Post 2: The math of quadrangles

In this section, I’ll try to explain the mathematical ideas behind my project topic. But to keep things readable (this is a blog post, not a textbook), I’m going to sacrifice some thoroughness in favor of clarity. That is to say, I’m going to gloss over a lot of details so you don’t need a math degree to read this.

Post 1: Where have I been?

So, I started my research on June 10. This, my first blog post, was written (is being written?) on July 3. That makes it… 23 days since I started. The question now becomes, “Where in the world have you been?”

Final Check-in — Placement of Numbers on a Grid

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:

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.