Tuesday Puzzle

Re: Tuesday Puzzle

using a sketchpad, I THINK I know the answer. But verifying it does not look easy. Something to work on at the lunch hour.

intuition failed me

Back to the scratch paper.

Timely answer

Well, lunch helped the thought process.

Answering in a riddle, if it can be done, one way involves the current time and an hour and a half from now, although another would have involved the time three hours ago and 4.5 hours ago.

In fact, the real challenge is to determine if it can be done, since assuming the possibility leads to a multitude of possible solutions. The consistency or inconsistency of those solutions provided the path to my answer.

hint

You can get it to two possible locations. If the first is not it, then the second will be.

That shoots down my answer

I thought I could walk right to it and start digging, with no trial and error or pacing off distances. Gotta think about this some more...

Two possible?

I can't figure out that part of your hint, and am becoming more convinced that there is only one possible location, which I can find. When you are ready, I will post it, and get you to show the other possibility.

Hmmm.

I haven't had much time for puzzles today (too many real-world puzzles to figure out). But with the rules as stated, I think there is only one place it could be.

My solution

Maybe you have a better one? If you can get it to one location, then you did better than I did.

Consider the oak tree to be at point (0,0), the palm tree to be at (p,0), and the gallows to be at (x,y). The location of the first stake is (p+y,p-x). The location of the second stake is (-y,x). The midpoint is ((p+y-y)/2,(p-x-(-x))/2) = (p/2,p/2)

The treasure chest can be in one of two places. Find the midpoint of the two trees and walk in either direction on the line perpendicular to the line that contains both trees. Stop after walking half the distance between the two trees and dig. If the treasue isn't there then walk the other direction from the midpoint of the two trees and it must be there.

My solution(s)

1) Stick a stake in the ground (anywhere) and label it "G". Then follow the original instructions, substituting "the stake labeled G" for "the gallows". IF there is a solution, that means that the location of the gallows is irrelevant, so putting it anywhere gives the same result. (Proved later)

2) This is the "timely solution" I mentioned earlier. It is based on case (1), with the gallows under the palm tree. Stand with your back to the palm tree, facing in the direction that sets the oak tree ahead and 45 degrees to the right of you (at your "1:30"). Pick a point on the horizon, and walk directly toward it until the oak tree is at your 3:00, stop and dig up the treasure.

Proving there is only one possible point, no matter where the gallows is. For simplicity of explanation and no loss of generality, assume that the Oak is directly east of the Palm. If the gallows is under the palm, then P' (the point staked out after walking form the gallows to the palm, then turning right) is the palm, while O' is directly north of the Oak at a distance equal to the distance between the trees. The treasure will be 1/2 way between the two trees on the W-E axis, and half the distance between the trees North of the W-E axis.

If you offset the gallows by x (Eastward), you will move P' Northward by x and O' Southward by x, not changing the treasure point.

If you move the gallows north by y, you will move P' West by y and O' east by y, again not affecting the treasure point.

Obviously, any relocation of the gallows can be done in two steps described above, so teh treasure point does not change.

Whoops. Got my trees reversed

Change oak to palm and vice-versa throughout my explanation. Was trying to follow Dan's explanation, and was having a tough time until I went back to the original and found out I was remembering the wrong direction to turn at each tree. But I think my explanation works with that change.

Sorry.

Re: My solution

I agree with your algebra, but it yields only one possible point,

(p/2, p/2).

If you walk from the palm tree half-way to the Oak tree, you have to turn right to get to (p/2,p/2) (unless I have gotten those directions all twisted around in my head again!)

Good puzzle!

Similar approach

My approach was similar to Alex's #2. You can start at either tree and follow the original directions. Depending on which tree you pick, half of the directions reduce to zero travel, so the starting tree becomes one of the endpoints also.

Alex's 'timely' solution simplifies finding the midpoint.

I believe that the starting point makes no difference, but I don't have a formal proof of that.

I'd Get a GPS

unfortunately ;-)

A GPS only helps if you know where you are going, or to tell you where you have been. Not much help for an unknown destination. ;-)

Re: unfortunately ;-)

How about a metal detector?

Re: hint

Due to a family emergency I missed the posting of the problem on Tuesday.

The only thing I have to add is that whoever "invented" the problem had the so called Vecten system in mind. It is a well known set of facts

in euclidean geometry stemming from a triangle with squares constructed on its sides.