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If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envelopes. Similarly, there must be a pigeonhole with at most a envelopes.

Cube point picking is the three-dimensional case of hypercube point picking. The average distance from a point picked at random inside a unit cube to the center is given by ...

The expected value B_n(s) of r^s from a fixed vertex of a unit n-cube to a point picked at random in the interior of the hypercube is given by B_n(s) = ...

Dynamical Systems

Cut a sphere by a plane in such a way that the volumes of the spherical segments have a given ratio.

Construct a 5×5×5 cube from thirteen 1×2×4 blocks, one 2×2×2 block, one 1×2×2, and three 1×1×3 blocks.

The slicing of a three-dimensional object by a plane (or more general slice).

The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) ...

Let two points x and y be picked randomly from a unit n-dimensional hypercube. The expected distance between the points Delta(n), i.e., the mean line segment length, is then ...

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