Origin Stories: Gerrymandering

The idea of using gerrymandering as the basis for a puzzle was not originally mine. The credit goes to my Imagine editor at the time, Carol Blackburn, who suggested it while discussing over e-mail the topics for the twelfth volume of the magazine (2004-2005). As soon as I read her idea, it immediately struck me as a great idea for a puzzle, so much so that I checked around first to see if anyone else had made a puzzle based on gerrymandering. To my surprise, no one had1.

This is a good opportunity to discuss puzzle suggestions and what I look for in a new puzzle concept. Carol’s idea for a gerrymandering puzzle was actually the first useful suggestion I’d ever received to make a new kind of puzzle. Over the years, I’ve received some feedback and a few fan letters from Knossos Games puzzle solvers. Suggestions for puzzles tend to go along the lines of, “Hey, you should do a puzzle about dinosaurs!” While I genuinely appreciate the feedback people give me, recommendations like these aren’t very helpful, since it’s not usually obvious to me what about (in this case) dinosaurs would make for a good puzzle. My response to these suggestions is always: “No, you should create a puzzle about dinosaurs, since you clearly have an idea of what a good puzzle about dinosaurs would look like.”

So what was different about Carol’s suggestion? To answer this question, I need to explain what I am looking for in a new puzzle concept. Puzzles are always based on rules.  The rules could be simple or complicated, few or many, obvious or obscure. Sometimes the stated rules for the puzzle conceal an entirely different set of rules that you can only discover by wading into the puzzle itself. Some of the hardest puzzles are those where solving the puzzle requires discerning the unstated rules by which the puzzle operates. Regardless, puzzles need rules, and coming up with new puzzles means coming up with new sets of rules. So I’m constantly on the lookout for real-world situations that contain some inherent rules or structure that I could build upon or transform into a puzzle.

This is what struck me about Carol’s suggestion of gerrymandering. All I knew about redistricting at the time was that it had to do with the process of forming political districts, and that it could be used nefariously2. But I immediately recognized that gerrymandering must have some underlying mathematical principles that could be utilized to make a puzzle. (In fact, actual gerrymandering could be considered a legitimate puzzle in and of itself. How no one else has thought to make a puzzle out of this is beyond me.)

The first step in exploring if my instinct was correct, that this would make for a good puzzle, was to learn more about gerrymandering. I found a lot of information concerning the history of gerrymandering, less on the mathematics of how it can be achieved. I’ve lost the reference where I first learned of the techniques of packing and diluting, perhaps it was this portion of the wikipedia entry. These mathematical techniques form the basis of how political redistricting can be used to affect election results, and as such they would need to form the basis of the puzzle.

The details surrounding the actual puzzles, like using square precincts (at first called “sub-districts”) in a square territory, and having yes and no votes (rather than democrat and republican, for example), were chosen to keep the rules simple, emphasize the mathematics of redistricting over the politics of it, and focus the difficulty of the puzzles on forming the districts themselves. I also recognized early on that the instructions would need to guide puzzle-solvers through the two main techniques they would be using to influence the vote, and thus example puzzles demonstrating those techniques would be necessary.

The final secret to these puzzles is that, once the gerrymandering cover story is wiped away, all that is left are pentomino tilling problems. The 5×5 grids can be divided up in many ways, and the precinct vote totals are what influences how one decides between multiple possibilities. So while I want solvers to learn about gerrymandering, on another level, there is something to be learned about tiling problems here as well.

Imagine has run two election issues, one in 2004 and another in 2008. When I made the first gerrymandering puzzle, I actually made several of them. Coming up with a cogent set of rules for myself as to how to design this type of gerrymandering puzzle, in which there is only a single solution, was very difficult. So in most cases, when I succeed in forming a new type of puzzle like this, I make more than one puzzle. I had hoped that Imagine would run another election issue in 2012 (and I’m lobbying for one in 2016), because I have several of these to share, in both the 5×5 size and a harder 7×7 size (where districts are septominos, containing seven precincts).

Update: Gerrymandering

Spoiler alert: a solution to one of the gerrymandering puzzles is shown at the end of this blog post. Go back to the puzzles if you haven’t finished solving them!

Even though a new gerrymandering puzzle hasn’t been published in the magazine since 2008 (that was the last time an issue was politically themed), that was right around the time the website had its last major update. Thus, there wasn’t a lot of work involved in updating the gerrymandering puzzles to the latest version of the website.

One choice that may disappoint puzzle solvers out there: I took down a few gerrymandering puzzles that I had posted to the website that were never published in the magazine. I’m trying to reorganize the puzzles I have (not just the gerrymandering puzzles) in order to plan for the future of Knossos Games, and in this case that meant removing a few puzzles. I’m lobbying for a politically-themed issue of Imagine to appear soon, so that the magazine can publish another gerrymandering puzzle (or two).

As for specific changes to the puzzles and instructions in this update, there were only two major ones. First, a colleague of mine suggested that I replace “sub-district” with “precinct”, since that’s what we (in the United States) call the territories that collectively make up political districts. As soon as he suggested it, it seemed so incredibly obvious. This rewording appears in the main instructions page (and corresponding PDF for printed instructions), instruction reminders for each puzzle, and the detailed solutions.

Second, I also updated how the solution is shown. The solution to each of these puzzles needs to communicate two pieces of information: how the grid of precincts is physically divided into districts, and how this dividing up of the vote count results in more no voting districts than yes voting districts. Before, I showed these two pieces of information separately by displaying a full sized grid with the districts outlined and, off to the side, a separate list of the no and yes voting districts. I always thought this was a little clumsy, since you need to scan back and forth between these two pieces of information to make sense of the solution. I can’t believe I hadn’t thought of this before, but coloring the districts and linking their (also colored) vote totals brings these two pieces of information into one graphic. I think it’s much more clear now, even before getting to the detailed solution, how each solution works.

original solution graphic
new solution graphic