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# HOW DID IT HAPPEN???

Best Answer bonanova, 24 September 2007 - 03:12 AM

Yep. I presume you mean 1 mile north of south pole?

doesn't work: when you get to the South pole, how do you run West?

There's an infinite number of circles around the South Pole where he could have started.

does: for example, any point on the circle (1 + 1/2pi) miles from the South Pole.

After going South 1 mile, you're (1/2pi) miles from the Pole,
which allows you to run West 1 mile [1 lap of a 1-mile circumference circle]
and be able to go a mile North to the starting point.

As Martini noted, there is an infinite number of starting distances:
1 + 1/2Npi miles North of the South pole where N is any positive integer.
N is then the number of circular laps in your westerly mile.

e.g. N=5280 - you'd run 5280 laps around a 1-foot circumference circle.

Here's a counter question - why can't N be negative?
i.e. start closer than a mile - you could still do N laps Go to the full post

70 replies to this topic

### #11 roolstar

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Posted 30 January 2008 - 01:45 PM

You allow 1 lap around a 1 mile circumference circle to serve as the westerly leg of the journey when in fact the direction of the run is changing from the moment the 2nd lap begins. It may start westerly but gradualy changes to NW, NNW, N, NNE, NE,ENE, E... etc.

Only the North Pole truly satisfies the constraints of this riddle.

I think I see what you mean here, but look at it this way:

If you're inside a helicopter about 50m from the ground and hovering just over the South pole.
If you are watching this guy trying to run 1 mile west at about 1/2pi miles from the south pole, you will see him go in a circle and would easilly feel he is moving from NW, NNW, N, NNE, NE,ENE, E... etc. just like you posted.

However, this is merely an illusion, in fact HIS compass will assure him he is going West all the time, no matter how many laps he ran...
In fact any perpendicular direction to any line issued from the south pole to any point on the globe is a EAST or WEST direction...
And he will be running in circle(s) around the north pole...

As for Bonanova's formula 1 + 1/2Npi it is a approximation (a very close one to reality), since he is assuming the small distance of 1 + 1/2Pi is too small to consider the earth's curvature. You can reach the same answer with a simple figure on a plane (piece of paper), and the rest is simple geometry...
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### #12 iamauen

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Posted 20 February 2008 - 06:08 PM

Bonanova,
I'm afraid you are inconrrect in your (1+1/2pi) miles. The circumferance of a circle is ∏D (Pi * the diameter of a circle).
After you walked south for one mile, then you would be 1/2∏ miles from the South Pole. This would leave a circle with a cicumfernce of 9.86miles. (1/2∏ = 1.57. 1.57*2 = 3.14. 3.14 * ∏ = 9.86.) You obviously can't walk 1 mile of your 9.86 mile circle and then go one mile north and end up back where you started. I would assume that you meant you need to start (1+ ((1/2)/∏)) miles from the south pole. After you walked one mile south, you would be left with a circle that has a circumference of 1 mile.

Adding your unknown interger (N) into the equation would only work if the circle your are left with, after walkgin south one mile, had a circumference of 1 mile or a non-complex, fraction of a mile with 1 as the numerator; i.e. 1/2 mile circumference, you would walk 2 laps to go one mile west. 1/3 mile you would walk 3 laps. 1/4 mile, 4 laps, etc...
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### #13 try n take me on

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Posted 20 February 2008 - 06:33 PM

Why would you change directions? Your compass stop working or something?

Starting from any point on earth other than the NP or SP, you can walk as far as you wish in a westerly direction.
The circular route in the answer is a path of constant latitude.

Well its obviously the north pole if he was running around the world-u see he runs south(south pole) runs west(pointless info.) and runs north(north pole)
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### #14 bonanova

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Posted 20 February 2008 - 10:50 PM

Bonanova,
I'm afraid you are inconrrect in your (1+1/2pi) miles. The circumferance of a circle is ∏D (Pi * the diameter of a circle).
After you walked south for one mile, then you would be 1/2∏ miles from the South Pole. This would leave a circle with a cicumfernce of 9.86miles. (1/2∏ = 1.57. 1.57*2 = 3.14. 3.14 * ∏ = 9.86.) You obviously can't walk 1 mile of your 9.86 mile circle and then go one mile north and end up back where you started. I would assume that you meant you need to start (1+ ((1/2)/∏)) miles from the south pole. After you walked one mile south, you would be left with a circle that has a circumference of 1 mile.

Adding your unknown interger (N) into the equation would only work if the circle your are left with, after walkgin south one mile, had a circumference of 1 mile or a non-complex, fraction of a mile with 1 as the numerator; i.e. 1/2 mile circumference, you would walk 2 laps to go one mile west. 1/3 mile you would walk 3 laps. 1/4 mile, 4 laps, etc...

Hint: Take a long look at the number 9.86 - you'll recognize it as pi2.
Put pi [and N] into the denominator of my fraction.
OK now? http://brainden.com/...O_DIR#/wink.gif
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### #15 Dragon

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Posted 21 February 2008 - 07:14 PM

He started at the north pole. Runs a mile south to the sout pole. Runs a mile west, then a mile north back to the north pole.
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### #16 Shaftius

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Posted 22 February 2008 - 07:54 PM

He could be on a treadmill with the treadmill oriented south, he runs a mile, gets off, turns the treadmill, runs a mile, turns the treadmill, runs a mile.
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### #17 Jase

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Posted 26 February 2008 - 04:58 PM

As far as we know, there are an infinite number of other planets, therefore an infinite number of north poles. Theoretically the runner could start from any north pole throughout the universe.
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### #18 Tyvlen

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Posted 26 February 2008 - 11:14 PM

well, north pole did not come to mind, but i thought of an island 1mile in diameter
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### #19 cranberries

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Posted 03 March 2008 - 04:55 AM

He could be on a treadmill with the treadmill oriented south, he runs a mile, gets off, turns the treadmill, runs a mile, turns the treadmill, runs a mile.

the treadmill is a possibility,all though im not so sure its that easy to move a treadmill, or he could have started at a bus stop, gotten on the bus, rode the bus for one mile east, gotten off, then ran 1 mile south, 1mile west, and 1 mile north.
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Posted 03 March 2008 - 12:04 PM

he gets a ride, rides a mile east.

takes the train east

rides a bike east

gets abducted from aliens and wakes up where he started.... ha
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