Polyhex prisms are shapes composed of multiple regular hexagonal prisms. Such a prism is obtained by extruding a regular hexagon. When several of these prisms are joined together, they form polyhex prisms — also referred to as stacked polyhexes.

The HexaPrism puzzle belongs to the family of polyhex prisms. Each of the 23 pieces consists of four regular hexagonal prisms, known as tetrahex prisms.

There are 2 dihex prisms, 5 trihex prisms, and 23 tetrahex prisms.

It is natural to try constructing a shape with four identical layers from the 23 tetrahex prisms, with each layer containing 23 hexagons (see Figure 1).
However, this is impossible for the following reason:

Consider first the total number of hexagons in the odd-numbered layers, that is, in layers 1 and 3. This total would be 46, which is an even number.

Now examine the individual tetrahex prisms. For 16 of the 23 pieces, the number of their hexagons that lie in odd-numbered layers—regardless of their position in the overall shape—is always 0, 2, or 4. The sum of these values is therefore always even (“16 × even = even”).

For the remaining 7 pieces, the number of their hexagons in the odd-numbered layers is either 1 or 3. The sum of these values is thus always odd (“7 × odd = odd”).

Adding both groups yields: even + odd = odd.
Consequently, the total number of hexagons in the odd-numbered layers is always odd—never even.
It is therefore impossible to form a four-layered shape in which all layers are identical.

Allgemein gilt:

In general, the following condition holds:
For any figure assembled from all tetrahex prisms to be solvable, the number of hexagons in both the odd-numbered layers and the even-numbered layers must be odd.

The symmetric shape shown in Figure 2 satisfies this requirement. It contains 49 hexagons in the odd layers and 43 hexagons in the even layers. This shape originates from the website polyforms.eu.

By contrast, the 3D-printing templates for tetrahex puzzles offered on that website consist of four identical layers and therefore contain fewer than all 23 tetrahex prisms.

All solutions were computed using the program BurrTools.

The 3D-print templates consist of 23 Tetrahexprism pieces and a box.