Question: How many points are there on the globe where, by walking one mile south, one mile east, and one mile north, you reach the place where you started?
Start by drawing a mental map: One mile south, one mile east, and one mile north covers three sides of a square. You ought to end up a mile east of where you started. The situation seems impossible, and you might think the answer is zero points.
Is ZERO the correct answer?Try again. The only way to make sense of the situation is to remember that the compass directions are relative ones applied to the surface of a sphere. At the North Pole, every horizontal direction is south. As long as you start precisely at the North Pole, you can walk a mile in any direction and that will count as walking south. Not only that, but a subsequent one-mile-east leg will curve in a circle centered on the North Pole. At any rate, it will if you interpret the puzzle to mean that you not only point yourself due east but constantly adjust your direction so that your bearing remains due east throughout the second mile. That then allows a final, straight, one-mile-north leg returning to the pole. The journey looks like a wedge of pie rather than an open square.
So the North Pole is one point where this could happen.Notice that it couldn't happen at the South Pole. At the South Pole, every direction is north. You can't go a mile south from the South Pole.
So is the answer 1?There is not just one point from which you can do this but an infinity of them. You can start from any point that is the correct distance from the South Pole. There is a complete circle, centered on the South Pole, of possible starting points.
What is the "correct distance"? The one-mile circumference circle must have a radius of l/2p miles. The starting point of the journey must be a mile farther from the pole than that, or 1 + 1/2 p miles, which comes to about 1.159 miles.
We're still not done. Suppose you started a little closer to the pole. You go a mile south, then travel continuously due east in a smaller circle, centered on the pole, of 1/2 mile circumference. You will go full circle twice. Then backtrack a mile north. This scheme nets another infinity of possible starting points, each 1 + 1/4 p miles from the pole. You can also manage a route in which you circle the pole three times, four times, or any whole number n of times. Each yields a new circle of starting points 1 + l/2n p miles from the pole. There is an infinite ensemble of ever-closer circles, each with an infinity of starting points.
So actual answer is infinite (Infinite*infinite + 1) points.PS: There is a variation of this puzzle also: "There is a point on the globe where, by walking one mile south, one mile east, and one mile north, you reach the place where you started; what will be color of bear at that place?"
Answer must be white, coz it's possible only at poles. And color of polar bears is WHITE.
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