A polycube wall is a placement of nonoverlapping polycubes in the cubic grid whose union is connected through shared square faces. Free cubes are connected if they meet through a face, edge, or corner. The wall volume is the number of free unit cubes in the largest bounded connected component (Langlois-Rémillard et al. 2026). This is the three-dimensional version of a polyomino fence.
The optimization problem uses each 
-polycube exactly once, counting
mirror images as distinct pieces. Each piece may be translated and rotated, but not
reflected. Thus the complete five-cube family contains 29 pentacubes,
rather than 23 when mirror images are identified (Bouwkamp 1981). Langlois-Rémillard
et al. (2026) proved that its maximum wall volume is 52.
