The box says "endless combinations", but this is obviously a false claim. My question is how many arrangements are actually possible? The OEIS has 49 sequences related to tetrominoes, but none of them have the relevant data. This is not surprising; the fact that they are held together only by gravity adds an unusual constraint that appears difficult to analyze, and since the pieces exist in three dimensions, the chiral tiles can be flipped over to produce two L- or S-tiles of the same type.
To make the question precise, I'd like to know how many valid arrangements are possible, where valid is defined as:
- The arrangement is stable under gravity.
- Every piece is lit. Note that this requires the presence of the long piece.
- The arrangement uses no more than one set of tiles. This means that there can be exactly one long piece, no more than one square piece, no more than one T-piece, no more than two S-pieces, and no more than two L-pieces. Note that the S- and L-pieces can be flipped over, so we can have (for example) two right-handed Ls, two left-handed Ls, or one of each handedness, and the same goes for the S-pieces.
- The arrangement need not use the full set. For example, the arrangement consisting of only the long piece laying flat on the floor is a valid arrangement.
- To keep the numbers finite, we impose some grid conditions — for example, we can achieve an uncountably infinite number of arrangements by sliding the three upper pieces in this setup horizontally (okay, maybe the combinations are endless...). We therefore require that the blocks be placed so that, whenever one block is above another, the centers of the pieces' constituent cubes be directly over/under each other. This condition rules out, for example, this arrangement because the square block doesn't fit the grid, as well as this one because why are those pieces angled‽