# Which state is the "most square"?

For reasons that I really can’t quite explain, I became determined to answer the question “which state is the most square?”.

How did this question come up in the first place? My wife’s dad asked my five year old son what he thought the most square state is while he was looking at his state map. I figured it had to be one of either Colorado or Wyoming, but my son (who loves states) immediately answered “New Mexico”. New Mexico hadn’t really crossed my mind, but the more I looked at it, the more I thought he might actually be right!

Regardless, I decided I would try to make a small project out of it.

# How do you define “most square”?

I don’t think there’s a right or wrong answer here; it’s going to be a bit subjective. I ended up doing something fairly simple: I just counted the fraction of the state’s area that overlaps with a square of roughly the same volume.

To lay that out in a bit more detail:

- Make a square that has roughly the same area as the state
- Overlay the square on top of the state and measure how much of the area overlaps
- Find the placement and orientation of the square that maximizes the overlap
- Measure the overlap as a percentage of the total area

# Methodology

First, I put each state on a lattice. Then, I scaled each state up or down so that the number of lattice points that fell within each state’s boundary was very close to 400 (+/- 2). In case that’s confusing, here’s a picture of me scaling California up and down until exactly 400 points fell within California’s state boundary:

Then, I found the position of the 20x20 square (so 400 points) that overlaps with the most points inside the state. Here’s a picture of me doing that for Vermont:

In Vermont’s case, the best square overlaps with 266 of the 400 points.

Then, I rotated each state in a bunch of different orientations to see if there was an orientation where I could find an even better 20x20 square. For Vermont, here is the best one (it overlaps with 290 points instead of just 266 above):

For fun, here’s a picture of me rotating Texas at 10 different angles to see which is best:

So, in the end, for every state, I found the rotation that had the 20x20 square that overlapped with the most points inside the state’s (rotated) boundary.

# Results!

First, the least square states (Hawaii and Florida):

I thought that the answer had to be either Colorado or Wyoming:

Here are the top 2. Notice the winner!