There and Back Again (Take 2)
Level
1
Sue travels South for 1 mile, then East for 1 mile, and then North for one mile, ending up exactly where she began. From how many different places on Earth's surface could Sue's journey have begun?
1
2
0
An infinite number
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1 solution
Well, Sue can start at the North Pole.
Her 1st leg takes her one mile South along a longitude line, then she goes East for a mile, then her final leg takes her North along a different longitude line, which after exactly one mile meets the first longitude line at the North Pole. Since the North Pole is the only point on Earth at which different longitude lines converge going Northward like this, we're done!
But wait - who said the two longitude lines have to be different ? If we could arrange for them to be exactly the same line, their North ends would necessarily coincide, and we would have another solution to the problem.
Is there a place on Earth where going East for a mile lands you at the same longitude you started from? Well, there is a latitude circle C near the South Pole which is one mile in circumference. So if Sue starts one mile North of C, her journey will take her South to C along a certain longitude line, then once around latitude circle C, then North back to her starting point along the original longitude line - success!
Now we have two solutions, right? No, we have an infinite number, because Sue's journey can begin anywhere on that latitude circle one mile North of C. And now we're done!
But wait - who said that Sue must go only once around the latitude circle? There's another circle a bit south of C whose circumference is 1/2 mile, and if Sue starts out 1 mile N of it , she goes around twice during her Eastbound leg, again returning to her starting point along the same longitude she set out on. And a bit further South there's another circle of circumference 1/3 mile, and - well, you get the idea.
So Sue's journey could have begun at the North Pole, or anywhere on any of a certain infinite set of circles centered on the South Pole.
...and now we're done:-)