Question

The diagram shows a river system with 7 segments (P, Q, R, S, T, U, V) splitting the land into 5 zones (Z1–Z5). We need to connect these zones using the least number of bridges. Which option is correct? \begin{center} \includegraphics[width=0.7\textwidth]{04.jpeg} \end{center}

• A. Bridges on P, Q, and T
• B. Bridges on P, Q, S, and T
• C. Bridges on Q, R, T, and V
• D. Bridges on P, Q, S, U, and V

Hint:

Always reduce such puzzles to a spanning tree problem: \(n\) regions require exactly \(n-1\) connections. Any fewer will fail, and any more are redundant.

Answer 1

The Correct Option is C

Step 1: Understand the problem as a graph.
Each zone (Z1–Z5) can be considered as a \emph{node}, and each river segment (P–V) as an \emph{edge}. Building a bridge is equivalent to making that edge traversable. Our task: ensure all 5 nodes are connected using the least number of bridges. Step 2: Apply spanning tree logic.
For \(n=5\) zones, the minimum edges required is \(n-1=4\). Hence, the optimal answer must contain exactly 4 bridges. Any option with fewer than 4 is insufficient; any with more than 4 is redundant. Step 3: Test each option.
- (A) Bridges on P, Q, T: Only 3 bridges. Since at least 4 are needed, this cannot connect all zones.
- (B) Bridges on P, Q, S, T: Connects Z1, Z2, Z3, Z5, but leaves Z4 disconnected.
- (C) Bridges on Q, R, T, V: \[ Q: Z1 \leftrightarrow Z2, \quad R: Z2 \leftrightarrow Z3, \quad T: Z3 \leftrightarrow Z5, \quad V: Z3 \leftrightarrow Z4 \] Together, this chain connects all zones: \(Z1 \to Z2 \to Z3 \to (Z4, Z5)\).
- (D) Bridges on P, Q, S, U, V: Contains 5 bridges. Though it connects zones, it is not minimal. Step 4: Conclude.
Thus, option (C) with bridges on Q, R, T, and V forms a spanning tree and is the correct answer. \[ \boxed{\text{Bridges on Q, R, T, and V}} \]