He actually travels east instead of west, but no matter, just assume he walked west instead. It looks like the circle he's walking in is about 20 feet in circumference, so let's assume it's precisely 20 feet. If, to satisfy step 2, he were to do that walk 264 times (exactly one mile), his end point would be exactly the same degree of longitude as where he started. He could then walk north one mile to satisfy step 3.

I don't do You Tubies. How about a latitude and a longitude?

The striped pole is the south pole. The yellow sphere is your beginning point. You follow the red path south. It's exactly one mile. Then you head west on the green circle. Its circumference is exactly one mile, which brings you back to the red path, which you take north for one mile back to your yellow sphere starting point.

J

North mean North a line of longitude. Longitudes all intersect at exactly two points. The geographic north pole and the geographic south pole. Walking north-west is NOT walking north.

I don't do You Tubies. How about a latitude and a longitude?

Argh!!!

Any longitude will work. The furthest latitude from the south pole is whatever is 6,120.34 feet away. The circumference of the circle (the degree of latitude you will follow when you go west) can be no larger than 5,280 feet in order for your last step to land you on the same degree of longitude that you started on. The formula for the circumference of a circle is 2πr. 5,280 / 2π is 840.34, that's the radius of the circle (with the south pole in the middle). 5,280 + 840.34 = 6,120.34, your starting point.

What degree of latitude is that? A degree of latitude is 69.407 miles at the pole, so it'll be 89.X South (I'm sure you can figure it out if you really need to know).

Other solutions rely on you making multiple westerly revolutions, so the next distance that'll work is one using a circle that's half the circumference of one mile. And on and on, until you're making 264 revolutions in the example already provided.

North mean North a line of longitude. Longitudes all intersect at exactly two points. The geographic north pole and the geographic south pole. Walking north-west is NOT walking north.

I'm not following you. I have no idea what you mean.

No one has suggested that walking north-west is walking north, so I don't know what you think you're arguing against.

North mean North a line of longitude. Longitudes all intersect at exactly two points. The geographic north pole and the geographic south pole. Walking north-west is NOT walking north.

Bob, here's the same image but with longitude lines added.

North mean North a line of longitude. Longitudes all intersect at exactly two points. The geographic north pole and the geographic south pole. Walking north-west is NOT walking north.

Bob, look at the picture in your own post. You start walking at the yellow point at the right, you walk along a meridian (the red line) southwards, to the South Pole (indicated by the barber pole). After 1 mile walking you arrive at the small, green latitude circle. There you start walking to the west, always keeping the South Pole to your left. After walking one mile westwards, you are back at the point where the red line meets the green circle, as the circumference of that latitude circle is exactly 1 mile (a very small latitude circle, you are really very close to the South Pole!). Now you walk back along the red meridian, and after another 1 mile you're back at your starting point. Voilà!

You have fulfilled all the conditions of the exercise: walked 1 mile southwards, then 1 mile westwards, then 1 mile northwards (along the same meridian as when you went to the south) and you are back again at your starting point. Any questions?

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, Merlin and Tony are mathies completely lacking in mech (and not necessarily all that great at math either), Bob's a bit short on mech but not as bad as Merlin and Tony, and then a few of you are a strong combo of both mech and math.

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, Merlin and Tony are mathies completely lacking in mech (and not necessarily all that great at math either), Bob's a bit short on mech but not as bad as Merlin and Tony, and then a few of you are a strong combo of both mech and math.

Cheers,

J

Tony isn't so lacking in mech. It's that he puts things into the mech which muddle up the problem as posed. I've seen no signs of his being any good at math. And he's atrocious at reading comprehension - See.

Bob, look at the picture in your own post. You start walking at the yellow point at the right, you walk along a meridian (the red line) southwards, to the South Pole (indicated by the barber pole). After 1 mile walking you arrive at the small, green latitude circle. There you start walking to the west, always keeping the South Pole to your left. After walking one mile westwards, you are back at the point where the red line meets the green circle, as the circumference of that latitude circle is exactly 1 mile (a very small latitude circle, you are really very close to the South Pole!). Now you walk back along the red meridian, and after another 1 mile you're back at your starting point. Voilà!

You have fulfilled all the conditions of the exercise: walked 1 mile southwards, then 1 mile westwards, then 1 mile northwards (along the same meridian as when you went to the south) and you are back again at your starting point. Any questions?

Since two legs of the walk, the first and the third are along lines of longitude one must end up where the lines of longitude intersect. Under the conditions of the puzzle that would be the north geographic pole. The three logs are along a line of longitude, a line of latitude and a line of longitude the conditions of the puzzle require that the journey begin and end at a pole.

Let me see if I can restate my comment better, then: Tony is less worse at math than he is at mech.

Yes. He's a scatterbrained ditz. Most of his time is spent constructing (mechanically deficient) straw men and knocking them down.

J

I won't argue about the degree of Tony's bad at mech. He isn't good. It's just that he does bring in some considerations - especially friction - which would have some effect in a physical instantiation.

I often wonder if Tony bothers to read what he's responding to.

I think that often Bob only sees some isolated feature of a post to which he responds, and that he does not follow the progression of a discussion, which is why he comes up with strange things, like taking your visual illustration of the walking instructions as being a northwest direction.

Bob, here's the same image but with longitude lines added.

Does this make it clearer? Understand now?

J

No. The problem state that the traveler started at a point, went a mile south then a mile west and then a mile north and ended where he started. That means his end point had to be on the intersection of two lines of longitude. Travelling North-South means travelling on a line of longitude. Travelling East West means travelling on a line of latitude parallel to the equator.

I won't argue about the degree of Tony's bad at mech. He isn't good. It's just that he does bring in some considerations - especially friction - which would have some effect in a physical instantiation.

I often wonder if Tony bothers to read what he's responding to.

I think that often Bob only sees some isolated feature of a post to which he responds, and that he does not follow the progression of a discussion, which is why he comes up with strange things, like taking your visual illustration of the walking instructions as being a northwest direction.

Ellen

Yeah, and Bob's latest post just now illustrates that. It appears that he paid no attention to what he was responding to. He didn't actually read and comprehend Max's post before replying.

That means his end point had to be on the intersection of two lines of longitude.

False. That does not logically follow, and we've shown it to be false with examples.

Bob, what you need to do is to slow down, actually read and comprehend what we've written and illustrated, and carefully consider what we've said. Our presentations of solutions near the south pole comply with the conditions of the exercise.

Since two legs of the walk, the first and the third are along lines of longitude one must end up where the lines of longitude intersect. Under the conditions of the puzzle that would be the north geographic pole. The three logs are along a line of longitude, a line of latitude and a line of longitude the conditions of the puzzle require that the journey begin and end at a pole.

In this case the line of longitude "intersects" itself over its whole length, you travel namely twice the same line of longitude. Again: the sameline of longitude. When it is the same line, it doesn't have to intersect another line of longitude to arrive at its starting point. That is the point!

Is the problem that Bob is misperceiving the illustration as representing the north pole despite being told otherwise? Verbal description is being overridden by a visual misunderstanding?

Maybe this will help?

Start at the yellow ball. Walk south 1 mile (the distance of the red line). Walk west 1 mile (the circumference of the green circle). Walk north 1 mile (the distance of the red line). You are back at your starting point, the yellow ball.

Is the problem that Bob is misperceiving the illustration as representing the north pole despite being told otherwise? Verbal description is being overridden by a visual misunderstanding?

Maybe this will help?

Start at the yellow ball. Walk south 1 mile (the distance of the red line). Walk west 1 mile (the circumference of the green circle). Walk north 1 mile (the distance of the red line). You are back at your starting point, the yellow ball.

J

The green circle of latitude is located X miles north of the South Pole.

The radius of the green circle is X miles, the diameter, 2*X miles.

The circumference of the green circle is 2*X*pi miles.

“Walk west 1 Mile is the circumference of the green circle” stated mathematically,

1 mile = 2*X*pi

From that we can solve for X:

divide each side of equal sign by 2.

1/2 = X*pi

divide each side by pi, and switch sides.

X = 1 / (2pi)

pi is 3.14159265 ... estimate of X:

X = 1 / 6.2831853

calculator

X = 0.1591549

So Jonathan’s yellow circle of latitude is located 1+ 1 / (2pi) miles, or 1.1591549 miles, north of the South Pole.

False. That does not logically follow, and we've shown it to be false with examples.

Bob, what you need to do is to slow down, actually read and comprehend what we've written and illustrated, and carefully consider what we've said. Our presentations of solutions near the south pole comply with the conditions of the exercise.

J

ALL northbound travel is along a line of longitude. Any other path or direction is NOT north or south. Since the last leg of this three part journey is north along a line of longitude different from the first leg the end point must lie on the intersection of the two lines of longitude, hence it is a pole. Given the conditions of the problem it is the north pole.

In this case the line of longitude "intersects" itself over its whole length, you travel namely twice the same line of longitude. Again: the sameline of longitude. When it is the same line, it doesn't have to intersect another line of longitude to arrive at its starting point. That is the point!

That last leg is NOT on the same line of longitude as the first leg. Why? Because the second leg is a traverse along a line of latitude which changes the longitude.

That last leg is NOT on the same line of longitude as the first leg. Why? Because the second leg is a traverse along a line of latitude which changes the longitude.

The last leg IS on the same line of longitude as the first leg, because following a line of latitude over its total length brings you back to the point were you started, lines of latitude are circles! If you follow the equator westwards for about 40000 kilometers you'll be back again at where you started. However, the green circle in the picture is very close to the South Pole, so in this case you have only to walk one mile, but the principle is the same.

According to the original problem Bob is right except for the one mile walk near the South Pole. Everybody is right about the North Pole.

Some busybody moved the goal posts.

--Brant

to make it more interesting

The "South Pole solution" is perfectly in accordance with the original problem, nobody moved the goal posts. Most people see only the first solution, the North Pole solution, but the South Pole solutions are just as good, only a bit less obvious.

According to the original problem Bob is right except for the one mile walk near the South Pole. Everybody is right about the North Pole.

Some busybody moved the goal posts.

--Brant

to make it more interesting

59 minutes ago, Max said:

The last leg IS on the same line of longitude as the first leg, because following a line of latitude over its total length brings you back to the point were you started, lines of latitude are circles! If you follow the equator westwards for about 40000 kilometers you'll be back again at where you started. However, the green circle in the picture is very close to the South Pole, so in this case you have only to walk one mile, but the principle is the same.

1 hour ago, Max said:

The last leg IS on the same line of longitude as the first leg, because following a line of latitude over its total length brings you back to the point were you started, lines of latitude are circles! If you follow the equator westwards for about 40000 kilometers you'll be back again at where you started. However, the green circle in the picture is very close to the South Pole, so in this case you have only to walk one mile, but the principle is the same.

The original problem was walk a mile south walk a mile west (was it east -- no matter) walk a mile north. The east-west walk is less than the length of line of latitude reached by the southword leg so that the return trip is along a different line of longitude. Let me give an example. The coordinates of the north geographic pole are (90, lon) where lon can be any angle between 0 and 360. The north geographic pole and the south geographic pole are the only two points on the earth sphere that do not have unique coordinate. Now let me widen the problem out Start at a point, walk to the equator in a southerly direction, walk east along the equator the same number of steps that one took to reach the equator then march in a northerly direction the same number of step.

Two cases:

Case 1 the starting point is the north pole. Assume the first leg is south along the Greenwich meridian, that is to say 0 longitude. This gets us down to (0, 0) on the equator. Walk west the same distance and we get to (0, 90). Now walk north the same distance and we get to (90,90) which is the same point as (90, 0) the north pole.

Case 2. The starting point is (x-lat, x-long) where x-lat is greater than 0 and less than 90. Assume x-long = 0 without loss of generality. Now leg 1: (x-lat, 0) to (x1, 0) where x1 < x-lat and greater or equal to 0. Leg 2 (x1,0) (x1, y1) where y1 > 0 but < 360. That means leg2 moved us to a different point with the same latitude. Now leg 3 northward by the same distance. This gets is to (x2, y1) because going north means following a meridian of longitude. Notice that x2 not = x1. The final destination is (x2, y1) which is different from (x-lat, 0). So we do not end up at the same place if we started out from a point that was not the pole.

Q.E.D.

Forget drawings. The proof is abstract and mathematical. Drawings are crutches for the logically feeble.

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Jonathan19 postsJon Letendre41 postsBaalChatzaf17 postsMax22 posts## Popular Days

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## Popular Posts

## Jonathan

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

## Ellen Stuttle

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

## Brant Gaede

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

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## BaalChatzaf

I don't do You Tubies. How about a latitude and a longitude?

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## BaalChatzaf

North mean North a line of longitude. Longitudes all intersect at exactly two points. The geographic north pole and the geographic south pole. Walking north-west is NOT walking north.

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## 9thdoctor

Argh!!!

Any longitude will work. The furthest latitude from the south pole is whatever is 6,120.34 feet away. The circumference of the circle (the degree of latitude you will follow when you go west) can be no larger than 5,280 feet in order for your last step to land you on the same degree of longitude that you started on. The formula for the circumference of a circle is 2πr. 5,280 / 2π is 840.34, that's the radius of the circle (with the south pole in the middle). 5,280 + 840.34 = 6,120.34, your starting point.

What degree of latitude is that? A degree of latitude is 69.407 miles at the pole, so it'll be 89.X South (I'm sure you can figure it out if you really need to know).

Other solutions rely on you making multiple westerly revolutions, so the next distance that'll work is one using a circle that's half the circumference of one mile. And on and on, until you're making 264 revolutions in the example already provided.

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## Jonathan

I'm not following you. I have no idea what you mean.

No one has suggested that walking north-west is walking north, so I don't know what you think you're arguing against.

J

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## Jonathan

Bob, here's the same image but with longitude lines added.

Does this make it clearer? Understand now?

J

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## Max

Bob, look at the picture in your own post. You start walking at the yellow point at the right, you walk along a meridian (the red line)

southwards, to the South Pole (indicated by the barber pole). After 1 mile walking you arrive at the small, green latitude circle. There you start walking to thewest,always keeping the South Pole to your left. After walking one mile westwards, you arebackat the point where the red line meets the green circle, as the circumference of that latitude circle is exactly 1 mile (a very small latitude circle, you are really very close to the South Pole!). Now you walk back along the red meridian, and after another 1 mile you're back at your starting point. Voilà!You have fulfilled all the conditions of the exercise: walked 1 mile southwards, then 1 mile westwards, then 1 mile northwards (along the

samemeridian as when you went to the south) and you are back again at your starting point. Any questions?## Link to comment

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## Jonathan

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, Merlin and Tony are mathies completely lacking in mech (and not necessarily all that great at math either), Bob's a bit short on mech but not as bad as Merlin and Tony, and then a few of you are a strong combo of both mech and math.

Cheers,

J

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## Ellen Stuttle

Tony isn't so lacking in mech. It's that he puts things into the mech which muddle up the problem as posed. I've seen no signs of his being any good at math. And he's atrocious at reading comprehension - See.

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## Jonathan

Sorry, but I disagree.

Yeah, he does so because he is lacking in mech.

Let me see if I can restate my comment better, then: Tony is less worse at math than he is at mech.

Yes. He's a scatterbrained ditz. Most of his time is spent constructing (mechanically deficient) straw men and knocking them down.

J

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## BaalChatzaf

Since two legs of the walk, the first and the third are along lines of longitude one must end up where the lines of longitude intersect. Under the conditions of the puzzle that would be the north geographic pole. The three logs are along a line of longitude, a line of latitude and a line of longitude the conditions of the puzzle require that the journey begin and end at a pole.

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## Ellen Stuttle

I won't argue about the degree of Tony's bad at mech. He isn't good. It's just that he does bring in some considerations - especially friction - which would have some effect in a physical instantiation.

I often wonder if Tony bothers to read what he's responding to.

I think that often Bob only sees some isolated feature of a post to which he responds, and that he does not follow the progression of a discussion, which is why he comes up with strange things, like taking your visual illustration of the walking instructions as being a northwest direction.

Ellen

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## BaalChatzaf

No. The problem state that the traveler started at a point, went a mile south then a mile west and then a mile north and ended where he started. That means his end point had to be on the intersection of two lines of longitude. Travelling North-South means travelling on a line of longitude. Travelling East West means travelling on a line of latitude parallel to the equator.

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## Jonathan

Yeah, and Bob's latest post just now illustrates that. It appears that he paid no attention to what he was responding to. He didn't actually read and comprehend Max's post before replying.

J

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## Jonathan

False. That does not logically follow, and we've shown it to be false with examples.

Bob, what you need to do is to slow down, actually read and comprehend what we've written and illustrated, and carefully consider what we've said. Our presentations of solutions near the south pole comply with the conditions of the exercise.

J

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## Max

In this case the line of longitude "intersects" itself over its whole length, you travel namely twice the

sameline of longitude. Again: thesameline of longitude. When it is thesameline, it doesn't have to intersectanotherline of longitude to arrive at its starting point. That is the point!## Link to comment

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## Jonathan

Is the problem that Bob is misperceiving the illustration as representing the north pole despite being told otherwise? Verbal description is being overridden by a visual misunderstanding?

Maybe this will help?

Start at the yellow ball. Walk south 1 mile (the distance of the red line). Walk west 1 mile (the circumference of the green circle). Walk north 1 mile (the distance of the red line). You are back at your starting point, the yellow ball.

J

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## Jon Letendre

AuthorThe green circle of latitude is located X miles north of the South Pole.

The radius of the green circle is X miles, the diameter, 2*X miles.

The circumference of the green circle is 2*X*pi miles.

“Walk west 1 Mile is the circumference of the green circle” stated mathematically,

1 mile = 2*X*pi

From that we can solve for X:

divide each side of equal sign by 2.

1/2 = X*pi

divide each side by pi, and switch sides.

X = 1 / (2pi)pi is 3.14159265 ... estimate of X:

X = 1 / 6.2831853

calculator

X = 0.1591549So Jonathan’s yellow circle of latitude is located 1+ 1 / (2pi) miles, or 1.1591549 miles, north of the South Pole.

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## BaalChatzaf

ALL northbound travel is along a line of longitude. Any other path or direction is NOT north or south. Since the last leg of this three part journey is north along a line of longitude different from the first leg the end point must lie on the intersection of the two lines of longitude, hence it is a pole. Given the conditions of the problem it is the north pole.

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## BaalChatzaf

That last leg is NOT on the same line of longitude as the first leg. Why? Because the second leg is a traverse along a line of latitude which changes the longitude.

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## Max

The last leg

ISon the same line of longitude as the first leg, because following a line of latitudeover its total lengthbrings you back to the point were you started, lines of latitude arecircles! If you follow the equator westwards for about 40000 kilometers you'll be back again at where you started. However, the green circle in the picture is very close to the South Pole, so in this case you have only to walk one mile, but the principle is the same.## Link to comment

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## Brant Gaede

According to the original problem Bob is right except for the one mile walk near the South Pole. Everybody is right about the North Pole.

Some busybody moved the goal posts.

--Brant

to make it more interesting

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## Max

The "South Pole solution" is perfectly in accordance with the original problem, nobody moved the goal posts. Most people see only the first solution, the North Pole solution, but the South Pole solutions are just as good, only a bit less obvious.

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## BaalChatzaf

The original problem was walk a mile south walk a mile west (was it east -- no matter) walk a mile north. The east-west walk is less than the length of line of latitude reached by the southword leg so that the return trip is along a different line of longitude. Let me give an example. The coordinates of the north geographic pole are (90, lon) where lon can be any angle between 0 and 360. The north geographic pole and the south geographic pole are the only two points on the earth sphere that do not have unique coordinate. Now let me widen the problem out Start at a point, walk to the equator in a southerly direction, walk east along the equator the same number of steps that one took to reach the equator then march in a northerly direction the same number of step.

Two cases:

Case 1 the starting point is the north pole. Assume the first leg is south along the Greenwich meridian, that is to say 0 longitude. This gets us down to (0, 0) on the equator. Walk west the same distance and we get to (0, 90). Now walk north the same distance and we get to (90,90) which is the same point as (90, 0) the north pole.

Case 2. The starting point is (x-lat, x-long) where x-lat is greater than 0 and less than 90. Assume x-long = 0 without loss of generality. Now leg 1: (x-lat, 0) to (x1, 0) where x1 < x-lat and greater or equal to 0. Leg 2 (x1,0) (x1, y1) where y1 > 0 but < 360. That means leg2 moved us to a different point with the same latitude. Now leg 3 northward by the same distance. This gets is to (x2, y1) because going north means following a meridian of longitude. Notice that x2 not = x1. The final destination is (x2, y1) which is different from (x-lat, 0). So we do not end up at the same place if we started out from a point that was not the pole.

Q.E.D.

Forget drawings. The proof is abstract and mathematical. Drawings are crutches for the logically feeble.

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## 9thdoctor

Bob, you're a knucklehead.

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