To determine the charge of ball 5 based on the given information about the electrostatic interactions between the balls, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Attraction and Repulsion Pairs:**
- Attraction pairs: (5,3), (3,4), (4,5)
- Repulsion pairs: (2,3), (4,1)
2. **Understand the Nature of Charges:**
- Attraction occurs between opposite charges (positive and negative).
- Repulsion occurs between like charges (both positive or both negative).
3. **Analyze the Attraction Pairs:**
- For the pair (5,3) to attract, if we assume ball 3 is positively charged, then ball 5 must be negatively charged.
- For the pair (3,4) to attract, if ball 3 is negative, then ball 4 must be positive.
- For the pair (4,5) to attract, if ball 4 is positive, then ball 5 must be negative.
4. **Check for Consistency with Repulsion Pairs:**
- The repulsion pair (2,3) indicates that both balls 2 and 3 must have the same charge. If ball 3 is negative, then ball 2 must also be negative.
- The repulsion pair (4,1) indicates that both balls 4 and 1 must have the same charge. If ball 4 is positive, then ball 1 must also be positive.
5. **Evaluate the Findings:**
- From the analysis, if ball 5 is negative, then it contradicts the attraction with ball 4 (which is positive) because ball 4 cannot be both positive and negative at the same time.
- Therefore, the assumption that ball 5 is negative leads to inconsistencies.
6. **Conclusion:**
- Since all possibilities lead to contradictions when assuming ball 5 is either positive or negative, the only consistent conclusion is that ball 5 must be neutral.
### Final Answer:
Ball 5 must be neutral.
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