In a certain country, 40 roads lead out of each city. When all roads are open, it is possible to travel from any city to any other. Each road leads from one city to another; there are no dead end roads.
If one road is closed for repairs, is it still necessarily possible to travel from any city to any other?
Yes. If Road 1 is closed between Cities A and B, then a person in A can take Road 2 to City C and Road 3 to City B.
To determine whether it is still possible to travel from any city to any other when one road is closed, we need to analyze the connectivity of the cities and roads in the country.
In this case, let's consider the city as a node and the road as a directed edge connecting two nodes/cities.
When all roads are open, there are a total of 40 roads leading out of each city. This means that each city has 39 outgoing roads connecting it to other cities.
Now, if we take any two cities, there is a path between them as all roads are open. We can travel from one city to another by following the roads.
When one road is closed for repairs, it means that one of the outgoing roads from a city is no longer accessible. So, each city now has only 38 outgoing roads.
To determine if it is still possible to travel from any city to any other, we need to check if there is an alternate route available between any two cities.
Since each city still has 38 outgoing roads, there are multiple paths available to reach any other city. Therefore, even with one road closed, it is still necessarily possible to travel from any city to any other.
To solve this problem, we relied on the fact that each city has multiple outgoing roads, and removing one road does not disconnect any part of the network. This ensures that there are alternate routes available, ensuring connectivity between cities.