If you put capacitive material in a unique pattern on the footprint of each piece, and the rest of the piece material was conductive enough to carry your body's charge to register a touch, the shape of that touch could be unique per-piece.
There's no mention of syncing pieces, charging pieces, keeping pieces in view of a wide-angle camera, anything like that, so that's my bet. (This would also mean moving a piece using a non-conductive material would be a way to cheat by having it not get registered!)
I just shared this on LI this morning, linking back to a video showing showing related touchscreen explorations I did for a colleague in early 2013, sensing different coins by their radii as you touch them: https://www.linkedin.com/posts/vmiliano_a-vertical-triptych-...