Explanation:

- Impossible: Means something that cannot happen or be done. It is often used broadly to describe tasks or events that cannot be achieved.

- Insoluble: Specifically refers to a problem or question that cannot be solved or explained. This is the most appropriate one-word substitute for a difficult, unsolvable problem.

- Correct Answer

- Terrible: This refers to something extremely bad or unpleasant. It doesn't necessarily imply that a problem can't be solved.

- Solvable: Indicates that a problem can be figured out or resolved, which is the opposite of what the group of words describes.