Editing Pentomino (section)

=== Box filling puzzle (3D) ===
[[File:Pentomino Cube Solutions.svg|thumb|upright=2|Sample solutions to pentacube puzzles of the stated dimensions, drawn one layer at a time.]]
A '''pentacube''' is a [[polycube]] of five cubes. Of the 29 one-sided pentacubes, exactly twelve pentacubes are flat (1-layer) and correspond to the twelve pentominoes extruded to a depth of one square.

A '''pentacube puzzle''' or 3D '''pentomino puzzle''', amounts to filling a 3-dimensional box with the 12 flat pentacubes, i.e. cover it without overlap and without gaps. Since each pentacube has a volume of 5 unit cubes, the box must have a volume of 60 units. Possible sizes are 2×3×10 (12 solutions), 2×5×6 (264 solutions) and 3×4×5 (3940 solutions).<ref>{{cite book |last1=Barequet |first1=Gill |last2=Tal |first2=Shahar |year=2010 |chapter=Solving General Lattice Puzzles |editor1-first=Der-Tsai |editor1-last=Lee |editor2-first=Danny Z. |editor2-last=Chen |editor3-first=Shi |editor3-last=Ying |title=Frontiers in Algorithmics |series=Lecture Notes in Computer Science |volume=6213 |url=https://archive.org/details/frontiersalgorit00leed |url-access=limited |pages=[https://archive.org/details/frontiersalgorit00leed/page/n132 124]–135 |location=Berlin Heidelberg |publisher=[[Springer Science+Business Media]] |doi=10.1007/978-3-642-14553-7_14|isbn=978-3-642-14552-0 }}</ref>

Alternatively one could also consider combinations of five cubes that are themselves 3D, i.e., those which include more than just the 12 "flat" single-layer thick combinations of cubes. However, in addition to the 12 "flat" [[Polycube|pentacubes]] formed by extruding the pentominoes, there are 6 sets of chiral pairs and 5 additional pieces, forming a total of 29 potential [[Polycube|pentacube]] pieces, which gives 145 cubes in total (=29×5); as 145 can only be packed into a box measuring 29×5×1, it cannot be formed by including the non-flat pentominoes.