Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ________.
Correct Answer
Solution & Step-by-step Explanation
To determine the minimum number of guards needed to watch all the walkways in the art gallery, we must consider the coverage provided by placing a guard at each junction (black dot).
The key is to ensure that every walkway is under surveillance without unnecessary overlap. This requires smart placement of guards at strategic junctions.
1. First, identify all the junctions in the diagram. Each junction is a point where two or more walkways meet.
2. The goal is to minimize the number of guard placements while covering all walkways connecting the junctions.
3. In this particular diagram, consider which junctions allow for maximal coverage: - Start by placing a guard at the top junction. This will cover the two walkways connecting to adjacent junctions. - Next, place a guard at one of the bottom junctions. This covers the walkways adjacent to the bottom row. - Finally, place a guard at the remaining junction to ensure all walkways are monitored.
4. Therefore, the minimum number of guards required is 3.
The correct answer is 3, as it ensures all junctions are covered with the least number of guards.
The key is to ensure that every walkway is under surveillance without unnecessary overlap. This requires smart placement of guards at strategic junctions.
1. First, identify all the junctions in the diagram. Each junction is a point where two or more walkways meet.
2. The goal is to minimize the number of guard placements while covering all walkways connecting the junctions.
3. In this particular diagram, consider which junctions allow for maximal coverage: - Start by placing a guard at the top junction. This will cover the two walkways connecting to adjacent junctions. - Next, place a guard at one of the bottom junctions. This covers the walkways adjacent to the bottom row. - Finally, place a guard at the remaining junction to ensure all walkways are monitored.
4. Therefore, the minimum number of guards required is 3.
The correct answer is 3, as it ensures all junctions are covered with the least number of guards.