Problem 3 · 2016 Math Kangaroo
Medium
Spatial & Visual Reasoning
sequence-of-figures
Maria wants to build a bridge across a river. This river has the special feature that from each point along one shore the shortest possible bridge to the other shore always has the same length. Which of the following diagrams is definitely not a sketch of this river?
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Answer: B
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Hint 1 of 3
A constant shortest crossing means the two shores stay a fixed distance apart everywhere.
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Hint 2 of 3
Look for a shore shape where a sharp corner would let you reach the far side by a shorter slanted bridge.
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Hint 3 of 3
At a sharp inside corner of a zig-zag, the nearest point on the far shore is closer than along a straight crossing, so the width can't stay constant.
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Approach: constant-width strip
- The condition says the two banks are everywhere the same perpendicular distance apart (a constant-width strip).
- Smooth parallel curves can keep a fixed gap.
- Banks made of straight segments meeting at sharp angles (the zig-zag) cannot: near an inside vertex the opposite bank is reached by a shorter slanted bridge.
- So B is definitely not such a river.
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