Here is the full list of all 16 Picross Fairytale achievements.
Mini-game complete
Tutorial finished
3 levels in raw without hints
Hamlet world complete
Forest world complete
Lake world complete
Mountain world complete
Castle world complete
Earned 100.000 coins
"Solved in less, then 60 seconds"
All Trophies collected
50 hints used
20 hints bought
All keys collected
World completed without hints
All mini-games complete