HOW DID IT HAPPEN???

[unreality]

A man runs a mile south, a mile west, and a mile north... and ends up back where he started!

How did it happen?

The North Pole

The Obvious Answer was the north pole, if you looked. Duh. Who knows how many times this problem has been redone.

But the real riddle is...

There are actually an infinite number of answers for where the man could have started from.Explain.

[Guest]

isn't it also possible to have 3 treadmills one pointed north one pointer south and one pointed west

[Magic_luver101]

he was at the North Pole <_<

[Guest]

For the south pole solution:

Let's say there's a latitude (a circle around the earth running east-west) that's exactly 1 mile in circumference. In other words, a point shortly north of the south pole that is 1 mile around the earth. If you start on any of the infinite points that are 1 mile north of that special latitude, then you can go 1 mile south and then 1 mile west to go all the way around the earth, then 1 mile north to be back where you started ;D

That works. And there are an infinite number of points exactly one mile north of that circle from which one could start.

Then if you take another circle which is one half mile in circumference, you would walk west one mile and make two trips around the circumference and then make your return trip north to your starting point. Again, there are infinite starting point for this solution.

And you could go on with a one third mile circle, one quarter mile circle, one fifth mile circle, and keep going for another infinite number of solutions.

This is all in addition to the obvious solution where you start at the North Pole.

Edited March 15, 2009 by Larry A

[Guest]

The riddle doesnt say the man is running on the earth just that he is running a mile in a given direction either north,south,east or west i think he is running on a sphere with a circumference of a mile and no matter what direction he runs in as long as he always runs an exact mile he will end up back where he started

[Guest]

he is hit by a bus which drags him back to where he started

[Guest]

he starts from a mile east of the north/south pole -.-

[Guest]

it is quite obvious to me... unless im wrong

he ran in a triangle!

[Guest]

WHO AGREES WITH MEH?

[Guest]

woah woah woah, you guys got it all wrong. If it is logically possible for him to be at the north pole then the answer to the true riddle is because when one runs over a spherical surface in the same manner described the same thing will happen regardless of where you started. Meaning of course that you have a infinite amount of origin points.

[unreality]

That works. And there are an infinite number of points exactly one mile north of that circle from which one could start.

Then if you take another circle which is one half mile in circumference, you would walk west one mile and make two trips around the circumference and then make your return trip north to your starting point. Again, there are infinite starting point for this solution.

And you could go on with a one third mile circle, one quarter mile circle, one fifth mile circle, and keep going for another infinite number of solutions.

This is all in addition to the obvious solution where you start at the North Pole.

nice! I hadn't considered the half-mile double-loop idea (and of course all the way down the reciprical integers)

[Guest]

This could not happen from the south pole as you cannot run 1 mile south when you are technically at the southernmost point of the globe. You could of course run in any direction from the North pole and it would be considered travelling south, then run 1 mile west and from here return to the north pole almost in a perfect equilateral triangle path of 1 mile per side. Simple.

[Guest]

it could actually happen anywhere

running north isn't necessarily u or straight instead of looking like a square w/out one side, the mans path would look like a equilateral triangle.

[Guest]

nice! I hadn't considered the half-mile double-loop idea (and of course all the way down the reciprical integers)

there is a problem with the half-mile part. if you start 1/2 mile north of the s. pole, and head south 1 mile, you wouldn't be going south after the pole, so u cant go one mile south, only 1/2 mile.

[Guest]

there is a problem with the half-mile part. if you start 1/2 mile north of the s. pole, and head south 1 mile, you wouldn't be going south after the pole, so u cant go one mile south, only 1/2 mile.

No no no, you're not starting half mile from the south pole.

Think of it this way:

There is a place on earth (close to the south pole) where the circumference of the earth is exactly 1 mile. That please is north of the south pole by x. Now, think of yourself starting 1 mile north of that place. You're now 1 mile + x north of the south pole. If you go south by a mile, you're not at the south pole yet (you're still x away). Then you go west for 1 mile, and are back to where you were before going west (since the circumference is exactly 1 mile) then you go north for 1 mile, and you are back to where you started. That's the "simple" south pole solution.

Then, to get more complex, think of a place on earth that has a circumference of exactly 1/2 a mile. That place would still be north of the south pole, but south of the circle we used in the previous example. Let's say that circle is y away from the south pole. If you start 1 mile north of that circle, you are 1 mile + y from the pole. Again, go south for a mile, and you're still y away from the pole. If you now travel west, you'll come back to your starting position after going half a mile. Since you have to walk a mile, you keep going, and return to that position AGAIN when you are done. Then you go north for 1 mile and you are back to fulfil the obligations of this problem.

So, any place that's a mile north of a whole fraction of a mile circumference, is fair game (1 mile, 1/2, 1/3, 1/4, etc). Each of these places are still north of the south pole (as every place on earth is) so you never have a problem with "running out of south".

[Guest]

There is something minor missing to the more precise "latitude solution" to the puzzle. I don't think it's been clarified yet. The spoiler contains the algebra to obtain an exact, mathematical answer, although the previous posts contains all the necessary ideas. Worthwhile to take a look if you're into math. So the same solution as some of the posters, just an attempt at an exact description of it.

Looking at the cross section of the Earth at the great circle connecting the north and south poles, we see that circumference of the latitudinal circles (i.e. the westing segment of the journey) near the south pole (actually, for all latitudes) is the sine of pi/2 minus the latitude, then times two pi radius of the Earth. Only some algebra is required to get to the solution below.

The "south pole solutions" are:

1 - R |arcsin (1/(2*pi*N*R))| [miles], north of the south pole as measured along the shortest path on the surface. N is an non-zero integer.

This solution is invariant with the sign of N because of the absolute value function.

I hope that this may be the correct and precise answer for a spherical Earth.

Edited March 19, 2010 by marsupialsoup

[Guest]

we can also consider a pyramid...he starts from the tip of the pyramid whose length of each side is greater than 1 mile....

although its only for the first part of the question...

Edited April 5, 2010 by fuser

[Guest]

[Guest]

There can be any place on earth that has a circle of radius 1 mile and 1 mile west would be walking on the circumference so there would be infinitely many answers.

[Mr. Chan]

Please forgive me if I am trying to act smart but it seems there's another possible answer, which doesn't involve the concepts of geography and calculations and gives this puzzle a kind of a new perspective.

Let us assume that this man is standing on a rotating disc. Let's take his start position as A and end position as B. Now it can easily occur that after the man went from A to B, the disc rotated in such a manner that original A and B coincide (the whole idea is that the platform he is running on, is mobile. Ain't it??)

Just a thought. Don't kill me on this guys!

[cakecookies2]

He ran a mile south, a mile west, and a mile north, but walked a mile east