# Where are you?

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I'm waiting for Tony to explain this.

--Brant

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Thanks for playing with the math, Jon and Max. It's fun to see. And I'm enjoying witnessing the spectrum that people fall on: mechies to mathies. I'm way over on the mechie side lacking in math, Merli

Imagine a circle surrounding the South Pole as its center and exactly a mile larger in radius than a concentric smaller circle with circumference of exactly a mile around the South Pole. Then the

The true north pole? This can't be the answer; it's too easy. Des Moines? --Brant

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30 minutes ago, Jonathan said:

And there are more! Many more.

Hint: if one is ordered to walk west, one doesn't run out of westwardness just because one has circled back to the point where one began walking west. If it takes two trips, or thirty two, around a circle to equal a mile, then that's what it takes.

J

Yeah I get it now.  Turning west means following a degree of latitude.

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1 hour ago, Jonathan said:

And there are more! Many more.

Hint: if one is ordered to walk west, one doesn't run out of westwardness just because one has circled back to the point where one began walking west. If it takes two trips, or thirty two, around a circle to equal a mile, then that's what it takes.

J

Yeah, I realized that a bit after I'd posted, but by then I'd left for the evening and I was too tired to post when I came home.

The smaller circle can be any division of a mile such that it can be circumnavigated X whole number times from the starting point.  A half mile, a third of a mile, a fourth, fifth, etc., etc.  Circumnavigate two times, three times, etc., etc.

As Jon says, the circle would soon get too small for a human to walk around it.  Eventually, it would be too small for even a dinky insect to walk around it.  But it could still be imagined as smaller and smaller, with the limiting case being the center point.

The mapping problem between the outer and inner circles is similar to that for the Aristotle's Wheel set-up.

Ellen

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1 hour ago, Brant Gaede said:

I'm waiting for Tony to explain this.

--Brant

HAHAHAHAHA.

And cycloids, anyone?

Ellen

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On 12/29/2018 at 8:31 PM, Jon Letendre said:

The North Pole works.

Other places work, too.

Nope.  The only places where two longitudes intersect are at the poles.  Heading North means moving on a line of longitude  in the direction of the North geographic pole.

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49 minutes ago, BaalChatzaf said:

Nope.  The only places where two longitudes intersect are at the poles.  Heading North means moving on a line of longitude  in the direction of the North geographic pole.

You ignore the possibility that going south and going north can be done on the same line of longitude, while going west between these two displacements.

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2 hours ago, Max said:

You [Bob] ignore the possibility that going south and going north can be done on the same line of longitude, while going west between these two displacements.

I bet that Bob didn't read the posts following the one by Jon to which he replied.

Ellen

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Extra points for some of the places where one can follow the walking instructions and end precisely one mile west of the start point.

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5 minutes ago, Jon Letendre said:

Extra points for some of the places where one can follow the walking instructions and end precisely one mile west of the start point.

Do you mean actual geographic locations?

Ellen

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32 minutes ago, Jon Letendre said:

Extra points for some of the places where one can follow the walking instructions and end precisely one mile west of the start point.

0.5 mile north of the equator. I'm too lazy to calculate the corresponding geographic location...

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48 minutes ago, Ellen Stuttle said:

Do you mean actual geographic locations?

Ellen

Yes.

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19 minutes ago, Max said:

0.5 mile north of the equator. I'm too lazy to calculate the corresponding geographic location...

Yes, that’s one.

I don’t know what you mean by corresponding location. Starting anywhere on that circle of latitude works.

There are more.

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14 minutes ago, Jon Letendre said:

Yes, that’s one.

I don’t know what you mean by corresponding location.

How many degrees, minutes, seconds,  north... I'm not familiar with miles (nautical? statute?).

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5 minutes ago, Max said:

How many degrees, minutes, seconds,  north... I'm not familiar with miles (nautical? statute?).

Ah shit, who cares? “The circle of latitude located precisely 0.5 mile north of the equator” works fine for our purposes. We uniquely identified it.

Both mile definitions are fixed lengths, so either can be used so long as the same one is used in both the walking instructions and our solutions.

There are more solutions that end precisely one mile west of start point.

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If the starting point is somewhere less than 1.5 miles north of the south pole it'll work.  It's a matter of solving for the degrees of longitude that are one mile apart at the appropriate latitude.

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41 minutes ago, Jon Letendre said:

There are more.

Similar to the solutions around the South Pole in the original puzzle, but now with the small circle with a circumference < 1 mile, such that after traversing 1 mile west you cross the meridian that crosses the larger circle 1 mile west of the starting point.

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45 minutes ago, 9thdoctor said:

If the starting point is somewhere less than 1.5 miles north of the south pole it'll work.  It's a matter of solving for the degrees of longitude that are one mile apart at the appropriate latitude.

Yes. This one is harder to specify.

I think it is less than 1.1592 or one and 1 divided﻿ by 2pi﻿﻿ (Ellen’s first solution, which made for one lap around the Pole) miles north of the South Pole. We must start closer to the Pole than that in order that the inner lap be smaller and one mile will exceed its circumference.

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3 minutes ago, Max said:

Similar to the solutions around the South Pole in the original puzzle, but now with the small circle with a circumference < 1 mile, such that after traversing 1 mile west you cross the meridian that crosses the larger circle 1 mile west of the starting point.

Yes, exactly. My current math skill sucks, but I haven’t given up yet. It is harder to specify.

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1 hour ago, Jon Letendre said:

Yes. This one is harder to specify.

I think it is less than 1.1592 or one and 1 divided﻿ by 2pi﻿﻿ (Ellen’s first solution, which made for one lap around the Pole) miles north of the South Pole. We must start closer to the Pole than that in order that the inner lap be smaller and one mile will exceed its circumference.

The second South Pole solution to the original problem makes for two laps around the South Pole. This circle of latitude is located 1.0796 mile or one and 1 divided by 4pi miles North of the South Pole.

So we are looking for a circle of latitude between 1.0796 and 1.1592 miles north of the South Pole.

We need the westery walk to exceed one inner lap by just a bit, so closer to 1.1592 than to 1.0796.

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2 hours ago, Jon Letendre said:

Yes, that’s one.

I don’t know what you mean by corresponding location. Starting anywhere on that circle of latitude works.

There are more.

I think that Max meant what I meant in asking if you meant "actual geographic locations."

Places where one can literally walk in compliance with the instructions.  Out in the ocean isn't such a place. Or in irregular terrain. And the going would be a bitch around the South Pole even if there was a smooth ice sheet.

Ellen

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51 minutes ago, Ellen Stuttle said:

I think that Max meant what I meant in asking if you meant "actual geographic locations."

Places where one can literally walk in compliance with the instructions.  Out in the ocean isn't such a place. Or in irregular terrain. And the going would be a bitch around the South Pole even if there was a smooth ice sheet.

Ellen

Oh, ok. Yes, they’re really walkable.

The half mile above the equator solution, for ending exactly one mile west of where you started, is walkable in Africa, South America, Indonesia. Ground can be flattened, ice can be Zambonied.

The truth is that at the populated places around the world the walking instructions end at almost exactly one mile west of start point and the deviance from one mile is undetectable with the usual measuring devices. But the deviance is real and precisely calculable, because the lines of longitude walked are not parallel. You go south on one line of longitude and return north on another, and the mile you took is now either a little less (northern hemisphere) or a little more (southern hemisphere) than it was, because the longitudinal lines you used got closer (northern hemisphere) or farther (southern hemisphere) to/from one another.

Max’s solution, 0.5 miles north of the equator, works to produce an end point precisely one mile west of the start point by having you walk south one mile to the latitude where the distance between the two walked lines of longitude is the same as at start.

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33 minutes ago, Ellen Stuttle said:

I think that Max meant what I meant in asking if you meant "actual geographic locations."

Places where one can literally walk in compliance with the instructions.  Out in the ocean isn't such a place. Or in irregular terrain. And the going would be a bitch around the South Pole even if there was a smooth ice sheet.

No, that would be like objecting to Aristotle's paradox by insisting that his wheel is not a good car wheel, it's not in the spirit of the puzzle. It's just the conversion from miles to latitudes etc. We can always assume that we're talking about Jesus or St Francis of Paola, who allegedly could walk on water.

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1 hour ago, Ellen Stuttle said:

And the going would be a bitch around the South Pole even if there was a smooth ice sheet.

Watch Michael Palin do it.

Down there, you certainly don't expect the Spanish Inquisition.

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3 hours ago, Max said:

Similar to the solutions around the South Pole in the original puzzle, but now with the small circle with a circumference < 1 mile, such that after traversing 1 mile west you cross the meridian that crosses the larger circle 1 mile west of the starting point.

I am embarrassed to admit I cannot solve it, but I think I have it. Maybe you can solve the equation.

I think it is 1 plus X miles above the South Pole such that 1–2Xpi equals 1 divided by X+1.

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I don’t think the above equation is right. I’m getting a bit tardfounded.