Here is the full list of all 12 Slide Stories: Bucky's Quest achievements.
Finish the first puzzle in any difficulty
Finish the second puzzle in any difficulty
Finish the third puzzle in any difficulty
Finish the fourth puzzle in any difficulty
Finish the fifth puzzle in any difficulty
Finish the sixth puzzle in any difficulty
Finish all puzzles in relaxing mode
Finish all puzzles in easy mode
Finish all puzzles in fun mode
Finish all puzzles in challenging mode
Finish all puzzles in pro mode