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 must determine the minimum number of colors required to color the train routes such that no two routes sharing a station have the same color. This is essentially a graph coloring problem, where we need to find the chromatic number of the graph. Firstly, let's identify the individual routes labeled from 1 to 7: - Route 1: Connects stations along the top-most horizontal line. - Route 2: Connects stations along the second horizontal line from the top. - Route 3: Connects stations along the third horizontal line from the top. - Route 4: Connects stations along the bottom-most horizontal line. - Route 5: Connects stations along the vertical line intercepting the middle horizontally. - Route 6: Connects stations along the vertical line intercepting the second horizontal line from the left. - Route 7: Connects stations uniquely as a distinct route. The following step-by-step will help us identify the minimum number of colors required: 1. **Construct a Graph Representation**: - Vertices represent different train routes. - An edge connects two vertices if the corresponding routes share at least one station. 2. **Identify Shared Stations**: - Route 1 shares stations with Route 5. - Route 2 shares stations with Route 5 and Route 6. - Route 3 shares stations with Route 5 and Route 6. - Route 4 shares stations with Route 5. - Route 5 shares stations with Routes 1, 2, 3, and 4. - Route 6 shares stations with Routes 2, 3, and 7. - Route 7 shares stations with Route 6. 3. **Graph Coloring**: - We can determine the minimum number of different colors needed by trying to color each route while ensuring no two adjacent routes share the same color. - Given our routes and their intersections: ``` - Color Route 1 with Color A. - Color Route 2 with Color B. - Color Route 3 with Color C. - Color Route 4 with Color A. - Color Route 5 with Color D (since it's adjacent to Routes 1, 2, 3, and 4, which are colored A, B, C, and A). - Color Route 6 with Color E (since it's adjacent to Routes 2 and 3, which are colored B and C, and Route 5 colored D). - Color Route 7 with Color B (different from Route 6 colored E). ``` 4. **Verification**: - Each route shares its intersections appropriately and does not violate the coloring rule where intersecting routes have the same color. Based on these steps, four colors are sufficient to satisfy the requirements. Therefore, the minimum number of colors Martin can use to ensure no two train routes sharing a station are painted with the same color is 4. The correct answer is: **B2: 4**.
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