# Tiling math: max number of ways the US states can be arranged to roughly fit the CONUS outline?

I found this:

Which prompted the pleasant response:

Puzzle enthusiasts and mathematicians study this type of stuff, and come up with classes of and maximum tilings all the time. Given the outline of the continental United States (CONUS), how many ways could the outlines of the states (handy separated list here: http://m.imgur.com/KfIhxkf?r) be apportioned to fit into it, given certain leeway?

How do you even approach the problem?

That “leeway” and how to come up with it–and approximating/classifying conditions is what I’m hoping to learn about. For example, the multiple sub tilings of the four-corner trapezoids (ok, one with a jigger)–can that be subsumed as part of the general analysis?

First, define “leeway”.

This is one of those “I know it when I see it” problems that doesn’t seem amenable to math or computation without doing a lot more work framing the problem in a rigorous way.

Definitions would be necessary to come up with an answer, but some of the ways to frame the problem would be pretty simple to define. For example, if we insist on using real outlines of the states and the US, you could say “No more than x% overlap/empty space.” With 100% allowance, any random position of the states would be allowed, and with 0% allowance, only the original map would be allowed.

The XKCD comic (which is the original source of the image, not the linked blog) takes enormous liberties with the shapes of the states. Just look at the state of Washington. XKCD completely misrepresents the Puget Sound and the Olympic Peninsula in order to make the state look square… and then puts Ohio in Washington’s place, when only one of Ohio’s borders is the right shape.

It’s an interesting result… but what it really illustrates is how flexible human memory and perception are. Take something out of context and we forget what it’s supposed to look like. It says very little about the capacity for the states to be rearranged.

Of course you’re correct…

Not for the first time, I saw a thing, realized its kind of like a real-life interesting area of science or math, and tried to finagle on the cheap from my SD friends “this is what real-life approaches” to that area of science or math may entail.

Wiki-ing “tiling” or “planar fill” or “Escher” gets me lost too quickly. A little knowledge delivered here I’ve found to be, contrary to the saying, less dangerous.

A few years ago I pitched an “analysis-query” on the dynamics of me taking a sharp left while stepping off an escalator, and I the drubbing I received still stings.
ETA: to DrCube.

You can always fill small gaps with sand or grout.

Indeed, it might be interesting to see a map with the states all in the positions shown in the comic, but retaining their correct shapes, with whatever overlaps or gaps that requires.