[Guest]

While in a flat, dense and uniformly wooded forest, you get lost. You come across an old man, who unfortunately can't help you get out. However, he talks about a treasure to deliver to his wife. He has been searching for her cabin for days, and has given up. Since you will be wandering the woods, he wants you to carry the treasure and deliver it to his wife if you happen to stumble upon her cabin. The old man starts going on and on about the history of it, and it gets pretty boring, so you doze off. You start staring off into space, and when you snap back, you realize he is gone.

You get confused about what just happen, shrug it off, and then continue walking in a straight path for 1 mile. At this point, you get tired and decide to fall asleep. When you wake up, you are reminded of the old man you met earlier and his treasure. Realizing you didn't take the treasure with you, you wonder if he left it at that spot. There is a chance that he didn't leave anything also. But you want to check and make sure nothing was left behind.

You know that you traveled 1 mile in a straight line from the point where you met him. However, you just woke up, and have been tossing and turning. The forest is flat, dense and the trees are all uniform, so you don't remember which direction you came from.

There is not much time to waste before the sun goes down, so you want to take the path with the shortest distance. Assuming the treasure was not left behind, what is the distance you travel before you realize this - if you were traveling the shortest distance?

The obvious guess is 2pi+1, where you travel in a straight line for 1 mile, and then start traveling around in a circle of circumference 2pi. But the answer is not 2pi+1

What is it?

[Guest]

720pi?????

[Guest]

I could be very wrong on this, and math is not my forte, but I am going to give it a try.

If you walk the one mile to the edge of the circle, you should be able to see a fair bit in either direction well enough to determine if the treasure was left within the line of sight. For the sake of easy numbers, which I still seem to be calculating incorrectly, let's assume that you can see the entire circle by walking in a square whose four corners touch the edges of the circle. If this particular forest is still too dense to see the entire circle from a square path, then you may have to chose a pentagon, hexagon, septagon, octagon, or whatever. In any case, walking a straight line between any two points on the circle from which path you can see the treasure if it were left on the edge of the circle should reduce the distance traveled over the option of walking the actual perimeter of the circle.

So, since I am assuming the square is sufficient, once you have walked one mile in a straight line, you would then face the way you came from and turn 30^o in either direction. From there, I believe you would walk a distance of:

2(Square-Root(1² - (Sin30)²)), all the while, keeping an eye toward the perimeter of the circle. Once you have traveled that distance, make a 90^o turn in the direction opposite of the previous 30^o turn, and continue walking in the square pattern around the circle.

If my calculations are correct (and I don't think they are), you would travel a total of 7.9282032 some odd miles. Unfortunately, 2pi + 1 = 7.28318 some odd miles.

In short, the theory seems sound to me, but the numbers seem off.

End of rambling.....

[Guest]

None

There is not much time to waste before the sun goes down, so you want to take the path with the shortest distance. Assuming the treasure was not left behind, what is the distance you travel before you realize this - if you were traveling the shortest distance?

If you are assuming the treasure was not left behind then you would have it right? So there would be no need to travel because you would look and see that you used the treasure for a pillow to nap on.

[Guest]

None

There is not much time to waste before the sun goes down, so you want to take the path with the shortest distance. Assuming the treasure was not left behind, what is the distance you travel before you realize this - if you were traveling the shortest distance?

If you are assuming the treasure was not left behind then you would have it right? So there would be no need to travel because you would look and see that you used the treasure for a pillow to nap on.

Realizing you didn't take the treasure with you, you wonder if he left it at that spot.

I believe that the phrase Assuming the treasure was not left behind is intended to mean that you assume you don't find the treasure halfway through your search, forcing you to follow your entire planned search pattern.

[Guest]

I believe that the phrase Assuming the treasure was not left behind is intended to mean that you assume you don't find the treasure halfway through your search, forcing you to follow your entire planned search pattern.

I'm assuming you can't see any clues like the tracks you left or the direction of the sun, right?

[Guest]

I may have found where my math was wrong in my first guess:

Instead of turning 30^o, you would turn 45^o, which would make the total distance walked

4(2(square-root(1-Sin45))) + 1, which works out to be 6.65685425.... something. Less than the 7.28318 miles if you walk the entire circle.

[Prof. Templeton]

I may have found where my math was wrong in my first guess:

Instead of turning 30^o, you would turn 45^o, which would make the total distance walked

4(2(square-root(1-Sin45))) + 1, which works out to be 6.65685425.... something. Less than the 7.28318 miles if you walk the entire circle.

I would expect walking a square inscribed inside a circle of radius 1 would

4*sqrt2 plus the initial 1, which is what you now have. Walking any polygon inscibed inside a circle will be shorter than the circle itself, we would just need to know how far from the original circle are we allowed to stray, if at all. Walking on a square we could be as far as .3 miles from the original circle. Hard to see through the trees at that distance.

Edit: Clarification

Edited October 17, 2008 by Prof. Templeton

[Guest]

720pi?????

You would need to walk a mile back to the treasure. But you dont know which direction, so you would have to walk a mile and back to the start for every degree in a circle. So it would be 2 miles per trip. Times 360 degrees until you got back to the last place you started. Would be 720 miles, not 720pi.

So my final answer is 720 miles.

Which could never be completed in a day, but would assure you didnt miss anything.

[Guest]

I would expect walking a square inscribed inside a circle of radius 1 would

Edit: Clarification

4*sqrt2 plus the initial 1, which is what you now have. Walking any polygon inscibed inside a circle will be shorter than the circle itself, we would just need to know how far from the original circle are we allowed to stray, if at all. Walking on a square we could be as far as .3 miles from the original circle. Hard to see through the trees at that distance.

You are absolutely correct. As I said, Math isn't my forte. I was making it too hard.

if the square took you too far from the circle, you may need to to an octagon, or something.

[Guest]

Just a couple of ideas...

You are (still) dreaming, and therefore you only realize that the treasure was not left behind once you wake up, after which you would have traveled no distance at all.

Evidence that you are dreaming includes:

• The old man is married to his wife, therefore lives with her, and thus would have no problem finding her cabin.
  
• A cabin is incompatible with the density and uniformity of the forest.
  
• You somehow miss the old man's departure, which is odd enough to confuse you.
  
• Despite having gotten yourself lost, you somehow know that you've traveled exactly one mile.
  
• Travel in a straight line is incompatible with the density and uniformity of the forest.
  
• Upon realizing that night will soon fall, for some reason your first priority isn't to continue trying to find your way out of the forest, but to look for a treasure that you realize may not even be there.
  
  
  
  The main problems with this are:
  
  

The shortest distance between two points is a straight line, and therefore one mile is the shortest distance to where the treasure would be. The problem, of course, is that you don't know the direction from which you came. This being, traveling one mile in any direction and then proceeding around the circumference of a circle is the best strategy for finding where the treasure was left if it was left behind. Assuming the treasure was not left behind, however, given that the forest is so dense and uniform that you don't know the direction from which you came, you would also never know that you had traveled the full 2pi miles, and would therefore continue around the circumference of the circle indefinitely, never realizing that the treasure had not been left.

It assumes you do not have the means to mark the point at which you started around the circumference of the circle.
If you somehow have the means to calculate that you've traveled one mile, you might also have the means to calculate that you've traveled 2pi (or just over six and a quarter) miles.