Math research is a fickle art, as I learned this summer. Each time I thought I was on one track, my research would lead me to another. As I wrap up my time with this research, I have found multiple patterns but disproved *many* others. However, all of my patterns are different than the original patterns for which I was looking. Of course, the few patterns I think I have proven may not be fully proven. Before the presentation in September, my advisor and I plan to more thoroughly review my proofs. However, I believe I have proven a few patterns that I will summarize below:

## Final Check-in — Placement of Numbers on a Grid

## Second Check-in — Placement of Numbers on a Grid

As the time for my project dwindles, I am still working with 3 by 3 grids. So far, I have been able to identify multiple specific scenarios that require more moves than are justified by the distance from original positions. I believe I have proven most of these scenarios, and with this, I have justified 90% of my examples. I have about 10 examples left that I have proven to require more moves than their distance, but I have not identified what patterns may cause this or if any of these examples have any bearing on other possible grid orientations.

## First Check-in — Placement of Numbers on a Grid

Beginning with 2 by 2 grids was surprisingly simple, with only 24 possible positions. Each grid contained four digits, and the goal was to return each digit to the original position using the same number of moves as the distance from the original position.

## Abstract — Placement of Numbers on a Grid

With increasing dependency on technology and data storage, efficient and searchable storage methods grow in importance. My project involves sorting data placement on a grid model. My project uses a 5 by 5 grid, whose 25 points are labeled 1 through 25. The goal is to return scattered numbers to their original positions in the fewest number of moves. I will also try situations in which each number is within a bounded distance from its original position. I will begin with a 2 by 2 grid and progress to grids with more points, looking for helpful patterns to efficiently return all numbers to their original positions. The ultimate goal is to find an optimal sorting method for grids of any size.