Problem 25 · 2015 Math Kangaroo
Stretch
Spatial & Visual Reasoning
path-tracing
Florian has seven pieces of wire of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm and 7 cm. He uses some of those pieces to form a wire model of a cube with side length 1. He does not want any overlapping wire parts. What is the smallest number of wire pieces that he can use?
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Answer: D — 4
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Hint 1 of 2
A cube frame has 12 edges of length 1; the wires can bend at the corners.
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Hint 2 of 2
Think of tracing all 12 edges with as few continuous strokes as possible without overlap - an Euler-path count.
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Approach: cover all 12 edges with the fewest continuous wires
- The cube wireframe has 12 edges meeting at 8 corners where 3 edges meet.
- Every corner has odd degree, so a single continuous wire cannot cover all edges without overlap.
- The minimum number of continuous strokes covering all 12 edges is 4.
- So the smallest number of pieces is 4 (D).
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