Problem 2 · AMC 8 Stretch
Core
Logic & Word Problems
Geometry & Measurement
symmetrylogical-reasoningconsidering-extreme-cases
A hunter leaves camp, walks 10 miles due north in a straight line, and stops for lunch. After lunch he again walks 10 miles due north in a straight line — and discovers he is back at camp! On a round Earth, where is the hunter's camp? ('North' always means 'toward the North Pole,' and his straight line follows the curve of the Earth like a taut string on a globe.)
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Answer: The South Pole
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Hint 1 of 3
On a globe, 'north' is not one fixed compass direction forever. It always means 'toward the North Pole.' Is there a special place where 'north' behaves strangely?
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Hint 2 of 3
Think about the South Pole. From the South Pole, which direction is north?
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Hint 3 of 3
From the South Pole, EVERY direction is north. So if the camp is at the South Pole, walking 'north' takes you out, and walking 'north' again along the same taut-string path on the far side brings you right back.
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Approach: Use the special symmetry of the pole
- 'North' means toward the North Pole. On a flat map that feels like one fixed direction, but on a round Earth it changes with where you stand. The trick is to find the one special spot where this matters.
- Stand exactly at the South Pole. The North Pole is straight up and over the globe in every direction, so from the South Pole every direction you face is 'north.'
- Put the camp at the South Pole. Walking 10 miles 'north' carries the hunter out along a great circle; walking 'north' again continues along that same circle, which loops back toward the pole on the far side and returns him to the South Pole.
- Because all directions from the South Pole are identical (all 'north'), a 'north then north' walk can close on itself. No ordinary spot has that symmetry, so the camp is at the South Pole.
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