You start off with $\$1.$ You are faced with $20$ doors, and in front of $10$ each are lying lookouts and honest guards. If you enter and completely walk through a door guarded by an honest hombre, you double your money; nothing happens if you slip past a conman. You cannot retrace your steps.
However, you are in a magical palace, where each door leads to a new, slightly different room. Let $(x, y)$ denote a room with $x$ liars and $y$ honest mens o that you start off at $(10, 10).$ Exiting through a liars door denotes the shift $(x,y) \to (x-1, y)$ while walking through a frank frame denotes the shift $(x,y) \to (x,y-1).$ In other words, you end up with a room with one less door, a person of the kind you passed by having been removed. The doors are also randomly shuffled, so you cannot assume anything about them a priori as you enter/exit.
Once you end up at the final room, you have merely one guarded door left, which ostensibly leads to The Accountant's Office. Fred hates visitors and has been grumpy ever since moving in, so he bills you on the spot for $\$c^q,$ where $q$ is the total number of questions you addressed to the guards along your journey and $c$ is a predetermined constant. Note that each question can only be addressed to one guard, and that each guard knows how the other guards behave.
For this riddle, I have prepared a set of questions ranging in difficulty, so that every puzzler may try their hand at analyzing this scenario. For the easy riddles, note that only one door leads into a room no matter how many exits it has.
Extremely easy: Explain this quote: "Fred hates visitors and has been grumpy ever since moving in..."
Very easy: assuming you may only enter the office by starting and proceeding in the aforementioned manner, and allowing an extra door for Fred to exit/enter, how many doors are connected to his office?
Easy: how many total doors are in the palace, assuming that only doors we may encounter count? Do not count Fred's secret escape hatch!
Medium: "Hi! Billy Mays here with the new mighty savings season. Call now and you can participate in this contest with $c=0.$" Despite the questions not costing anything, you're still impatient, so you want to get through the fortress as fast as possible. What's the minimum number of questions you need in order to nab the maximum prize?
Hard: Frank Lank the Gedankedank wants revenge since I didn't include him in the rest of the puzzle; he plans to really ramp up the costs. To top it off, he adds a $10$ cent tax after you walk through every door, but before you might potentially double. What is the minimum value of $c$ that ensures that you cannot guarantee a positive profit?