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.

While I originally planned to focus on methods of solving, I have found more meaningful patterns about what justifies additional moves. I will focus the remaining time refining these conclusions and searching for patterns in the methods of solving. I have solved most of the concerns I mentioned in my last post by proving why many of those examples require extra moves. My next post will be my final post, in which I will summarize all of my research and include the scenarios that require additional moves.

I do wish I had had the time to work with both 4 by 4 and 5 by 5 grids, but I acknowledged when beginning my research that I may not progress as quickly as I hoped. I think I made substantial progress, though it turns out to be different than my original expectations. Math research is similar to other research in that it can definitely involve unexpected results, but its approach is vastly different. Throughout my project, it has been mainly trial and error. I had to work through many, many examples to even create theories to test, and then I had to work through more examples while attempting to disprove my own theories. It is much different than any science project I’ve ever done, but I am certainly glad for the experience. I think this was a good start on my journey through undergraduate math.

See you in my summary post!