|Arcane Turret||Fire||Cut, Stab, Ice, Wind||--|
|King Castle||Elec, Dark||Cut, Stab||Light|
|Mind Dice||Dark||Cut, Stab, Fire, Ice, Elec, Wind||Light|
|Purple Sigil||Fire||Ice, Elec, Wind, Light, Dark||--|
Upon entering this floor, you'll be greeted by two Watchers in the first room, so study their patterns and move past them to the door in the north. One step forward, you'll step on a plate that has "12" written on it, which opens a gate on the west side of the room. Basically, the number written on the plate (there's another you'll come to shortly) is how many steps you can take before the gate will come crashing down. For now, ignore this plate and continue towards the west side of the room.
You won't be able to make it past the gate, which the party will notice when it closes, so you'll have to find another way. On the south end of the western side of the room with another plate with "6" written on it. Step on it, then move three steps west, one north and two more west to pass under the gate, with it crashing down behind you. Just past the gate is another of the chests for The Battle of Wits request, so ignore it and go through the nearby door. You'll view a cutscene here, and will also find the 100% map chest (Repulse Alloy key item) in the same room.
Open the shortcut north of the chest before going through the door, then open the shortcut that leads into the room with the two Watchers. There's another gate blocking your progress in the previous room and the "6" plate will not allow you to make it in time, which means you'll have to use the other plate on the floor. Head there via the shortcut and step on the plate, then follow the next steps carefully, as you only afford a one-step mistake before having to go back to the plate. Go south through the door and along the north edge of the room with the two Watchers, then through the shortcut. Here, go west three tiles and then south through the gate.
Past the gate, you'll find a Power Spot and another shortcut that opens from this side that'll lead to the room with the Watchers. Go through the door and you'll see two conveyor belts straight ahead, making it so you can't go that way, so continue south. There'll be a short scene here where you press a switch that will move a piston out of the way, while also blocking a path on the fourth floor (which you'll see shortly). Don't worry about it right now and open the chest south of the stairs to find a Cosmos Dog-Suit, before ascending the stairs to the fourth floor.