Here is the full list of all 16 Puzzle Showdown 4K achievements.
Complete First Puzzle
Complete a Puzzle With Scoring Disabled
Complete two Bonuses each worth 30 Points or more during a single Showdown
Complete Three Puzzles
Compete Against Another Player
Complete 3 Speed Streak Bonuses in a Row
Complete 5 Accuracy Bonuses in a Row
Complete a Line Bonus of 5 or more pieces
Complete an Isthmus Bonus of 15 or more pieces
Complete the Another's Treasure Bonus twice in one Showdown
First Competitive Win
10 Competitive Wins
Come Back from a 1000+ point deficit to achieve sweet victory in a Showdown
Complete a 510 piece Puzzle
Complete a 510 piece Puzzle in under two hours
Complete 50 Puzzles