I love this question, because it illustrates in a sense, Einstein's equivalence principle, which is a fancy word for saying that you cannot distinguish between moving in an accelerated reference frame (such as an elevator) and a gravitational field itself.
In order to solve this problem, I first created a free-body diagram of the forces acting on the box. What force is acting on the box from the floor of the elevator? What force is pulling the box towards the center of the earth?
After the forces are identified and a free-body diagram is drawn, now, we need to write down Newton's 2nd Law (F=ma) for each of the three scenarios:
1) in the first scenario, is the box accelerating in the upward direction? What then is the sum of the forces?
2) in the second scenario, is the box accelerating? If so, what is the sum of the forces equal to given Newton's 2nd law?
3) in the third scenario, repeat the same questions in (2)
4) Now in order to solve the problem, we need to find a relation between the frictional force, which is in the horizontal direction, and the forces in the vertical direction. Can you think of how the frictional force is related to another forcein the vertical direction ? (Hint: Think normally)
5) if you have thought about the above four steps, I think you should be able to figure out on your own the answer to the problem. Good luck!
