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If I have blocks of size $1\times 1\times 2$ cubes, how many ways there are to stack them isohedrally in 3D space? I now have pretty robust system for solving this kind of problem in 2D, but 3D escapes me.

I presume that it could be solved by simply enumerating various kinds of planar layers composed of $1\times 2$ rectangles and/or $1\times 1$ squares and stacking them on top of each other, but that seems like it would have a lot of false solutions and couldn't be easily generalized for other shapes/geometries.

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  • $\begingroup$ Stack them to make what shape? What size/shape of base? $\endgroup$
    – Jaap Scherphuis
    Dec 7, 2020 at 11:21
  • $\begingroup$ Stack them to fill space. $\endgroup$
    – Marek14
    Dec 7, 2020 at 18:08
  • $\begingroup$ Are you counting "ways" up to isometries of space? E.g., does stacking them all in vertical "columns" along the $z$-axis count as a distinct tiling from doing so along the $x$-axis? $\endgroup$
    – RavenclawPrefect
    Mar 7, 2021 at 7:15
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    $\begingroup$ No. Two arrangements that can be transformed into one another by isometry are considered the same. $\endgroup$
    – Marek14
    Mar 8, 2021 at 8:12

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