The hallmark of the problem-solving approach is the presence of an implicit hint given in the problem statement. Let's understand why this characteristic is important and how it distinguishes the problem-solving approach from other approaches.

1. Implicit Hint:
The problem-solving approach often presents a problem statement that includes an implicit hint or clue. This hint may not be explicitly stated but is crucial in guiding the problem-solving process. It requires the problem solver to carefully analyze the problem, identify relevant information, and make connections between different concepts. The implicit hint acts as a guide, leading the problem solver towards the correct solution.

2. Multiple Approaches:
Contrary to option (a), the problem-solving approach does not rely on one specific approach to arrive at the right answer. Instead, it encourages individuals to explore multiple approaches and strategies. The problem solver may utilize different problem-solving techniques, such as trial and error, logical reasoning, or creative thinking, to find a solution. This flexibility allows for a more comprehensive and innovative problem-solving process.

3. Broad Topics:
Option (b) states that the problem in the problem-solving approach is based on only one principle or topic. However, the hallmark of the problem-solving approach is its ability to address a wide range of topics and principles. Problems can be related to various disciplines, such as mathematics, science, literature, or social issues. The problem-solving approach is not limited to a single domain but can be applied universally.

4. Originality:
While option (d) suggests that the problem in the problem-solving approach is original, this characteristic is not exclusive to the problem-solving approach. Problems can be original or derived from existing scenarios. What truly distinguishes the problem-solving approach is the presence of an implicit hint, which guides the problem solver towards the solution.

In conclusion, the hallmark of the problem-solving approach is the presence of an implicit hint in the problem statement. This hint acts as a guide, allowing individuals to explore multiple approaches and apply their problem-solving skills to arrive at the right answer. The problem-solving approach is not limited to a single principle or topic and can be applied to a wide range of problems across different domains.