Here is how many are possible, starting with 1 cell.
A000162 - OEIS - Number of 3-dimensional polyominoes (or polycubes) with n cells.
1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480
A038119 - OEIS - Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).
1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663
Pieces A and B are mirror images of each other.
How many solutions: 240, within rotations and reflections (24 rotations, 24 reflections)
There are some constraints that one can find for solutions.
One of them is that the long direction of the T must be at an edge, meaning that the two ends in that direction are at corners.
Here is a proof. The T can contain either 0 or 2 corners, the L 0, 1, or 2 corners, and the five remaining pieces 0 or 1 corner.
The best case is 9 corners, more than 8 corners, but if the T is not at the edge, then the best case is 7 corners. So the T must be at an edge.
Another constraint is parity. Imagine an infinite 3D grid of Soma cells, and give one of them the value +1. Its neighbors across from each face have a reversed of its sign, or -1. Extend this to every cell. All the cells with +1, or else -1, form a face-centered cubic lattice, much like the sodium and chloride ions in table salt, NaCl.
For our target, a 3-cube, let us set one of the corners to +1. Then:
- Corners (8): +1
- Edge centers (12): -1
- Face centers (6): +1
- Overall center (1): -1
Total: +1
So this cube's parity sum is 1.
The Soma pieces have parity sums
which can be + or - depending on how they are placed relative to the grid.
Since the long direction of the T must lie along an edge, the T makes a contribution +2. That means that the others must make contribution -1, and that is only possible for P -2 and V +1. That means that the P's center must be at an edge center or the overall center, while the V's center must be at a corner or a face center. The others' locations are not constrained by this method.
This might be easier to picture in two dimensions. Consider dominoes. Try to make shapes with them flat on a surface and touching each other. What shapes are possible? A domino has 2 cells, so a domino shape must have an even number of cells. Using this parity argument, a domino has a parity sum of 0, and every constructible domino shape has a party sum of 0.
Thus, this is constructible:
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but this isn't:
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