Questions tagged [packing-and-covering]

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How many unit cubes are needed to 'hide' a unit cube fully in 3D?

Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
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Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

Consider the problem of packing an upwards-pointing unit equilateral triangle "efficiently" by downwards-pointing equilateral triangles, where "efficiently" means that there is ...
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Covering triangles with mutually congruent planar regions - optimally

We continue from this old post: From a given triangle, to cut 2 mutually congruent convex pieces that together 'use' maximum area of the triangle and go from partitioning to covering. Given ...
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On rigid packings of the plane with a constraint

This post continues Thinnest rigid packings of the plane A packing of the plane with copies of any shape is called rigid (or "stable") if every unit is fixed by its neighbors, i.e., no unit ...
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For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s?

Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...
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Which pentagon gives least packing density?

We extend Which convex pentagon gives least packing density? by going from convex pentagons to general ones. Question: Which pentagon gives the least packing density on the Euclidean plane? Note: All ...
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