Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An algorithmic approach to Rupert's problem   A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a
hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this
hole. There are several works investigating this property for some specific
polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13
Archimedean solids admit Rupert's property. A commonly believed conjecture
states that every convex polyhedron is Rupert. We prove that Rupert's problem
is algorithmically decidable for polyhedra with algebraic coordinates. We also
design a probabilistic algorithm which can efficiently prove that a given
polyhedron is Rupert. Using this algorithm we not only confirm this property
for the known Platonic and Archimedean solids, but also prove it for one of the
remaining Archimedean polyhedra and many others. Moreover, we significantly
improve on almost all known Nieuwland numbers and finally conjecture, based on
statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.

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The categories of this paper are: math-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph