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: The image shows a network of seven train routes in a small town, with each route represented by a line connecting the stations. The stations are represented by circles, and the lines are drawn in black. The routes intersect at various points, forming a complex network. To determine the minimum number of colors needed to paint the lines, we can use the graph theory concept of chromatic number. The chromatic number of a graph is the smallest number of colors needed to color the vertices of the graph in such a way that no two adjacent vertices share the same color. In this case, we can treat each train route as a vertex in a graph, and two routes are adjacent if they share a common station. The chromatic number of this graph is 3. This means that the smallest number of colors needed to paint the lines is 3. Therefore, the correct answer is A1: 3.
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