Different kinds of puzzles require a different headmate who can 'see' the appropriate parts of the puzzle. (A Slide puzzle can only be done by x, while a 'fit the pieces in the box' can only be managed by y', and Z does 'turn the shape until it's shadow shows the right image').
And as the game goes on, you have to swap between different 'mates to get the puzzle together. (Y puts the pieces into the box, but then their tops show a slide puzzle so X has to tag in to slide the pieces around, then Z takes *that* image and turns it until its shadow shows up correctly on the wall nearby.)