The Solution

To solve the puzzle we started out by replacing the cloth with numbers.  Originally, we did that because it was easier than trying to draw up another pattern and I didn't want to cut apart my original one.  Jamie and I figured out a layout pretty quickly that would require us to throw away three or four squares.  I was fine with that, I didn't want to spend a ton of time trying to find a solution.

I knew that a computer could solve the problem, but I don't know how to do that.  Jamie, being a mathematician and wanting the most elegant answer, took the problem to work and got a solution from a co-worker.

The numbers that we assigned to each cloth type now became critical in figuring out the solution.  Putting the tiles into a computer and telling the computer to match them up I'm sure is a pretty simple thing to do, something I wish I knew how to do, but don't.  I ended up with a list of numbers.  From that point I was able to reconstruct what the nine patches should look like:

From the solution, I was also able to figure out what the most probable identity of the thrown away square was.

 The key was the every cloth type had nine squares, except the blue dot and line piece.

So, when your kids ask you why they have to learn math, you can tell them that some day they might want to make a quilt and that they might have to figure out the easiest way to do that, and that a little bit of math might save them a whole lot of time and frustration.