QuestionThe figure shows the plan of the seven train routes of a small town. The circles indicate the stations. Martin wants to paint the lines in such a way that if two lines share a common station, then they are painted with different colors. What is the smallest number of colors that he can use? |
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| A: 3 | B: 4 | C: 5 | D: 6 | E: 7 |
LLM Response: To solve this problem, we'll use a graph coloring approach. Let's go through the steps: 1. First, we need to understand that each train route represents a node in our graph, and shared stations represent edges between nodes. 2. We'll start coloring the routes one by one, ensuring that no two adjacent routes (those sharing a station) have the same color. 3. Let's start with route 1: - Color route 1 with color A 4. Now, route 2: - Route 2 shares stations with route 1, so it needs a different color - Color route 2 with color B 5. Route 3: - Shares stations with routes 1 and 2 - Needs a new color C 6. Route 4: - Shares stations with routes 1, 2, and 3 - Needs a new color D 7. Route 5: - Shares stations with routes 1, 2, 3, and 4 - However, it doesn't share a station directly with route 3 - We can use color C for route 5 8. Route 6: - Shares stations with routes 1, 2, 3, 4, and 5 - We can use color A, as it doesn't directly share a station with route 1 9. Route 7: - Shares stations with routes 1, 2, 5, and 6 - We can use color D, as it doesn't directly share a station with route 4 After this process, we find that we needed 4 colors (A, B, C, and D) to ensure that no two routes sharing a station have the same color. Therefore, the smallest number of colors Martin can use is 4. B2
Final Prediction: Puzzle #143: Solve this question with explanation of the intermediate steps. Your response must end with one of the selected answer options from A1, B2, C3, D4, or E5. The figure shows the plan of the seven train routes of a small town. The circles indicate the stations. Martin wants to paint the lines in such a way that if two lines share a common station, then they are painted with different colors. What is the smallest number of colors that he can use? Options: A1: 3, B2: 4, C3: 5, D4: 6, E5: 7. Predicted Answer = B Correct Answer = A