GATE 2025 PH – Question 4
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?
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Correct answer: (A) 3
Explanation
**Model.** The cube frame is a graph with 8 vertices (corners) and 12 edges. At each corner **3 edges meet**.
**A continuous piece of wire** that runs through a corner uses 2 of the 3 edges there. The third edge at that corner must be covered by a different piece of wire (or by the end of the same piece). So in a single piece of wire, each corner with an odd number of edges (an odd vertex) must be an **end** of some piece.
**Counting ends.** All 8 corners have 3 edges, so all 8 corners need a wire end. A piece of wire has only 2 ends, so at least
$$\frac82=4\text{ pieces}$$
are needed. Four pieces can indeed cover all 12 edges (for example four pieces of 3 m each along suitable paths, each path starting and ending at corners), and soldering joins the pieces at the corners.
**Number of cuts.** Going from 1 piece of wire to 4 pieces needs
$$4-1=\mathbf{3\ cuts}\quad(\text{option A}).$$