Chapter 1: Where is the arcade going?

My grandma like a small child on Christmas morning painfully awaited for an earthquake to see where the arcade is going. Then it happened, the ground shook, placing herself under the kitchen table my grandma grinned uncontrollably, the anticipation was even more unbearable. She ran out the door and measured how far the arcade had moved. She sent me the following image with this letter:

"The arcade has moved 3.36 miles from where it was before. I have explored a couple of things with the equation x

^{n}+ y^{n}= 1 and I have written separetly why n cannot be zero. So unfortunetly I will never know where the arcade was before I began my observations. Since this equation has worked so far for me, I decided to use geometry sketchpad to make a conjecture on the movement of the arcade. I created another image with the n being from 1 to 10 and I was able to make the following chart:

n Distance Moved in Miles2 3.363 1.404 0.775 0.486 0.337 0.248 0.189 0.1410 0.12

Click here to see a video of how the Arcade is moving

Because of this information that my grandma collected she was able to make a conjecture that the arcade is getting closer and closer to the point (1,1) on the coordinate plane. This point is called a . She had a simple proof for why x and y cannot be (1,1). With this proof my grandma felt confident that she will be able to live off of the arcade for many more years.