Puzzle events appear randomly behind blue barriers. Complete the puzzle within these puzzle rooms to add one to the count. As far as I could remember, the following all count towards the achievements:
1. Room with four buttons on the floor
You'd need to step on the buttons in the correct sequence. The numbers are random. If you step on a wrong button, enemies appear. Since there are only four buttons, you should be able to guess the correct sequence with some trial and error.
2. Room with three buttons on the floor
A game of chance. You simply have to step on the one correct button to win this. As far as I know, this is random. There are some theories about if this room appears in certain maps, the answer is fixed, but I haven't actually found an actual, comprehensive list for this. Even the narrator says that this is a game of chance when you enter the room, so I'm inclined to think that this is basically RNG.
3. Room with Pong
You'll control a peg to bounce a ball against another peg. Hit the opponents five times to win this. You've three lives. Don't die.
So, basically just win the above random puzzle events when you encounter them for 20 times, and you'll unlock the achievement.