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### On packings of squares and rectangles

Derek Jennings
1995 Discrete Mathematics
The numbers k on the edges of the diagram indicate that the edge of the adjacent rectangle is within a distance on 1/k from the edge of the enclosing rectangle. []To analyse the efficiency of this packing  ...  Improvements on Theorem 2, by the methods of this paper, may be difficult, since the squares \$3, \$5 and \$7 would have to be aligned.  ...

### On Packing Unequal Squares

Karen Ball
1996 Journal of combinatorial theory. Series A
A 68 (1994), 465 469], on the smallest rectangle into which all of the squares of side length 1Ân, n=2, 3, 4, ... can be packed.  ...  The question of whether a packing with an arbitrarily small excess area is possible remains unanswered. 1996 Academic Press, Inc.  ...  He packs the first r&1 rectangles, where r 16 and is a multiple of 4, into the unit square and the remaining rectangles into a rectangle of sides (6Â5r)_1.  ...

### Packing unequal rectangles and squares in a fixed size circular container using formulation space search [article]

C.O. López, J.E. Beasley
2018 arXiv   pre-print
Here we pack rectangles so as to maximise some objective (e.g. maximise the number of rectangles packed or maximise the total area of the rectangles packed).  ...  In this paper we formulate the problem of packing unequal rectangles/squares into a fixed size circular container as a mixed-integer nonlinear program.  ...  In that section we give results both for maximising the number of rectangles/squares packed and for maximising the total area of the rectangles/squares packed.  ...

### An Algorithm for Packing Squares

Marc M. Paulhus
1998 Journal of combinatorial theory. Series A
An algorithm is presented that can be used to pack sets of squares (or rectangles) into rectangles.  ...  The algorithm is applied to three open problems and will show how the best known results can be improved by a factor of at least 6_10 6 in the first two problems and 2_10 6 in the third.  ...  Apply Rule 1 and find that square 1Â7 goes on top of square 1Â6 in Fig. 4 and into rectangle S (which is actually a square!).  ...

### On packing unequal rectangles in the unit square

Derek Jennings
1994 Journal of combinatorial theory. Series A
Theory 5 (1968), 126-134] concerning the smallest square into which all of the rectangles of size 1/n × 1/(n + 1), n = 1,2,3 .... can be packed.  ...  This paper improves a previous bound, due to Meir and Moser in [J. Combin.  ...  Meir and Moser showed that they can be packed into a square of side 31/30 [1] . This result is improved by showing that they can be packed into a square of side 133/132.  ...

### Optimal rectangle packing

Richard E. Korf, Michael D. Moffitt, Martha E. Pollack
2008 Annals of Operations Research
In two sets of experiments, we find both the smallest rectangles and squares that can contain the set of squares of size 1 × 1, 2 × 2, . . . , N × N , for N up to 27.  ...  In addition, we solve an open problem dating to 1966, concerning packing the set of consecutive squares up to 24 × 24 in a square of size 70 × 70.  ...  Thanks to Satish Gupta and the IBM corporation for providing the machine our experiments were run on.  ...

### Two packing problems

Vojtech Bálint
1998 Discrete Mathematics
The demonstration of a packing of many such rectangles into a unit square requires either one very great sketch or many smaller figures.  ...  What is the rectangle of smallest area A into which the squares of side 1 2,÷1, n --1,2,3 ..... can be packed? Let us denote H = 15182 and pack the squares \$3,\$5 .....  ...

### Perfectly packing a square by squares of nearly harmonic sidelength [article]

Terence Tao
2022 arXiv   pre-print
This was previously known (if one packs a rectangle instead of a square) for 1/2 < t ≤ 2/3 (in which case one can take n_0=1).  ...  In this paper we show that for any 1/2 < t < 1, and any n_0 that is sufficiently large depending on t, the squares of sidelength n^-t for n ≥ n_0 can be packed perfectly into a square of area ∑_n=n_0^∞  ...  As one measure of partial progress towards these results, Paulhus [11] showed 2 that one could pack rectangles of dimensions 1 n × 1 n+1 for n ≥ 1 into a square of area 1 + 1 10 9 +1 , and squares of  ...

### Rectangle packing with additional restrictions

Jens Maßberg, Jan Schneider
2011 Theoretical Computer Science
We formulate a generalization of the NP-complete rectangle packing problem by parameterizing it in terms of packing density, the ratio of rectangle areas, and the aspect ratio of individual rectangles.  ...  Then we show that almost all restrictions of this problem remain NP-complete and identify some cases where the answer to the decision problem can be found in constant time.  ...  Ulrich Brenner and Christoph Bartoschek for valuable discussions on the topic of rectangle packings as well as the anonymous referee for his helpful comments.  ...

### On packing squares into a rectangle

Stefan Hougardy
2011 Computational geometry
We prove that every set of squares with total area 1 can be packed into a rectangle of area at most 2867/2048 = 1.399 . . . . This improves on the previous best bound of 1.53.  ...  Also, our proof yields a linear time algorithm for finding such a packing.  ...  We try to find a packing of the squares with sides x 1 , . . . , x k−1 and one of the a 1 × a 2 rectangles with (a 1 , a 2 ) ∈ S into a rectangle R of area α.  ...

### Anchored rectangle and square packings

Kevin Balas, Adrian Dumitrescu, Csaba D. Tóth
2017 Discrete Optimization
Given n points, an (1 − ε)-approximation for the anchored square packing of maximum area can be computed in time n O(1/ε) ; and for the anchored rectangle packing of maximum area, in time exp(poly(log(  ...  The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.  ...  In particular, the preliminary approximation ratio 1/6 in Theorem 12 incorporates one of his ideas.  ...

### On packing of squares and cubes

A. Meir, L. Moser
1968 Journal of Combinatorial Theory
It is possible to pack the cubes into the paraUelepiped if aj ~> xl, ] = 1, 2, ...,k and xlk + I~ (at-x0 > V, where V denotes the volume of the cubes.  ...  The main result of the paper is the following: Suppose xl ~> xz > "" are the sides of cubes in the k-dimensional space and al, a2 ,..., a~ are the edges of a rectangular parallelepiped.  ...  On the v-th layer we can pack the squares of As ~ \$ 0 we see that the left-hand side of (2.1) approaches the right-hand side.  ...

### Improved Packings of \$n(n − 1)\$ Unit Squares in a Square

M. Z. Arslanov, S.A. Mustafin, Z.K. Shangitbayev
2021 Electronic Journal of Combinatorics
Let \$s(n)\$ be the side of the smallest square into which we can pack \$n\$ unit squares. The purpose of this paper is to prove that \$s(n^2-n)<n\$ for all \$n\geq 12\$.  ...  Besides, we show that \$s(18^2-17) < 18, s(17^2-16) < 17,\$ and \$s(16^2-15) < 16.\$  ...  The square S 1 has a vertex on the side of the rectangle (5,10), one on a side of S 2 , and one on a side of S 3 . The right vertex of S 1 is on the bottom side of S 4 .  ...

### Near-Optimal Solutions to Two-Dimensional Bin Packing With 90 Degree Rotations

José R. Correa
2004 Electronic Notes in Discrete Mathematics
We consider a multidimensional generalization of the bin packing problem, namely, packing 2−dimensional rectangles into the minimum number of unit squares, where 90 degree rotations are allowed.  ...  These include minimum rectangle packing, two dimensional strip packing, and the z-oriented 3-dimensional packing problem.  ...  The author thanks Claire Kenyon for introducing him to multidimensional packing problems and for asking the question about rotations.  ...

### Resource augmentation in two-dimensional packing with orthogonal rotations

José R. Correa
2006 Operations Research Letters
We consider the problem of packing two-dimensional rectangles into the minimum number of unit squares, when 90 • rotations are allowed.  ...  Additionally, we show near-optimal packing results for a number of related packing problems.  ...  The paper greatly benefited from insightful discussions with Claire and from the useful comments of Michael Wagner and an anonymous referee.  ...
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