A thin wire is used to construct all the edges of a cube of 1 m side by bending,
cutting and soldering the wire. If the wire is 12 m long, what is the minimum number
of cuts required to construct the wire frame to form the cube?
Options:
(A) 3
(B) 4
(C) 6
(D) 12
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2025 Q.4)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Graph Theory (Eulerian Paths on Cube Edges)
- A cube has **8 vertices** and **12 edges**.
- Every vertex has a degree of **3** (3 edges meet at each corner).
- By Euler's theorem on traversability, a continuous non-overlapping path (Eulerian path) can only exist if a graph has at most **two vertices of odd degree**.
- Because all 8 vertices of a cube have odd degree (degree 3), it is impossible to trace all 12 edges without retracing edges or cutting the wire.
- To construct all 12 edges by bending and cutting with the minimum number of cuts:
- Retracing edges minimizes cuts, or minimal disjoint paths $= 8/2 = 4$ paths.
**2. Option Analysis**:
- **Option (A) is CORRECT**: Accurately derives the minimal wire cutting/bending solution dictated by odd-degree vertex constraints.
- **Options (B), (C), and (D) are INCORRECT**: Violate graph theoretical edge-coverage theorems.
**3. Final Answer**:
- **Correct Choice**: **A**
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