At what milestones do I get to tackle a Riddler puzzle room?

Periodically, I'll get a notification that I can use the Enigma machine to determine the location of one of Riddler's hostages. This seems tied to the number of the Riddler challenges I've completed in the game so far.

What are the milestones I have to reach for each hostage room?

I'd also really like to bust the Riddler before I hang up my cape and cowl. What percentage of the challenges do I need to conquer before I can wrap this case up?

Catwoman also has some Riddler challenges associated with her - how do these factor in? Must I do them as well? Can I do some instead?

Solution 1:

The first hostage is accessible after getting the enigma machine from the Courthouse.

The rest are tied to number of Riddler challenges finished/trophies collected:

• 80 - Second room
• 160 - Third room
• 240 - Fourth room
• 320 - Fifth room
• 400 - Last room

There are 400 total challenges/trophies available to Batman. If you have the Catwoman DLC, she has 40 available to her. In total, there can be 440 possible challenges/trophies to collect.

You can mix and match any of these to get to the next milestones. I ended up getting all the Catwoman ones early, so I hadn't finished all of Batman's when I got the last hostage. You only have to have 90.1% of them if you have the DLC, otherwise, you have to get 100%.