Here is the full list of all 14 Animated Puzzles achievements.
Complete all tiers.
Begin a battle.
Get a diamond medal in Daily challenges.
Complete the "sunset house" puzzle upside down.
Get a medal in Daily challenges.
Complete the cow puzzle (normal) in stripes, by placcing pieces only on every second row. Begins with top left corner.
Complete the "black cat" puzzle within 2 minutes. (Normal difficulty)
Complete the dandelion puzzle (normal) like a checker board, by placing only every second piece of the puzzle. Begins with top left corner.
Complete tier 1.
Complete a tier puzzle in all difficulties.
Complete all tier puzzles in hard.
Complete all tier puzzles in all difficulties.
Complete all tier puzzles with three different puzzle piece shapes.
Win a battle.