For those going for the final puzzle challenge

As long as you have a save point prior to leaving the care center, you can reload back to that save to input the code for the final puzzle.

You still have to complete the correct playthrough of course once you have the doll. But I read some guides saying you have to start a new game and get to the same point again to unlock the puzzle. However, if you reload any save, the doll will be in your storage.