地球行走问题
Earth where you can walk
题目详情
地球上有多少个地点满足：从该点出发向南走 1 英里、向东走 1 英里、向北走 1 英里，最终恰好回到出发点？
假设地球是完美球体，且每一段行走时指南针方向保持恒定；地球表面任意位置都可行走；你的脚可以无限小。
英文原题
How many places are there on the Earth where you can walk one mile south, one mile east, one mile north, and end up exactly where youstarted? Assume the Earth is a perfect sphere, that your compass bearing is constant on each leg of the walk, that all parts of the Earth are able to be walked upon, and that your feet are arbitrarily small.
解析
一个显然的地点：北极。
从北极出发向南走 1 英里到某条纬线；再向东走 1 英里（沿该纬线绕一段弧）；最后向北走 1 英里会回到北极。由于在北极处“向东”的方向不影响最终向北回到同一点，因此北极满足条件。
除此之外还有无穷多个地点（在南极附近）。
思路：希望“向南 1 英里后到达的纬线”，其周长能被 1 英里整除，使得向东走 1 英里正好绕行整数圈回到同一点。
对任意正整数 ，考虑一条纬线，其周长恰好为 英里。若你在这条纬线以北 1 英里处出发：
- 向南 1 英里：到达该纬线；
- 向东 1 英里：沿纬线走了 圈，回到同一点；
- 向北 1 英里：回到出发点。
这样的纬线在南极附近都存在（纬线周长从 0 连续增大），因此起点有 无穷多个。
英文解析
An obvious location: The North Pole.
Starting from the North Pole, walk 1 mile south to a certain latitude; then walk 1 mile east (along an arc of that latitude); finally, walk 1 mile north and you will return to the North Pole. Since the direction "east" at the North Pole does not affect the final northward return to the same point, the North Pole satisfies the condition.
There are also infinitely many other locations (near the South Pole).
The idea is that the latitude reached after walking 1 mile south should have a circumference divisible by 1 mile, so that walking 1 mile east allows you to complete an integer number of laps and return to the same point.
For any positive integer , consider a latitude whose circumference is exactly miles. If you start 1 mile north of this latitude:
- Walk 1 mile south: You arrive at this latitude;
- Walk 1 mile east: You travel laps along the latitude and return to the same point;
- Walk 1 mile north: You return to your starting point.
Such latitudes exist near the South Pole (as the circumference of latitudes increases continuously from 0), so there are infinitely many starting points.