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October 18th, 2013, 03:41 AM  #1 
Member Joined: Sep 2013 Posts: 84 Thanks: 2  You are standing somewhere on the earth...
Hi You are standing somewhere on the earth and go 10 km south then go 10 km east then go 10 km north and you get back to the start. Where is the start? There should be 2 solutions, one of them is the north pole(?). Can someone explain this to me? I just can't imagine it. And what is the other solution? 
October 18th, 2013, 04:15 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: You are standing somewhere on the earth...
When you are at the north pole any direction you take is SOUTH. The rules given in the problem will make you move in a triangle (non  flat) Starting at the NORTH POLE , you walk backwards for 10 km always facing north , then you turn 90º clockwise to the right and face EAST and walk 10 km , your distance from the north pole remains 10 km because you are walking along a line of latitude , then you face NORTH and walk 10 km , you're back at the pole where you started , a feature of the geometry on the sphere , NON  EUCLIDEAN. The other answer has to be the SOUTH POLE , but ... how to go south from the south pole? 
October 18th, 2013, 06:13 AM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Re: You are standing somewhere on the earth... Hello, ricsi046! Quote:
If you can't visualize the North Pole solution, you probably won't understand the South Pole solution either. Somewhere near the South Pole, there is a circle of latitude which is exactly 10 km in circumference. Select a point 10 km north of that circle and start there. You walk 10 km south and reach that circle. You walk 10 km east and walk around the earth. Then you walk 10 km north to your starting point. There are many other starting points. Somewhere near the South Pole, there is a circle of latitude which is exactly 5 km in circumference, Select a point 10 km north of that circle and start there. You walk 10 km south and reach that circle. You walk 10 km east and walk twice around the earth. Then you walk 10 km north to your starting point.  
October 18th, 2013, 10:05 AM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: You are standing somewhere on the earth...
Absolutely fantastic [color=#00FF00]Soroban[/color] , I never would have thought of that ... so ... there are an infinite number of ways to do this ... you just have to make sure that walking East for 10 km brings you back to the same latitude position so North (or South) won't be offset. I nominate [color=#00FF00]Soroban [/color] for the MMF Apollonius of Perga Award 2013 THE GREAT GEOMETER http://wwwhistory.mcs.stand.ac.uk/Bio ... onius.html 
October 18th, 2013, 10:19 AM  #5 
Member Joined: Sep 2013 Posts: 84 Thanks: 2  Re: You are standing somewhere on the earth...
Thanks, I know it now; the problem was that I thought when you need to go north you go straight upwards not straight to the northern pole (point). http://tavmjong.free.fr/INKSCAPE/MANUAL ... Sphere.png This picture helped me imagining it, I just couldn't see it in plane. I still need to learn really much. 
October 18th, 2013, 10:49 AM  #6  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: You are standing somewhere on the earth... Quote:
 

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