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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Geometry question about a six-pack of beer

On a hot summer day like today, I like to put a six-pack of beer in my cooler and enjoy some cold ones outdoors. My cooler is in the shape of a cylinder. When I place the six-pack in the cooler against the wall, with three beer cans touching the…
Dan
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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 $0.0001$.…
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.: Q .…
Joseph O'Rourke
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For what values of $n$ can coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ be held rigidly in a circular tray of radius $1$?

For what values of $n$ can circular coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ ( at least one of each , and no other kind of coin) be held rigidly in a circular tray of radius $1$ ? By "held rigidly", I mean that no coin can move, except…
Dan
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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
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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 that…
Dominik
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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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What is the minimum area of a rectangle containing all circles of radius $1/n$?

What is the minimum area of a rectangle containing all (non-overlapping) circles of radius $1/n$ , $n\in\mathbb{N}$ ? The total area of the circles is finite: $\sum\limits_{n=1}^\infty \frac{\pi}{n^2}=\frac{\pi^3}{6}\approx5.168$ . Below I show the…
Dan
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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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Five circles in a rectangle: can the circles move?

Five unit circles are in a rectangle. In the beginning, their centres are the vertices of a regular pentagon, and each circle is tangent to two other circles and one edge of the rectangle. Can the circles move without overlapping? I will post my…
Dan
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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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