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An extension to a traditional puzzle.
chmarr
When I was a kid, I was a real smartass. Yes, okay, perhaps I haven't changed much :)

Anyway... one of my math teachers in high school gave us this puzzle:

"Assume the Earth is a nice, smooth globe. If you were to walk one kilometer South, then one kilometer West, then one kilometer North, and you finished where you started, where are you now?"

You can imagine miles instead of kilometers if you really like; it doesn't matter.

The answer, of course is "the North Pole". Sometimes the puzzle ask "what colour are the bears?" instead, to which the answer is "white" :)

Me, being my smartassy self, thought about it for a bit, and posed a counter-question:

"Okay, name one other location on the Earth where that is also true."

Can you? There's no trick to this; there really is at least one other location :) See if you can figure it out before clicking through, in case someone posts the answer as a reply.

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Heehee :)

No, there's no tricks to this; just cleverness. Assume nice standard polar coordinates.

Ok. I don't know if I would have ever figured out those answers. :)

I actually ran into this question growing up. I don't recall if I came up with the second answer on my own, but I did see mention in a textbook.

It's 1.159 (1 + 1/2pi) km north of the South Pole. Walk one km south, then 1 km west, a full circle around the South Pole, then 1 km north back to your starting point.

Yep, that's it :)

Now for part three:

Describe ANOTHER point, or set of points, where the condition is also true.

Errr...whatever point out from the center of the earth falls on a circle 1 km in circumference?

That's actually the same answer as nerfcoyote's, said a different way :)

*grin* No, no--I mean from the core of the earth. Rather than a pole, there's got to be a spot way down in the hot-'n-squishy core that falls on a circle 1km in circumference with the dead center of the planet as the center point, so that circumscribing that 1km circle made you orbit the dead center of the core. (You'd need special equipment to DO it, mind you...)

So you pick a spot 1 km north of the spot out from the core, go 1km south, describe the 1km circle around the core of the planet, then go back 1km north.

I don't know if anything where you have to dig counts though.

(Okay, okay, geekery forces me to add that the core is not really hot'n'squishy but probably hot'n'nickel-irony.)

OKay, haha, very clever :) That's under the "trick" category, though :)

The set of points on the circle at (1 - 1/2pi) north from the south pole. For which case your "north" and "south" walks cross over the south pole, and you go round the same 1km circle.

Additionally, all the points on an infinite number of circles around the south pole where your north walk ends up at a point where the "west" circle around the pole is exactly 1/n km.

That works, and its very close to the third part of the puzzle. But, you need to start 1mile north of that circle, as you can't touch the south pole :)

That was what I meant, actually :)

And "walk one km south" could be read to include the cases where you start off south and walk one km, even if the last part is no longer south.

Almost! Unfortunately, once you cross the south pole, you're no longer walking south :)

if one defines a set of circles around the South Pole that the circumference is the reciprocal of a whole number, one km north of those circles. Essentially you'd walk south, then go around the South Pole a whole number of times and end up on the same longitudinal line you started, then 1 km north back to the start.

Yep. That's it. And there's no more parts.

Congratulations. You win the nerdy part of the Internet :)

1KM north of the South Pole? Although your end location is "anywhere on a circle 1KM in radius centered on the South Pole" in that case.

That won't work, since once you've walked your 1km to the south pole, you can no longer walk west, since the only direction available is "north".

When I was a kid, I was a real smartass

LOL...you're telling me cuz...I had to put up with it every xmas and mother's day!

I hear all that brainiac stuff you used to torture me with is being put to good use over there in the land of plenty?

There's something scary in the fact that I just checked your journal for the first time in a year, and you only started re-posting for the first time in a year...freaky huh?

Cheers,

Justin.

Hey man... good to hear from you :)

Will try and visit there in the next week or two and give you your yearly dose of "brainiac" :)

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