The answer to a riddle (is not what you would expect)

Now, you may have seen the riddle I posted yesterday, the one that I've been claiming to everyone that I had worked on all day yesterday. I just posted a link, but here is the riddle:

An old parchment describes the location of buried treasure: "On the island there are only two trees, A and B, and the remains of a gallows. Start at the gallows and count the steps required to walk in a straight line to tree A. At the tree turn 90 degrees to the left and then walk forward the same number of steps. At the point where you top drive a spike into the ground. Now return to the gallows and walk in a straight line, counting your steps, to tree B. When you reach the tree, turn 90 degrees to the right and take the same number of steps forward, placing another spike at the point where you stop. Dig at the point exactly halfway between the spikes and you will find the treasure." However, our hero when he gets to the island finds the gallows missing. Is there any way he can still get to the treasure?

I'm going to give you the website's answer in the next paragraph, so fair warning if you're coming here today and you're interested in solving this yourself. But first, a "funny" story: The reason I became so obsessed with this is because I was convinced that the traditional answer to this puzzle is bullshit. And it was the first thing I sort of proved yesterday. Then I realized I did something wrong, the answer was right, and spent an entire day working on trying to get a rule set up. The last thing I did yesterday before giving up on the post and posting what I did instead was realize that my initial hypothesis was true. Yup, it took me nine hours to go in a full circle. This is type of stuff that needs to be taught to aspiring scientists and mathematicians.

Now that I've made some space and allowed some of you to work on the puzzle, I'll give you both answers to the puzzles. Everyone's answer to this is: Don't think too hard on this. A simple experiment with paper, pencil and a ruler will show that the treasure is always in the same spot no matter where the gallows is. Which is true, sort of.

You are about to witness the reason why I did so poorly in high school math. On a piece of paper, label two points, A and B.  They can be at any distance, they can be at an angle, whatever floats your boat. With any random third point, following the rules (or algorithm if you prefer) will bring you to the exact same point. That is correct.

Try to actually visualize this in the real world though. Imagine yourself on an island, with only two trees. Which tree is A, and which tree is B? The riddle doesn't describe a map, it just says it "describes the location of the treasure." After working with the real answer all day, I discovered a fairly simple rule that bypasses all the crap of the directions.

Stand in a straight line with the trees, in such a way the tree in front overlaps the tree in back. With this setup in mind, the tree in front will be tree A. The treasure will always be to the right of tree A. More specifically, find the distance halfway between the trees, turn to the right of tree A, and walk that distance again. No matter what the distance is, as long as you follow the rules in the parchment, the treasure will always be here. But do you see the practical problem here?

How do you tell which is tree A, and which is tree B? A simple experiment with pencil and paper will demonstrate what I am talking about. 2 points on a graph, approximately 5 units apart. The one on the right is A, and the one on the left is B. According to the rules set out by the parchment, the treasure will be above the trees. But take the same dots, and change the names, so lefty is A and righty is B. Now the treasure is south of the dots. Same distance, same rules. 2 different answers.

The reason why I did poorly in math class was because I was the smartass who asked "How am I going to us this in the real world?" and because I thought the real world examples they used were poorly designed. I have a story about triangles and men who put up circus tents that I like to tell. Anyways, there is their answer, and there is my answer. I have a ton of math to explain why that I'm working on at the moment, but there is still a lot of unknowns in it that I'm trying to explore. For example, squares, right triangles and the fact the quadratic formula is the only formula I can think of with a positive and negative answer keeps popping up in this work time and time again. Until then, polka!