Spoiler free :) To get the Perfect Knight - Day 2 achievement, you'll need to have done the following:
*Beat the main game on either NORMAL or HARD. You cannot unlock New Game Plus on Easy so I don't think you can even get this achievement. Beating the main game shouldn't be a problem, even on Hard. you'll have beaten the story when you get the Exit Stage Right achievement.
*Now that we have beaten the main story, we have unlocked a mode called New Game Plus (NG+) this is a game mode where you'll restart the story and in which you'll need to beat it again. still, this should not bring too many pain-in-the-ass moments. once you've beaten NG+ you will get the Twice Nightly achievement.
*Side missions. There's a lot of confusion about these. That has a good reason thus I will split it up in three parts.
The straight forward ones. These are the regular side-missions, which include AR-training, the detective missions, etc.. You will get an achievement for everyone you beat and you WILL need to beat every single side mission on NG+ as well
The Riddler's collectibles, upgrades, art concepts and character trophies. You will need to collect all Riddler trophies in EITHER regular game or NG+. but these DO carry over, so that's some good news. every riddle or trophie only needs to be collected once. After having collected enough trophies, one of the Riddler's hostages' locations will be shown to complete. you DO need to save every hostage one both regular game and NG+. during your search for these collectibles you'll get enough XP to get all your upgrades, every story, art concept, character trophies, you name it. you will get the Genius achievement after saving all hostages.
Credit goes to NinjaDave04 for confirming this. You will also have to complete all the physical challenges (found on the same menu as The Riddler's trophies. these were not too hard for me even after beating the main story. If you have a question about a physical challenge, just leave a comment.
the most irritating things in this game I found to be were the random acts of violence side missions. these are random events, where a political prisoner is attacked by a thug and you will need to fullfill your task as the Dark Knight and save him. the annoying thing is that you, obviously, have to complete all 16 events twice (in regular game and NG+) you will find most of them during your hunt for riddles, but i ran out of things to do in NG+ and I was left with only these things to do. a proper and not-so-boring way to complete them is to fly around the city, complete a event, save and go back to the main menu. then load up your game again and you'll find it much easier to find all these events. repeat until you're done. i completed 2 events in 30 minutes without this technique, and 6 in 12 minutes with.
*Ranked Challenges. these are the regular Predator and Combat challenges you'll be familiar with if you played Batman: Arkham Asylum. they made the combat challenges a little easier, as it's not as hard to get the high combos, which are the key to getting a lot of points. but the Predator challenges are a lot harder this time. Luckily you are on Trueachievements right now and someone's posted some excellent videos of how to beat both Predator and Combat challenges. you will find them in the link of the Gold Revenge achievement
*Campaign Challenges. these are new in the series. the name "campaign" is false. campaign is just three ranked challenges with choseable modifiers, that will either make your life easier or harder. i once again refer to the amazing videos that you will find in the link of the Gold Revenge achievement. you get 3 retries in one campaign challenge, so i would reccommend getting used to the challenges first in ranked, before going for the Campaign Gold achievement.
If you did everything above, you should get the achievement. If you don't know what you missed, you can always go to the tracker list in the main menu that tells you exactly what you still need to do.
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