South, east and north
- Round
- Theory
- Points
- 20
- Level
- Gold
- Taught in
- Section 1.7
Adapted from IOAA 2019 (Keszthely), theory problem 10. We have restated it in our own words and kept the original data. The solution was last checked on 15 October 2026.
Treat Earth as a rigid sphere of radius 6378 km. From some points on its surface we can travel 6378 km due south, then 6378 km due east, then 6378 km due north, and arrive back where we started. Find all such points and the paths, give the geographic coordinates of the points where the path turns, and draw the paths. Paths that differ only by a turn about Earth's axis count as the same.
- A distance of 6378 km, one Earth radius, is an arc of 1 radian, or 57.296°, along a great circle.
- The southward and northward legs run along meridians and cancel in latitude, so the path ends at the starting latitude. It closes only if the eastward leg ends on the meridian where it began, or if the start is the North Pole, where all meridians meet.
- From the North Pole, the first leg reaches latitude
- Otherwise the eastward leg must go round its parallel a whole number of times
- Each larger
AnswerOne path starts at the North Pole and turns at 32°42′ N. Every other path starts in the southern hemisphere and runs south to a parallel near the South Pole, goes round that parallel a whole number of times and comes back north. Going round once, it starts at 23°33′ S and turns at 80°51′ S. Going round twice, it starts at 28°08′ S and turns at 85°26′ S, and going round three times, it starts at 29°40′ S and turns at 86°58′ S. The more times the path goes round, the closer its start comes to 32°42′ S.
This problem also appears among the olympiad problems of Section 1.7, Problems for Chapter 1, which teaches the ideas it uses. Equation and figure numbers refer to the course.