# Questions tagged [packing-problem]

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

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### Fractional covering and packing problems.

I found the fractional covering number of the graph in the figure. And as a result, I came across the result of 5/2. Next I wanted to show its equality with fractional packing number. I found the n-...
• 29
477 views

### You have $n$ rectangles of area $1$ (and variable height). Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ (and variable height). Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...
• 25.8k
190 views

### Do perfect squared rectangles with corners of sizes 10, 12 and 13 exist?

A squared rectangle is a rectangle dissected into squares. squared rectangles are called perfect if the squares in the tiling are all of different sizes and are positive integers. The smallest perfect ...
70 views

### How many spaced squares fit in outer square

Say I have an outer square like this, which is 14m x 14m. Each inner square is separated by a space of 3m, and one square is 1m$^2$. I of course have the image up, but how would I mathematically ...
• 41
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### Find a sphere tangent to four other

I am working in an algorithm to order a bed of close-packed spheres. In the case where I have got four spheres, I understand that the fifth sphere position and radius is determined by the positions ...
1 vote
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### What does the + sign mean at the end of a calculation result?

I discovered yesterday the Square packing in a square problem, and I ended up on this page that shows the proven answers for the first 89 values of $n$. But at the end of some answers, we can see a ...
• 11
1 vote
468 views

### Using algebra to figure out how many boxes fit in an area without over estimating

I'm trying to figure out how many boxes fit in a specific area without over-estimating. The problem is the width of the space isn't a whole number multiple of the width of the boxes, the depth isn't ...
• 189
52 views

### Sum of fractions in the form $\frac{1}{n}$ from $\frac{1}{2^n}$ to $\frac{1}{3*2^{(n-1)} - 1}$ less than $\frac{1}{2}$?

https://mathoverflow.net/a/278290/501460 I've been trying to figure out why this works, and why the tiles don't go past the middle, considering all the squares together have an infinite side length. ...
462 views

### Total placement number of battleship game

The board is 4x4 and there are three types of battleships: 3x1, 2x1, 1x1. One for each type. How many total placements are possible? Notice, the ships cannot overlap and we must use all ships. (there ...
• 151
1 vote
59 views

### Fomin et al., Mathematical Circles Chapter 4- Pigeon Hole Principle Problem 12. Max. no. of kings that can be placed so no two put each other in check

I found this problem in Mathematical Circles in the Pigeon Hole Principle chapter: What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in ...
• 5,316
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### Does a (5,3,4)-code exist?

I am a bit confused on whether a binary (5,3,4)-code exists. As far as I am aware, this code exists if and only if a binary (4,3,3)-code exists according to Theorem 2.7 in Raymond Hill's book "A ...
• 339
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### Marjorie Senechal (2011): "The question of packing tetrahedra is still unsolved" Is it still unsolved?

At about 15:49 in her 2011 talk Prof. Marjorie Senechal - "Quasicrystals Gifts to Mathematics": But Hilbert understood that groups aren't everything and maybe not even the main thing. And ...
• 1,893
1 vote
### $1$-Lipschitz $f:[0,1]^k\to[0,1]^n$ With Maximally Dense Image
Is there anything known about which $1$-Lipschitz functions $$f:(X:=[0,1]^m)\to(Y:=[0,1]^n)$$ for $m$ < $n$ fill the codomain maximally dense, i.e. I want to minimize $\sup_{y\in Y}d(y, f(X))$ ...