Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [packing-problem]

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

Packing is distinct from tiling in that the given shapes may have gaps between them; the goal is often to minimise the relative area of those gaps, or maximise the density. For example, the best packing of equal circles in the plane is $\pi/\sqrt{12}=0.907$, and that of equal spheres $\pi/(3\sqrt2)=0.740$ (the content of Hales's theorem). Packing within a bounded region poses very different challenges due to the boundaries and is an active research topic. is often paired with this tag.

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The Mathematics of Tetris

I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game. Background: The Tetris…
Eric Naslund
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Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian Art Problem. For…
Ed Pegg
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The most effective windshield-wiper setup. (Packing a square with sectors)

I was on the bus on the way to uni this morning and it was raining quite heavily. I was sitting right up near the front where I could see the window wipers doing their thing. It made me think "what is the best configuration of window wipers for…
Harambe
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Putting many disks in the unit square

Consider a square of side equal to $1$. Prove that we can place inside the square a finite number of disjoint discs, with different radii of the form $1/k$ with $k$ a positive integer, such that the area of the remaining region is at most…
Beni Bogosel
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Why can't three unit regular triangles cover a unit square?

A square with edge length $1$ has area $1$. An equilateral triangle with edge length $1$ has area $\sqrt{3}/4 \approx 0.433$. So three such triangles have area $\approx 1.3$, but it requires four such triangles to cover the unit square, e.g.:    …
Joseph O'Rourke
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Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$?

This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The…
Rob Bland
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Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this? Note that the 2014 points do not have to lie on the circle.
Ayesha
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If two non-overlapping squares inside a unit square have side lengths $a$ and $b$, prove that $a+b \le 1$.

According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdős, but I cannot find the solution: Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side lengths $a$ and $b$, respectively. Prove…
Dominik
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Kissing number for equal ellipses on the plane

It's quite easy to determine maximum kissing number for circles (disks) on a plane. Since we have circumference $C=2 \pi r$, we just take the integer part of $2\pi$, which is $6$. However, for ellipses the problem seems to be much more complicated.…
Yuriy S
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Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an approximation to how many fit (given the radius of the…
M.B.M.
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How many circles of a given radius can be packed into a given rectangular box?

I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out. Let's say I have a rectangle of length $l$ and width $w$. Is there a simple equation that can be used to show me how…
Justin L.
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How many balls of radius 1 can be packed into a sphere of radius 10?

How I can calculate the maximum number of balls of radius 1 that can be packed into a sphere of radius 10?
Zakharia Stanley
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Why are these geometric problems so hard?

I was surprised to learn that both for the Moving Sofa Problem and Packing 11 Squares solutions have been proposed, but in either case the optimality of the proposed solution is, as of yet, only conjectured. At first glance, these problems appear…
Řídící
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How many colors are necessary for a rectangle to never cover a color more than once?

If we have an infinite grid, and we color each cell, how many colors do we need so that a $m \times n$ rectangle always covers at most 1 of each color no matter how it is placed? (Rotation of the rectangle is allowed.) It must be at least $mn$, but…
Herman Tulleken
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How densely can the :...: polyomino fill the plane?

This is a follow-up to the question How good can a "near-miss" polyomino packing be?. Let $P$ be the heptomino shown below: I am interested in the packing density of $P$ on the square grid. (Unlike all polyominoes on $6$ or fewer cells, $P$ does…
RavenclawPrefect
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