So the puzzler goes: A person spots a bear and follows it 1 mile due south. He looses the bear but follows the tracks 1 mile due east where he comes to the bear which chases him 1 mile due north at which point he is right where he started. What color was the bear?
The answer
The only place this could happen is at the north pole, so it’s a polar bear, so the answer is, “white.”
Could it also occur anywhere on the ‘latitude’ that is 1.25 miles north of the the south pole? If the person walked 1 mile due east (or west) at a point where the circumference is 1 mile then he ends up where he started and you can add 1 mile north to get the puzzler start, right?
Assuming I didn’t make an egregious error, that’s 1 1/4 mile north of the S pole. The radius of the earth where the diameter is 1mi is equal to the vertical distance from the pole to that level through the earth’s mantel (assuming a spherical earth) so the surface distance from the pole to the latitude we’re talking about is 1/4th of the circumference, or 1/4th of mile. Right?
Let’s take a flat earth as an approximation. You want to know how far from the South pole you have to get to have a circle whose circumference is 1 mi.
1 mi. = C = 2 pi r
So r = 1 mi/(2 pi) = approx 0.16 miles
If you were 1 + 1/(2 pi) miles from the South pole, and traveled one mile south, one mile east and then one mile north you would end up back where you started.
You could also start one mile north of a circle whose circumference is 1/n miles and it would work; you’d just travel the circle n times. This would be 1 + 1/(2 pi n) miles north of the South pole.
Except of course, there are no polar bears at the South pole.
… and I would like to point out that whenever there is a brain teaser that asks for the colour of a bear, the answer is always “white”. You don’t need to read the rest of the teaser
That was my first approximation, but wouldn’t a sphere be a better one? If you draw a line from pole to pole, would the distance from the line to the surface (along a line at right angles to the pole line- i.e. radius of the circumference) be equal to the distance along the polar line from the pole to the radius? Which should mean that the surface distance is actually 1/4 of the circumference rather than 1/2pi, right?
And how exactly does one go south from the South Pole? Well, besides the fact that the South Pole and magnetic south are not the same, but then again these teasers never specify that part.
Not to be picky, but polar bears’ fur isn’t white - it’s actually transparent… the “white” occurs because of the refraction of light through the hairs… or somesuch
Naw, the orignial point of my OP was to figure out how far north of the south pole you have to be to fulfill the 1mile south/1mile east/1mile north and end up at the same place.
So polar bear fur is white from a distance and translucent up (really) close. And that’s why we pay those McMurdo Station scientists good tax payer money to stay at the frigid north pole studying polar bears and penquins.*
PC
*Next you’re going to tell me penquins aren’t really black & white!
You people are making this much more difficult than it is. Read the OP- You start by going 1 mile south, then 1 mile east, then 1 mile back north to starting point. The ONLY place this is possible is the north pole. Once you go 1 mile south from the north pole, then no matter how far you go east or west you will always be 1 mile from the north pole. Going east or west any distance would describe an arc (or a circle if you go far enough) that is 1 mile equidistant from the pole. So you could go 1 mile south, 100 miles east (going around the earth a little under 16 times) and 1 mile north will take you back to the pole. The question has nothing to do with the south pole so penguins vs. polar bears is irrelevant. The only nitpick I have with the OP is the misspelling: you don’t “loose” a bear, you “lose” it.
No, of course not. I mean, the approximation is going to be somewhat close, right? 1/4 is .25, whereas 1/2pi is .16
On a sphere (which the Earth is not–it bulges 20 km farther at the equator than the pole, but we’ll ignore that) of radius R, at a latitude (we can assume that the geodetic latitude is the same as geocentric latitude since we are assuming the Earth is a sphere) L south, the radius of the circle formed by following a constant easterly direction is R times the cosine of L. The circumference is 2pi x R x cos(L)–and you want that to be an even fraction 1/N of 1 mile. In other words, L is the arccos(1/(2pi x R x N)). Mean radius of the Earth is about 6371 kilometers or 3959 miles–which we can use for our purposes.
The distance from the pole to that circle is (90-L) x pi/180 x R, where the pi/180 is necessary to convert degrees into radians. Then, you’d add a mile to get the starting point.
I calculate that starting point (for N equal one–that is, you’d only go around the circle once) to be 1.1592 miles from the South Pole, very close to DrMatrix’s answer. As you’d expect, since the Earth is so large the curvature is not going to make much of a difference. In fact, if we hadn’t rounded off our answers, his answer and mine would agree for ten decimal places.
I deleted the first sentence, but when I said
