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:

The most general pattern is a possible equation for the maximum number of required moves, denoted “M,” in relation to the bandwidth (or the largest distance of any number to its original position), denoted “B”. My advisor had mentioned that there is conjecture that M=2B (or that the maximum required moves to solve any example would be twice the largest distance from the original position). However, I believe I have two examples that do not fit this theory. I have been able to solve all of my examples in 5 moves or less. Two examples that (I have found) require five moves only have a distance of 2 from the original position! This suggests that perhaps **M=2B+1**. However, these two examples are my shakiest proofs. Because I am trying to prove that numbers with the distance two can only be solved in five moves, there are a *lot* of possibilities. As such, I am unsure that I exhausted every possible route to solving the examples within four moves. However, I am still in the process of proving that the following examples require 5 moves:

1 6 7 3 6 9

8 9 2 8 5 2

3 4 5 1 4 7

You can see that all of the numbers are only two away from the original position of:

1 4 7

2 5 8

3 6 9

Yet, I have only been able to solve these examples in 5 moves. Thus, I believe **M=2D+1 **may be the more accurate equation.

I also found other patterns about what kinds of positions may require additional moves:

First, if two numbers are reflected across the same middle position, the solution requires at least **3** moves. Examples of this position include:

3 _ _ _ 6 _ _ 6 _

_ _ _ or _ _ _ or 8 _ _

1 _ _ _ 4 _ _ _ _

Second, if three evens are reflected, the solution requires at least **4** moves. Examples of this position include:

_ _ _ _ 6 _

8 _ 2 or _ _ 2

_ 4 _ _ 4 _

Third, if all four corners are at a distance of 4, two opposing evens are in their original position, and either the five is misplaced or the two remaining evens are reflected, the solution requires at least **5** moves. Examples of this position include:

9 **5** 3 9 **6** 3 9 4 3

2 _ 8 or 2 5 8 or _ _ **5**

7 _ 1 7 **4** 1 7 6 1

(This last pattern is difficult to explain, so please bear with me.) The last pattern involves two numbers (one corner and one even) that are next to each other in the original position. If, in the example, these two numbers are next to each other on an adjacent side (as if translated along the perimeter or as if the perimeter had rotated) and also are adjacent to the 5 or any of the numbers that belong on the opposite side of the current position of these numbers, then the solution requires at least **3** moves.

Examples of this position include:

3 5 _ 3 7 _ 3 8 _ 3 9 _

6 _ _ or 6 _ _ or 6 _ _ or 6 _ _

_ _ _ _ _ _ _ _ _ _ _ _

I have many other potential patterns and unjustified solutions, but these are the most significant patterns I was able to find in my research. Thank you for reading, and I hope to see you at the Summer Research Showcase!