Here is the full list of all 39 Cubicity: Slide puzzle achievements.
Reach Level 1
Reach Level 5
Reach Level 10
Reach Level 15
Reach Level 20
Reach Level 25
Reach Level 30
Reach Level 35
Reach Level 40
Reach Level 45
Reach Level 50
Reach Level 55
Reach Level 60
Reach Level 65
Reach Level 70
Reach Level 75
Reach Level 80
Reach Level 85
Reach Level 90
Reach Level 95
Use 3 ice blocks at a time
Use 5 ice blocks at a time
Use 100 ice blocks
Use 3 bombs at a time
Use 5 bombs at a time
Use 100 bombs
Cancel 3 moves at a time
Cancel 5 moves at a time
Cancel 100 moves
Use teleport 100 times
Complete 10 levels with a rating of at least 2 stars
Complete 50 levels with a rating of at least 2 stars
Complete 95 levels with a rating of at least 2 stars
Replay any level and improve the result
Open a new pack of characters
Open all the characters in the set
Use any elevator 10 times
Fall off the level