Unfortunately it's impossible to construct a prism using all pieces, because the number of layers is even or the area of the cross-section is even, contrary to the conditions for constructions..

For some cross-sections I tested, whether a prism of given height is possible to construct with the stacked tetrahexes. If a solution was found, it's indicated by a +. If a solution is impossible, the prism got a -. The prisms with a ? seem to be impossible, too, but I haven't got a proof.

## Triangular Cross-section | ||
---|---|---|

Shape of Cross-section | Area of Cross-section | Hight / Number of Hexagons / Solvable |

T2 | 3 | 4/12/+, 8/24/+ , 12/36/- |

T3 | 6 | 2/12/+, 4/24/+, 6/36/+, 8/48/+, 10/60/+, 12/72/+ , 14/84/- |

T4 | 10 | 2/20/+, 4/40+, 6/60+, 8/80/+ |

T5 | 15 | 4/60/+ |

T6 | 21 | 4/84/+ |

T7 | 28 | 1/28/-, 2/54/+, 3/72/+ |

T8 | 36 | 1/36/-, 2/72/+ |

## Hexagonal Cross-section | ||

Shape of Cross-section | Area of Cross-section | Hight / Number of Hexagons / Solvable |

H2 | 7 | 4/28+, 8/56/+ , 12/84/- |

H3 | 19 | 4/76/+ |

## Rhombic Cross-section | ||

Shape of Cross-section | Area of Cross-section | Hight / Number of Hexagons / Solvable |

RH2 | 4 | 1/4/+, 2/8/+, 3/12/+, 4/16/+, 5/20/+, 6/24/+, 7/28/+, 8/32/+, 9/36/+, 10/40/+ , 11/44/?, 15/60/- |

RH3 | 9 | 4/36/+, 8/72/+ |

RH4 | 16 | 1/16/+, 2/32/+, 3/48/+, 4/80/+ |

RH6 | 36 | 1/36/-, 2/72/+ |

## Rectangular Cross-section | ||

Shape of Cross-section | Area of Cross-section | Hight / Number of Hexagons / Solvable |

R 3 4 3 | 10 | 2/20/+, 4/40/+, 6/60/+, 8/80/+ |

R 7 8 7 | 22 | 2/44/+, 4/88/+ |

R 9 10 9 | 28 | 1/28/+, 2/56/+, 3/84/+ |

Z 11 12 11 | 34 | 1/34/-, 2/68/+ |

R 3 2 3 | 8 | 1/8/-, 2/16/+, 3/24/+,...,10/80/+ , 11/88/- |

R 4 3 4 | 11 | 4/44/+, 8/88+ |

R 11 10 11 | 32 | 1/32/-, 2/64/+ |

R 13 12 13 | 38 | 2/76/+ |

R 2 3 2 3 2 | 12 | 1/12/+, 2/24/+,...,7/84/+ |

R 3 4 3 4 3 | 17 | 4/68/+ |

R 4 5 4 5 4 | 22 | 2/44/+, 4/88/+ |

R 2 1 2 1 2 | 8 | 1/8/-, 2/16/+, 3/16/+,...8/64/+ , 9/72/?, 10/80/- |

r 6 5 6 5 6 | 28 | 1/28/+, 2/56/+, 3/84/+ |

R 8 7 8 7 8 | 38 | 1/38/-, 2/76/+ |

The maximum height of the prisms in the table is 12. If you allow an asymmetric cross-section even a height of 14 can be achieved.

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