In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.

Gensane^[1] traces the origin of the problem to work of J. Schaer in the mid-1960s.^[2] Reviewing Schaer's work, H. S. M. Coxeter writes that he "proves that the arrangements for ${\displaystyle k=2,3,4,8,9}$ are what anyone would have guessed".^[3] The cases ${\displaystyle k=7}$ and ${\displaystyle k=10}$ were resolved in later work of Schaer,^[4] and a packing for ${\displaystyle k=14}$ was proven optimal by Joós.^[5] For larger numbers of spheres, all results so far are conjectural.^[1] In a 1971 paper, Goldberg found many non-optimal packings for other values of ${\displaystyle k}$ and three that may still be optimal.^[6] Gensane improved the rest of Goldberg's packings and found good packings for up to 32 spheres.^[1]

Goldberg also conjectured that for numbers of spheres of the form ${\displaystyle k=\lfloor p^{3}/2\rfloor }$, the optimal packing of spheres in a cube is a form of cubic close-packing. However, omitting as few as two spheres from this number allows a different and tighter packing.^[7]

• Packing problem
• Sphere packing in a cylinder

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