Thanks!

Thoughts on representing the state of a partially completed puzzle:

There are 8 pieces, each of which has a position on the board, a rotation, and a chirality. If the piece is not on the board yet, those are null.

There are 12+31 = 43 positions.

Each piece has a focal cell, chosen arbitrarily. Like the orange piece's focal cell is the topmost one in the image below, on the position marked "3". Now we try:

for each position pos on the board:

for each piece pie not yet on the board:

for each rotation rot of pie:

put the focal cell of pie at pos;

count it as a valid successor state if it can be placed there with no overlaps or overhang

So that would be a branching factor of up to 43×8×4×2 = 2752 but most of those are pruned for hanging off the board or overlapping another piece or being a duplicate (e.g., the green piece has just 2 distinct rotations and 1 chirality).