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The number of "prime" boxes is always finite, where a set of boxes is prime if it cannot be built up from one or more given configurations of boxes.

An arrangement of overlapping circles which cover the entire plane. A lower bound for a covering using equivalent circles is 2pi/sqrt(27) (Williams 1979, p. 51).

Maximize the number of cookies you can cut from a given expanse of dough (Hoffman 1998, p. 173).

Let four lines in a plane represent four roads in general position, and let one traveler T_i be walking along each road at a constant (but not necessarily equal to any other ...

A honeycomb-like packing that forms hexagons.

Given n circles and a perimeter p, the total area of the convex hull is A_(Convex Hull)=2sqrt(3)(n-1)+p(1-1/2sqrt(3))+pi(sqrt(3)-1). Furthermore, the actual area equals this ...

Finding the densest not necessarily periodic sphere packing.

An a×b rectangle can be packed with 1×n strips iff n|a or n|b.

The polyhedron resulting from letting each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces.

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is ...