Cube Packing

Cube packing asks for the smallest cube that can contain unit cubes with arbitrary orientations and pairwise disjoint interiors. Let denote its side length. Comparing volumes and arranging the unit cubes in a cubic lattice gives

where is the ceiling function. In particular, for every positive integer .

For , the trivial construction starts with eight unit cubes in a cubic lattice arrangement and removes of them. The containing cube has side length 2, which is the smallest known for every in this range (Friedman 2026).

The nontrivial best known packings for as of October 2026 are illustrated above and listed below, together with their discoverers and discovery dates (Friedman 2026). The illustrations are reproduced from Friedman's GitHub repository. The panels labeled 9-10 and 29-33 show the configurations with 10 and 33 cubes, respectively, from which cubes may be removed to obtain the other orders. The decimal side lengths are approximate reported values, and optimality is not asserted. For the remaining positive values , the best known construction is the cubic lattice arrangement with side length (Friedman 2026).

	exact side length	approximate side length	discoverer	date
9			E. Friedman	1998
10			E. Friedman	1998
11			S. Sønderland	Oct. 2026
12			Y. Nakajima	Oct. 2026
13			R. Hayes	Oct. 2026
14			E. Friedman	1998
28			H. Nakano	Oct. 2026
29-33			E. Friedman	1998

The packing of 10 unit cubes illustrated above consists of two horizontal layers, each with four corner cubes and a central cube rotated through about a vertical axis. The left panel shows the square packing in one layer. Removing any cube gives a packing of nine unit cubes in the same containing cube.