# A Real Bear of a Problem

A hunter rises early one morning and walks one mile due south from his camp and spots a bear. He shoots the bear, wounding it and the bear goes due east one mile where the hunter finally catches up with him and finishes him off. The hunter drags his quarry exactly one mile due north back to his camp. What color was the bear?

You must explain your reasoning.

*Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.* - Ernst F. Schumacher

**Does the blood count?**

He shot him with a cauterizing laser. No blood, no mark.

**depends on which color is the light.**

Well that depends on the season.

*Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.* - Ernst F. Schumacher

Sheriff

[Looks like Jessica got it before me.]

[ June 02, 2003: Message edited by: Jason Menard ]

A Real Bear of a Problem

That's what they call UI issues here at work that I'm expected to solve!

**Devo 1:**Hey, I'm having a wierd problem with this servlet doing something-really-bizarre-or-other.

**Devo 2:**Sounds like that's a real Bear of a problem.

**Bear:**Sigh! (writes it down as an action item)

bear

**That's what they call UI issues here at work that I'm expected to solve!**

We all have our crosses to

**Bear**

**In a related question: if the hunter goes f-=39 and t+=2899, where does he end up?**

Hate to show my ignorance, but I'm not following your nomenclature. Remember the the last circular triangle I solved was probably circa 1970.

*Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.* - Ernst F. Schumacher

Sheriff

**I'm not following your nomenclature.**

That's OK, I just made it up. But there is a meaning, which will be reasonably self-evident if you look at it the right way. Well, maybe. Let's just say they're not the sort of spatial coordinates you're likely to be thinking of. Gotta think outside the box. Or sphere in this case.

"I'm not back." - Bill Harding, *Twister*

Sheriff

Originally posted by Jim Yingst:

Also, for anyone who hasn't seen this problem before: take the bear out of the picture. From the rest of Michael's problem statement, whereelsecould the hunter be?

If polar bears were not involved, his camp could be one mile north of the South Pole. Of course his walk one mile east would be a bit monotonous, and he would have to make sure he went one mile north on the correct heading.

Sheriff

At those points, the hunter walks south a mile, then east in a circle having a one mile circumference, then retraces his steps north to his starting point.

If you're having trouble picturing it, think of those wire easter egg dippers you find in easter egg coloring kits.

[ June 04, 2003: Message edited by: Roy Tock ]

Sheriff

Sheriff

Sheriff

Originally posted by Jim Yingst:

Well, almost. Where exactly do you end up for n = 1000, for example?

For those who were perplexed by the mysterious coordinates in my previous post - take a look at the URL of this page.

Here?

Sheriff

Sheriff

Sheriff

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Sheriff

Sheriff

What I'm 'trying' , to do is map all the circles whose circumference is 1/n miles. Then, once you've walked south a mile, to a point on one of these 1/n circles, you walk around the circle n times and then head north again.

Spot false dilemmas now, ask me how!

(If you're not on the edge, you're taking up too much room.)

Sheriff

**(Let's see if I can do it now!)**

1 + 1 / (2n * pi )

1 + 1 / (2n * pi )

*That's*what I was hoping to see. We don't want that hunter ending up on the other side of the world, do we? (Even if that's just a few miles away.)

**[M^2]: Ok, I'm getting lost here (pun intended). I don't think Roy's solution is correct and I'm not even sure what Bert is trying to do.**

Roy was doing the same sort of thing Bert has just explained - but also overlooked a factor of 2.

**At first thought it would seem that you would have to be 1 mile north of the circle that is cut by the plane parallel to and the same distance from the equator as the one in the northern hemisphere that was the path of our hunter going east. That circle should have a radius of R*sin(1/R). The circumference of that circle is 2*PI*R*sin(1/R) which is very close to 2*PI not 1**

as Roy stated. But would that circle do the trick? As you wander from the South Pole then north has a specific direction, from the South Pole all directions are north. What am I missing?

as Roy stated. But would that circle do the trick? As you wander from the South Pole then north has a specific direction, from the South Pole all directions are north. What am I missing?

Ummm... I'm not sure I followed that. As Bert said, we're looking for circles whose exact radius is 1/n, which means their exact (flat) radius is 1/(2n*PI). Due to the curvature of the earth there's technically an extra x/(sin x) factor here, where x is the angle from the south pole (in radians) but 1/(2n*PI) is close enough, all things considered. Let's just flatten the pole out and not worry about it.

[ June 29, 2003: Message edited by: Jim Yingst ]

"I'm not back." - Bill Harding, *Twister*

**Ummm... I'm not sure I followed that. As Bert said, we're looking for circles whose exact radius is 1/n, which means their exact (flat) radius is 1/(2n*PI). Due to the curvature of the earth there's technically an extra x/(sin x) factor here, where x is the angle from the south pole (in radians) but 1/(2n*PI) is close enough, all things considered. Let's just flatten the pole out and not worry about it.**

Maybe I don't understand what problem we are trying to solve. Are we trying to find any alternate location on the earth where a person can walk south one mile, east or west one mile and return to the

*exact*place of beginning by walking one mile north? If that is the problem, there is no alternate spot. To show this we need to define some criteria. First can we agree that in order to move in a true south or true north direction we will remain on the same longitude, in other words if I walk due north from Greenwich 100 miles, my lattitude will change but I will still be on the prime meridian? The same is true for east and west except that the lattitude will remain constant while the longitude changes. Next, there is an infinite set of longitudes on the face of the earth and they intersect at

*only*two points: the North and South poles. With that in mind, let's take a trip to Antarctica. Even though we can find a circle near the South Pole where we can

*almost*pull this off it becomes obvious that we cannot do the two things required: return to the

*exact*beginning and remain on the same longitude. The last leg of the journey would be slightly longer and ever so slightly west of north. This becomes much more apparent when the arc length of each leg of the spherical triangle is sufficiently large.

[ June 29, 2003: Message edited by: Michael Morris ]

Sheriff

Sheriff

**Maybe I don't understand what problem we are trying to solve. Are we trying to find any alternate location on the earth where a person can walk south one mile, east or west one mile and return to the exact place of beginning by walking one mile north? If that is the problem, there is no alternate spot. To show this we need to define some criteria. First can we agree that in order to move in a true south or true north direction we will remain on the same longitude, in other words if I walk due north from Greenwich 100 miles, my lattitude will change but I will still be on the prime meridian? The same is true for east and west except that the lattitude will remain constant while the longitude changes. Next, there is an infinite set of longitudes on the face of the earth and they intersect at only two points: the North and South poles.**

I agree with everything so far.

**With that in mind, let's take a trip to Antarctica.**

Well OK, if you're paying for it.

**Even though we can find a circle near the South Pole where we can almost pull this off it becomes obvious that we cannot do the two things required: return to the exact beginning and remain on the same longitude. The last leg of the journey would be slightly longer and ever so slightly west of north. This becomes much more apparent when the arc length of each leg of the spherical triangle is sufficiently large.**

Eh, I just don't see where this is coming from. Are you asserting that we can't find a circle such that after traveling 1 mile east (let's stick with east for the sake of argument) we end up exactly where we were before traveling 1 mile east? Or are you saying that even if our circle takes us right to the start of the circle again, the subsequent 1 mile north will diverge from the earlier 1 mile south?

Let's standardize some terminology here. The position the hunter starts at is point A. Traveling 1 mile south takes him to B. Traveling 1 mile east takes him to C. Traveling 1 mile north again takes him to D. OK - we're looking for solutions where A and D overlap exactly. For the solutions Bert, Roy and I are talking about, this is achieved by having B and C overlap exactly, which means all of AB and CD overlap exactly. In between is BC - this is a circle along a "parallel" (meaning a path of constant latitude, as it doesn't look very parallel when it's a circle). The circumference of this circle is exactly 1 - or more generally, 1/n. So that if you travel 1 mile "east" along it, you go all the way around the circle exactly n times, and arrive where you started, B = C.

So, where do these statements differ from your own mental picture?

"I'm not back." - Bill Harding, *Twister*

**Eh, I just don't see where this is coming from. Are you asserting that we can't find a circle such that after traveling 1 mile east (let's stick with east for the sake of argument) we end up exactly where we were before traveling 1 mile east? Or are you saying that even if our circle takes us right to the start of the circle again, the subsequent 1 mile north will diverge from the earlier 1 mile south?**

OK, I'm slow somtimes. Now, I see where you're going. Our hunter is taking a trip around the world. Let's hope it doesn't take him 80 days.

Sheriff

Sheriff

Originally posted by Jim Yingst:

Woo Hoo! I was afraid we were going to have to send a polar bear to Texas to take care of M^2 for us, but I guess not.

A polar bear wouldn't stand a chance against the fire ants and mosquitos in Texas.

Sheriff

The solution accepted was

1 + 1 / (2n * pi ) correct?

Is this the distance from the South Pole?

So the first 1 is the distance traveled north/south? But 1/(2n* pi) is the radius from the Earth's axis isn't it and not distance traveled from the South Pole?

Sorry I am still a bit confused I guess.

thanks

Sheriff

**So the first 1 is the distance traveled north/south? But 1/(2n* pi) is the radius from the Earth's axis isn't it and not distance traveled from the South Pole?**

You are correct. However they are very, very close to each other considering we're less than two miles from the south pole; the curvature of the earth is not that great. As I noted above:

Due to the curvature of the earth there's technically an extra x/(sin x) factor here, where x is the angle from the south pole (in radians) but 1/(2n*PI) is close enough, all things considered. Let's just flatten the pole out and not worry about it.

"I'm not back." - Bill Harding, *Twister*

**A hunter rises early one morning and walks one mile due south from his camp and spots a bear. He shoots the bear, wounding it and the bear goes due east one mile where the hunter finally catches up with him and finishes him off. The hunter drags his quarry exactly one mile due north back to his camp. What color was the bear?**

Here is my variation:

A hunter rises early one morning on Jan 1st, 2002 and walks one mile due north (using a regular compass) and then one mile due east. The hunter stops to rest, and a hungry bear attacks and kills the hunter. Just before the hunter is killed, he phones his son at the camp and screams "Son, I walked one mile due north and then one mile due east, here is where you can find me if am killed". The bear tastes the hunter, but decides not to eat him. Assume that the body of the hunter stays highly visible and well-preserved for a year.

On Jan 1st, 2003, the hunter's son decided to find his father and give him a proper burial. So he leaves the same camp, walks one mile due north (using a regular compass) and then one mile due east, just like his father instructed him. But his father is not there!

Why?

[ July 06, 2003: Message edited by: Eugene Kononov ]