Here is the full list of all 21 Rooms: The Unsolvable Puzzle achievements.
Make the toymaker's dream come true.
Escape the toymaker's mansion.
Earn 432 puzzle pieces.
Clear all levels in Mansion 4.
Clear all underground levels in Mansion 4.
Earn 324 puzzle pieces.
Clear all underground levels in Mansion 3.
Clear all levels in Mansion3.
Earn 216 puzzle pieces.
Rotate rooms often.
Clear all underground levels in Mansion 2.
Clear all levels in Mansion 2.
Earn 130 puzzle pieces.
Move rooms often.
Clear all underground levels in Mansion 1.
Earn 80 puzzle pieces.
Don't worry about failing.
Destroy as many puppets as you can.
Use the lamp often.
Clear all levels in Mansion 1.
Earn 30 puzzle pieces.