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: 1. **Identify the most connected station:** The station in the center of the figure is connected to all six other stations. 2. **Assign the first color:** This central station needs its own unique color since it shares a route with every other station. Let's call this color 1. 3. **Assign the second color:** Now, look at the outer stations. Each outer station is connected to the central station and two other outer stations. Therefore, we can use a second color (color 2) for alternating outer stations. 4. **Assign the third color:** The remaining outer stations, which are not connected to each other, can share a third color (color 3). 5. **Verify:** We have now colored all the lines using only three colors, and no lines sharing a station have the same color. **Therefore, the smallest number of colors Martin can use is 3.** **Answer: A1**
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 = A Correct Answer = A