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.

## Modular Matrices

My research will be on one of the many facets of matrix manipulation seen in linear algebra. All matrices have a rank, which is the amount of independent vectors that make up its column or row space. A vector is independent if it a not a linear combination of previous vectors or in simpler terms there are no combination of other vectors in the matrix that can be added together to equal this independent vector. A quantifiable rank for a single can be obtained after some minor calculations and manipulations. More specifically I will see what happens when a given n x n matrix, has all of its values reduced to the entry mod(n). To find the rank of a single matrix as stated is just a matter of some minor calculations, but the description of all of the possible different n x n matrices for any n is significantly more complex. When the modular operation occurs, it causes changes in nearly every property of the matrix including but not limited to rank, the determinant of sub matrices, coefficients of the characteristic polynomial, and the eigenvalues. These properties appear to be described by an equation of group of equations. These equations will be the focal point of my research and will be specifically focused on determining the rank of all square matrix reduced mod(n) for any n.