[Guest]

You begin your journey starting at the equator. You travel 400 miles due south, then 400 miles due east, then 400 miles due north and finally 400 miles due west. Amazingly, despite travelling on completely level ground, you are not back where you started.

1. Why?

2. How far away are you from your starting point?

No point trying #2 if you havn't figured out #1 .

Can be done with simple trigonometry and good spatial awareness.

[Guest]

depending on where on the equator, if you were to follow exact magnetic north/south, etc, you would actually travel slightly SE or SW. Therefore making it impossible to calculate exactly how far from the start you were without knowing where on the equator you are.

[Guest]

You begin your journey starting at the equator. You travel 400 miles due south, then 400 miles due east, then 400 miles due north and finally 400 miles due west. Amazingly, despite travelling on completely level ground, you are not back where you started.

1. Why?

2. How far away are you from your starting point?

No point trying #2 if you havn't figured out #1 .

Can be done with simple trigonometry and good spatial awareness.

1) Curvature of the earth.

2) Sorry, my skills have collected WAY too much dust to bother with that one.

Also, is this a repost? This one sounds kinda familiar, or else I probably wouldn't have even gotten #1

[Guest]

27.66 miles East of your starting point along the equater

[Guest]

Sorry ^---- That one is wrong.

6.557 Miles due east on the equater from starting point.

[bushindo]

1) You won't return to the point of origin because the earth is a sphere, or close enough to it. When you travel along the second leg of the trip (400 miles due east), it takes a shorter amount of time to circumnavigate the earth. The problem is more obvious if we exaggerate the number of miles travel. Suppose we travel 2000 miles in each leg, then it is clear why we won't return to the point of origin.

2) The radius of the earth at the equator (latitude = 0) is r = 3963.2 miles, with diameter d = 24901.6 miles

After going 400 miles south and then traveling due east, we'll be traveling along a smaller circle at a constant latitude theta = 5.78 S. This is because

length of first leg = r * theta * pi / 180

400 = r * theta * pi / 180

5.78 = theta

The radius h of a circle at constant latitude 5.78 S is

h = r sin(90 - theta )

h = 3943.036

The distance x needed to return to the origin on the last leg can be found using the proportional relationship

x/r = 400/h

x = 402.046

So you need to travel about 2.046 miles extra on the last leg to get home.

[Guest]

ok first I horable at math so i am sitting this one out but I just want to understand why the starting point on the equator is so important. why does it mater where in the equater you start?

[Guest]

That was just my answer that I still like better. Requires far less geometry/trig. :-) "Impossible" is my favorite solution to math problems.

[Guest]

I don't understand the relation of distance to time. 400 miles is 400 miles no matter how long in time it takes.

[Guest]

even with the math i still do not understand how you would reach a different point from which you start if you are traveling in a square? Assuming the math is right of course. I still dont understand how this riddle is possible.

[Guest]

1. North Pole

2. 400 Miles

[Guest]

1) You won't return to the point of origin because the earth is a sphere, or close enough to it. When you travel along the second leg of the trip (400 miles due east), it takes a shorter amount of time to circumnavigate the earth. The problem is more obvious if we exaggerate the number of miles travel. Suppose we travel 2000 miles in each leg, then it is clear why we won't return to the point of origin.

2) The radius of the earth at the equator (latitude = 0) is r = 3963.2 miles, with diameter d = 24901.6 miles

After going 400 miles south and then traveling due east, we'll be traveling along a smaller circle at a constant latitude theta = 5.78 S. This is because

length of first leg = r * theta * pi / 180

400 = r * theta * pi / 180

5.78 = theta

The radius h of a circle at constant latitude 5.78 S is

h = r sin(90 - theta )

h = 3943.036

The distance x needed to return to the origin on the last leg can be found using the proportional relationship

x/r = 400/h

x = 402.046

So you need to travel about 2.046 miles extra on the last leg to get home.

Ecellent .

[Guest]

But the circumference of the earth 400 mile south is smaller then it is at the equater. Therfore you travel further in degrees. walking north expands the distance.

[Guest]

ok that would make a difference thank you for clearing that up.

[Guest]

But the circumference of the earth 400 mile south is smaller then it is at the equater. Therfore you travel further in degrees. walking north expands the distance.

That is exactly why you get a bigger answer than 0 miles... because when you walk back north the angle stays the same but because the circumference is bigger at the equator this angle is now equivalent of more than 400 miles.

[Guest]

The circumference of the earth around the equator is 24783mi.

The circumference of the earth north to south pole is 24901mi.

Therefore, unless we know the exact dimensions of the earth (which is contradicted by "completely level ground," meaning a flat plane) this riddle is impossible.

It would be different if the riddle was "on a perfect, planet-sized sphere measuring 24783 miles around".

[Guest]

The difference in diameters of the earth is so small that the earth can be assumed spherical and the results will be almost exactly the same.

[Pickett]

even with the math i still do not understand how you would reach a different point from which you start if you are traveling in a square? Assuming the math is right of course. I still dont understand how this riddle is possible.

Since no one has really answered your question on how this is possible...it's simply because you're thinking in Euclidean Geometry (which is understandable, since that's what we are all taught from grade school), but since we are talking about the surface of the earth (assuming a perfect sphere)...we have to use

Spherical Geometry (this link has a good image showing a triangle in spherical geometry with sum of angles greater than 180 degrees...which is impossible in Euclidean Geometry). In short, you are not travelling in a "square" so to speak...you are traveling south (assuming a meaning of heading directly towards the south pole) around a curved surface 400mi, then east (assuming parallell to the equator) around it 400mi, then north (again, assuming directly towards the north pole) 400mi, and because of the curvature of the surface, you are actually more than 400mi to the east of your original starting point.

It's very similar to the old riddle saying, you're somewhere on the earth and you travel south 100 miles, then east 100 miles, and finally north 100 miles...and somehow you ended up exactly where you started...how is this possible. The answer is you are at the north pole (use the picture on that link above to a visual reference)

Oh, how I miss my non-Euclidean Geometry class...Hope this helped.