Here is the full list of all 47 Knowledge is Power trophies - there are 35 bronze, 9 silver, 2 gold and 1 platinum.
Obtain all trophies
Complete a Two-player game
Complete a Three-player game
Complete a Four-player game
Complete a Five-player game
Complete a Six-player game
Play five games
Play fifteen games
Play thirty games
Finish a 3-6 player game with all players within four steps of each other
Use all of the game's characters
Use a Power Pick twenty times
Use a Power Pick on a door choice, play Points Party and then be the only player to get the question wrong
Answer first but be the only player to get a multiple-choice question wrong in a 3-6 player game
Play five Power Plays on one character on the same question
Come from last in the Finale to win a 3-6 player game
Play every Power Play with the same player in a single game
Get 100% correct answers on a Linking question
Get 100% correct answers on a Sorting question
Answer first and correctly after being hit by a Gloop, Freeze and Bomble in the same question
Apply three or more Freeze Power Plays to one player on the same question
Win a game with each character
Answer all multiple-choice questions in a single game correctly with all players
Apply three or more Gloop Power Plays to one player on the same question
Place a Bet on a player that answers correctly first, but get the wrong answer yourself
Power Pick a door when everyone else chose a different door in a 3-6 player game
Everyone Freezes everyone else on a question in a 3-6 player game
Everyone Power Plays everyone else on a question in a 3-6 player game
When Glooped, remove ALL the Gloop from your answer screen
When Bombled, tap two Bombles on your answer screen
When Frozen, remove ALL the Ice from your answer screen
Choose an incorrect answer to a multiple-choice question with only one player in a 3-6 player game when all other players get it right
Answer first and correctly after being hit by two Nibblers in the same question
Win three games as the Magician character
Win three games as the Mountaineer character
Win three games as the Fairy character
Win three games as the Duchess character
Win three games as the Cowgirl character
Win three games as the Hotdog Man character
Win three games as the Scientist character
Win three games as the Spaceman character
Choose the same incorrect answer to a multiple-choice question with all players in a 3-6 player game
Freeze six opponents as the Cowgirl character
Choose all four doors, with at least one character on each door, in the door choice screen
Get the correct answer to a Finale question after trying all the wrong ones
Play a Power Pick with every player in a single game
Be the only person to miss out on a Points Party