Problem 28 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
spatial-reasoningcareful-counting
The ant Tanti starts an adventure at a vertex of a cube with side length 1. She wants to walk along each edge of the cube at least once and return to the starting point at the end. What is the minimum possible length of her walk?
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Answer: D — 16
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Hint 1 of 2
The ant must traverse all 12 edges and return; every cube vertex has 3 edges (an odd number).
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Hint 2 of 2
To close the walk, some edges must be repeated — pair up the 8 odd vertices to add as few repeats as possible.
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Approach: route-inspection on the cube graph
- A cube has 12 edges; each of its 8 vertices has degree 3 (odd).
- A closed walk covering every edge needs all even degrees, so the 8 odd vertices must be fixed by repeating edges.
- Pairing them needs 4 extra unit edges, so minimum walk = 12 + 4 = 16.
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