Here is the full list of all 22 Rooms: The Unsolvable Puzzle trophies - there are 5 bronze, 9 silver, 7 gold and 1 platinum.
Rotate rooms often.
Clear all levels in Mansion 3.
Earn 216 puzzle pieces.
Clear all levels in Mansion 2.
Earn 130 puzzle pieces.
Move rooms often.
Earn 80 puzzle pieces.
Clear all levels in Mansion 1.
Earn 30 puzzle pieces.
Earn all trophies.
Earn 288 puzzle pieces.
Earn 432 puzzle pieces.
Clear all levels in Mansion 4.
Don't worry about failing.
Use the lamp often.
Destroy as many puppets as you can.
Clear all underground levels in Mansion 1.
Clear all underground levels in Mansion 2.
Clear all underground levels in Mansion 3.
Clear all underground levels in Mansion 4.
Make the toymaker's dream come true.
Escape the toymaker's mansion.