Discussion
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?
Answer: A
The minimum number of cuts is 3.
Think of the cube frame as a graph: 8 corners, each joined to 3 edges. Each piece of wire, bent at corners, traces a continuous path along edges. A path can pass through a corner using two of its edges, so every corner where an odd number of edge-ends meet must be an end of some piece. All 8 corners have 3 edges (odd), so the frame needs at least 8 piece-ends, that is at least 4 pieces.
Four pieces are also enough: pair the eight corners up, and the edges of any connected frame with 2k odd corners can be split into exactly k continuous paths. Cutting a single 12 m wire into 4 pieces takes 3 cuts.
4 cuts would give 5 pieces, more than needed; 6 and 12 correspond to cutting the wire much more finely, up to one piece per edge.
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