[Math] Walk on Earth: Math Puzzle

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Here's the famous math puzzle posted by Prof. Walter Lewin about a person walking on earth, quoted below for posterity:

A person stands on the North Pole. She walks 10 miles South, then 10
miles East, then 10 miles North and she is back at her starting point
(the North Pole). There are more points on Earth which meet this same
condition: You walk 10 miles South, then 10 miles East, then 10 miles
North and you are back at the point where you started. Question
Describe briefly one or more points on Earth (not the North Pole)
which meet these conditions.

The solutions are very easy to find, and Prof. Lewin outlines them very well in his comment which I also quote here:

In addition to the North Pole, there are infinite locations on Earth that meet the condition: (i) you walk 10 miles South, then (ii) 10 miles East, then (iii) ten miles North and you are back where you started. All of them are in the vicinity of the South Pole.

Choose a latitude circle around the South Pole that has a circumference of 10 miles. The radius of that circle is about 1.6 miles (about 2.5 km). Pick a random point on that circle (point B). Walk 10 miles North and you arrive at point A. Point A is about 11.6 miles from the SP, it meets the required conditions. Starting at A, (i) you walk 10 miles South – that brings you to point B. (ii) You now walk 10 miles East, that means you walk a full circle around the South Pole on your chosen latitude circle. Thus you are back at B. (iii) You now walk 10 miles North and you are back at A.

There are infinite points A that are about 11.6 miles from the SP; they lie on a latitude circle.

BUT THERE ARE MORE. Choose a latitude circle around the South Pole whose circumference is 10/n miles. n = 1,2,3,4,5,6,7 etc. Let's choose n=5 as an example (but you may choose any other value for n). You are now on a latitude circle with a circumference of 2 miles. You choose any point, B5, on that circle. You now walk 10 miles North and you arrive at point A5. A5 meets the required conditions. (i) You walk 10 miles South and you arrive at B5. (ii) You now walk 10 miles East, that means you walk 5 times full circle around the South Pole and you are back at B5. (iii) You now walk 10 miles North and you are back at A5.

When n becomes very large, for instance 500, then the situation may become not very practical (but in principle still possible). The circumference of the latitude circle would then be 10/500=0.02 mile. The radius is then about 0.0032 miles which is about 5.1 m. Thus all B500 points on this circle would be about 5.1 m from the South Pole. All points A500 would then lie about 10.0032 miles North of the South Pole and the infinite number of A500 points would meet the required condition.

However, he does not show that those solutions are the only ones. Anybody have any hints for the proof of uniqueness? This is a famous puzzle whose answer is well-known but a proof of uniqueness is not well-known.

Best Answer

We travle from $A$ ten miles south to $B$, then ten miles east to $C$, then ten miles north to $D$. Clearly $B$ and $C$ and hence also $A$ and $D$ have the same latitude. Also, for definiteness, $A$ must be more than ten miles away from the south pole (walking sout beyond the pole makes no sense, nor does walking east at the pole). If $A$ is not the north pole, then $B$ has the same longitude as $A$ and $C$ has the same longitude as $D$, so that we require $B=C$. The road from $B$ to $C$ is along a longitude circle and must consist of an integer number of full rounds, as in the suggested solutions.

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