# Questions tagged [packing-problem]

Questions on the packing of various (two- or three-dimensional) geometric objects.

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### Is it impossible to fully pave this imaginary bathroom with some conditions?

Bob and Sarah decide to pave their bathroom floor with ceramic tiles. The bathroom measures $1.4$m by $3.7$m. A tile is a square slab with side length $30$cm. Tiles not closest to the walls of the ...
1answer
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### In an $8\times 8$ square, what's the min number of dots to be placed so that there's always a pair with distance apart at most $\sqrt8$?

By the Pigeon Hole Principle (PHP), we know that when we are to place $17$ dots in an $8 \times 8$ square, then there will always be a pair with distance $< \sqrt8$. However, does PHP actually ...
0answers
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### Circle packing in any shape

Is there a heuristic for packing a "maximum" amount of circles of radius at least 1 into any shape? Hex packing seems to be the most efficient, but sometimes there might be a strange shape ...
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### How many spheres can fit inside a cylinder container?

So this is on the behalf of another friend for a school assignment. The assignment is: Find how many soccer balls fit inside a cylindrical building. Obviously, since a soccer ball is a sphere, the ...
0answers
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### Same generator matrix implies same lattice?

Say there exists some lattice $L$ with a generator matrix $A$, As well, there exists some lattice $J$ with a generator matrix $B$, If $A=B$, is it necessarily the case that $L=J$? I know that the ...
2answers
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### Linear Integer Programing: fill the package with products $A, B, C, D$

A linear integer programming problem ask to consider the next restriction: we want to fill as much as possible a package that has a capacity of $1m^3$ and we have to choose between a variety of ...
1answer
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### Is the packing density of an ellipse the same as that of a circle?

It is well-known that the densest packing of circles in the plane is the close hexagonal packing, with a density of $\frac{\pi\sqrt{3}}6\approx0.9069$: By applying an affine transformation, we obtain ...
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### Algorithm for Box Selection / Space Optimization

So, I have an optimization/space management problem. This same question is posted on two SE sites simultaneously, Stack Overflow and Mathematics, since I think it is fitting for both. Let's say I have ...
0answers
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### Are “loose” circles typical in optimal circle packings?

There has been much work done on finding optimal packings of $k$ congruent circular discs in a larger circle; see here or here for a tabulation of many results. Scrolling through the results, one sees ...
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### How to count the number of selected element in each slot within the packed ciphertext

Given an encrypted ciphertext (n slots, packed n elements into a single ciphertext), such as $ct=\{(2,0,1,2),(3,2,1,3),(3,4,0,4),(5,1,4,2)\}$. Formally, $n$ slots can be expressed as $m$ blocks, each ...
1answer
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### How to create successive layers of Hexagonal Closed Packing?

How do I create a HCP using the bottom A layer? In other words, if I take the first layer and make a second layer, how much do I shift it vertically (z direction) and how much do I shift it backwards ...
3answers
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### Minimum number of trips for a truck with weight limit of $200$ to transport boxes with weights 81, 73, 67, 49, 37, 34, 30, and 26

My 9-year-old son had the following math problem to solve: A truck can carry $200$kg or less. We have 8 different boxes with given weights: $81$kg, $73$kg, $67$kg, $49$kg, $37$kg, $34$kg, $30$kg, and ...
1answer
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### Polyhedra which can be perfectly split into self-similar pieces

A cube can be perfectly split into smaller equally sized cubes. Similarly, a triangular prism can be perfectly split into smaller equally sized triangular prisms. Is there a name for or list of the ...
1answer
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### How many generator blocks can I fit inside a $3\times3\times3$ cube?

In my Minecraft world I have a $3\times3\times3$ cube of space which I want to fill with $1\times1\times1$ generator and wire blocks. I can install a single outlet as part of the room's wall; it takes ...
3answers
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### How good can a “near-miss” polyomino packing be?

Given a polyomino $P$ with $n$ cells, we can ask about its maximal packing density $\delta_P$ in the plane (perhaps the limsup if we are concerned about convergence issues, though I don't think this ...
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### Unique question about packing problem

I added the related pages from part 3 of the book: combinatorial geometry by János Pach,Pankaj K.Agarwal (1995) (which is not available on net so I added them as pictures). A. Prove that one can ...
2answers
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### How to calculate new box dimensions given a reduction in volume

I have a rectangular box with given dimensions $l, w, h$ with available volume $v = lwh$. If I can only pack up to 80% of $v$, how would I go about determining the new, smaller dimensions such that ...
0answers
23 views

### Summing squares within an irregular polygon

I am trying to know how many rectangles with a constant base and height (3mx1.8m) can fit inside an irregular polygon. To solve this, I'm taking the irregular polygon and putting small squares (0.6mx0....
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### Tiling a unit square with rectangles of area $\frac 1k \times \frac{1}{k+1}$ but not with those side lengths - any references (Web, book, etc.)?

A unit square can be tiled with rectangles in the following manner (please refer to the accompanying Figure). The first rectangle A is half the square. The next rectangle B is one-third of A; the next ...
1answer
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### Maximum number of balls included in one ball

I guess this question has received an answer since a long time, but I was not able to find it (bad queries on Internet, I suppose): Take a ball $\mathcal{B}$ of radius $r$ in $\mathbb{R}^3$, for ...
1answer
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### When packing disks into a square, is it best to be greedy?

The problem is to pack $n$ non-overlapping disks (not necessarily of the same size) of greatest total area into a unit square. The case $n=1$ is obvious: just place a disk of radius $\frac12$ ...
2answers
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### What is the packing number of the unit cube?

The $\varepsilon$-packing number of the unit cube $[0,1]^d$ with respect to the infinity norm is the biggest number of $\varepsilon$-strictly-separated points, i.e., the biggest cardinality of a set ...
1answer
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### Whats the maximum number of points inside a rectangle such that no two points have a distance less than one

Find the maximum number of points inside a 3 by 4 rectangle (the points CAN also lie on the perimeter) with the constraint that no two points have a distance less than one. someone suggested the ...
1answer
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### Given fixed number of bins with equal capacity and variable number of items how many bins will I need?

I'm slightly embarrassed to ask this question as I feel that the answer might be exceedingly simple but I sat down to think about it and can't seem come up with a formula. I have a fixed number of ...
1answer
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### Bin Packing Problem with fixed size of bins

I'm studying Bin Packing Problem for my thesis and I meet this definition of the decision verson of the problem in the book "Computers and Intractability" by Michael R. Garey and David S. ...
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### optimal swarm plot packing

A swarm plot is a recently popular form of a scatter plot for one-dimensional data. Its basis is a set of $n$ real numbers, and each value is visualized by a marker, usually a circle of given radius. ...
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### Packing L's in Tans and L's in L's

I'm a young researcher and I'm pretty new in this field. I want to work on packing problem "L's in Tans" and "L's in L's" as presented on https://erich-friedman.github.io/packing/...