Filling Boxes
A pentacube is a polycube of five cubes. Twelve of the 29 pentacubes 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 these 1-layer pentacubes, i.e. cover it without overlap and without gaps. Each of the 12 pentacubes consists of 5 unit cubes, and are like 2D pentominoes but with unit thickness. Clearly the box must have a volume of 60 units. Possible sizes are 2×3×10, 2×5×6 and 3×4×5. Following are several solutions.
Alternatively one could also consider combinations of five cubes that are themselves 3D, i.e., are not part of one layer of cubes. However, in addition to the 12 extruded pentominoes, 6 sets of chiral pairs and 5 pieces make total 29 pieces, resulting 145 cubes, which will not make a 3D box.
Read more about this topic: Pentomino
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—For the State of California, U.S. public relief program (1935-1943)