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Minkowski's conjecture states that every lattice tiling of R^n by unit hypercubes contains two hypercubes that meet in an (n-1)-dimensional face. Minkowski first considered ...

What is the sofa of greatest area S which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that S>=pi/2+2/pi=2.2074... (1) ...

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of a topological space X is ...

Connecting the centers of touching spheres in a three-dimensional Apollonian gasket by edges given a graph known as the Apollonian network. This process is illustrated above ...

In 1803, Malfatti posed the problem of determining the three circular columns of marble of possibly different sizes which, when carved out of a right triangular prism, would ...

Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist ...

Given a unit disk, find the smallest radius r(n) required for n equal disks to completely cover the unit disk. The first few such values are r(1) = 1 (1) r(2) = 1 (2) r(3) = ...

Keller conjectured that tiling an n-dimensional space with n-dimensional hypercubes of equal size yields an arrangement in which at least two hypercubes have an entire ...

Find the plane lamina of least area A which is capable of covering any plane figure of unit generalized diameter. A unit circle is too small, but a hexagon circumscribed on ...

Given the closed interval [0,x] with x>1, let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number M(x) of cars which can fit (without ...