod_{v\mid p} U_v/U_v^2\) is injective.
Yan Li, Xianke Zhang
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Packing 10 or 11 Unit Squares in a Square [PDF]

The Electronic Journal of Combinatorics, 2003
Let $s(n)$ be the side of the smallest square into which it is possible pack $n$ unit squares. We show that $s(10)=3+\sqrt{1\over 2}\approx3.707$ and that $s(11)\geq2+2\sqrt{4\over 5}\approx3.789$. We also show that an optimal packing of $11$ unit squares with orientations limited to $0$ degrees or $45$ degrees has side $2+2\sqrt{8\over 9}\approx3 ...
W. Stromquist
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Unit squares intersecting all secants of a square [PDF]

Discrete & Computational Geometry, 1994
Let S be a square of side length s > 0. We construct, for any sufficiently large s, a set of less than 1.994 s closed unit squares whose sides are parallel to those of S such that any straight line intersecting S intersects at least one square of S. It disproves L.
P. Valtr
semanticscholar   +4 more sources

Totally positive units and squares [PDF]

Proceedings of the American Mathematical Society, 1983
Let K K be a finite cyclic extension of the rational number field Q Q , with Galois group G ( K / Q ) G(K/Q) of order p a {p^a} for an odd prime p p . Armitage and Fröhlich
Hughes, I., Mollin, R.
openaire   +2 more sources