Thanks for the comment.
No puzzles are impossible -- S symbols don't invert the pentacircle, and the practice tablet for the S symbols should include every other symbol (eventually) if you keep retrying it long enough since those are also randomly generated.
(Also red and blank are treated the same for all symbols... the ability to turn spaces red was just for player sanity to mark off things that couldn't be blue.)
I don't *think* I made any mistakes in the above puzzle spoilers? It's possible you're confusing the Situation Room and the Engine Room... The Engine Room is the one next to the bridge elevator. (The solutions above are arranged in order you'll encounter new symbols in the door puzzles.)