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"Don't cut yourself on this Sharply observed portrait." is the riddle the Riddler will give. Basically, you just need to look for the portrait of Warden Sharp that's hanging right by you.
In the Utility Corridor:
"You don't know Jack about Gotham. Tune in to find out." is what he'll say. You'll be in a weird room like a shower, and you'll see a radio in the middle of the room. Just find the name on it and you'll solve this one.
In the Transfer Loop:
"A top hat and tails is the only dress code for this party in the North" will be your next riddle. When you look at the map, you need to find the corridor that goes left and right. Follow this one and about half way through it you will see a half open door, in here you just need to find the Penguin poster that's hanging on the wall.
In the Holding Cells:
"Where would you find my home sweet home?" is your next riddle and you will need the Cryptographic sequencer for this one. You'll find a cell nearby that has "?" all over it at the beginning of this area.
In the Secure Transit Area:
"Dr. Jonathan Crane plans on elevating fear to new depths" is the riddle you'll get here. Just find the elevator that Harley will crash around the beginning of the game. You'll find a vent toward the bottom, and you just follow it to the Scarecrow's hideout.
In the Cell Block Transfer area:
"Hook up with the relatives before you're transferred out of here" is what he'll say to you here. At the end of this level, by where you finish up defeating Scarecrow, there is a hallway on your right. Take it and stop the barrier at the bottom of the set of stairs that is the farthest from you. This will take you into Aaron Cash's office. Now, just zoom in on those photos that are on the desk.
In the Patient Pacification Chamber:
"Even I was shocked when I saw how Maxie Zeus was treated" is your riddle here. You will need the Explosive Gel to do this one. Go to Maxie's cell and go to the wall that is in the corner, opposite of the grate that you used to get away from Zsasz. Blow up this part of the wall with the gel.
In the Intensive Treatment Lobby:
"A puzzle has many sides, but only some are visible" is what he'll say here. Go into detective mode and you'll find a "?". Stand so that it lines up perfectly and don't forget to dot it underneath. Now, to do this, you have to be in the structure in the very center of this room. When you look at the map, you want to make sure that you're in the area with the glass roof on the left side. The "?" will be on the window that faces the south wall.
