Interesting trivia question, if you were a gym leader, what would your type, badge name, and puzzle be?
Type: Dark (I still don't think there has been a dark gym)
Badge Name: Affluence Badge
Puzzle: The gym would take the appearance of a mansion. In the gym, there is a huge crowd of people in the way of the leader. The camera is changed to be a behind the back view of your trainer to simulate actually being in the middle of a crowded place. The only trainers that battle based on sight here are gamblers/PIs, so one can avoid all trainer battles if they recognize them. You can shove people out of the way, but doing so will trigger a battle that would have been avoided otherwise. There are also waitresses that move through the crowd, and can optionally be battled. If you battle them, you can purchase Fresh Water, Soda Pop, or Lemonade from them after for half of the usual price. Upon beating the leader, the crowd will bend to allow a straight path out of the mansion. The crowd constantly moves, so there is no one way to travel through the gym, and on multiple playthroughs, certain routes may not work at all, and trainers may be in different spots.
As for why the badge is called the Affluence Badge in a dark type gym? It's a secret. You will never know how the leader obtained his vast wealth, but it's heavily implied through speaking to certain NPCs within the crowd that it was probably illegally obtained.
I've always liked the idea of a dark type billionaire trainer who probably obtained their vast fortune illegally. Too bad it will likely never happen.