The diagram below shows a river system consisting of 7 segments, marked P, Q, R, S, T, U, and V. It splits the land into 5 zones, marked Z1, Z2, Z3, Z4, and Z5. We need to connect these zones using the least number of bridges. Out of the following options, which one is correct?

The correct option is Bridges on Q, R, T, and V.

Step-by-Step Explanation:

1. Understanding the Problem as a Graph:
We can model this geographical problem using graph theory. Let the land zones (Z1 to Z5) represent the nodes (vertices) of a graph, and the potential bridge locations crossing the river segments (P, Q, R, S, T, U, V) represent the edges connecting these nodes.

Based on the network connections in the diagram:
• There are 5 zones: Z1, Z2, Z3, Z4, and Z5.
• There are 7 potential bridge locations: P, Q, R, S, T, U, and V.

2. Finding the Minimum Connections Needed:
To connect any group of n isolated nodes so that every node is reachable from every other node without creating redundant loops (forming a spanning tree), the minimum number of connections (edges) required is:

$n-1$

$n=5$

$5-1=4$

3. Evaluating the Connections:
Let us examine the connections provided by the bridges in the correct option (Q, R, T, and V):
• Bridge Q connects zone Z1 and zone Z3.
• Bridge R connects zone Z2 and zone Z3.
• Bridge T connects zone Z3 and zone Z5.
• Bridge V connects zone Z3 and zone Z4.

By building bridges at these four locations, Z3 acts as a central hub. From Z3, you can directly access:
• Z1 (via Q)
• Z2 (via R)
• Z5 (via T)
• Z4 (via V)

Thus, all five zones Z1, Z2, Z3, Z4, and Z5 are fully connected, allowing us to visit every zone by crossing exactly these 4 bridges.