Problem 20 · 2009 Math Kangaroo
Stretch
Spatial & Visual Reasoning
path-tracingsymmetry
There are three great circles on a sphere that intersect each other at right angles. Starting at point S, a little bug moves along the great circles in the direction indicated. At each crossing it turns alternately to the right or to the left. How many quarter circles does it crawl along until it is back at point S?
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Answer: A — 6
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Hint 1 of 2
The three perpendicular great circles meet at six points (the axis tips); each arc between them is a quarter circle.
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Hint 2 of 2
Track the turns: with the right/left alternation the path closes after surprisingly few quarter-arcs.
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Approach: trace the alternating-turn path between the six crossing points
- Three mutually perpendicular great circles cross at six points (like the six face-centres of a cube); each arc from one crossing to the next neighbouring crossing is a quarter circle.
- From S the bug reaches a crossing after one quarter arc, turns, takes the next quarter arc, and so on, alternating right and left.
- Tracing this alternating walk closes the loop after visiting six such crossings, so it covers six quarter circles before returning to S.
- So the answer is 6.
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