Territorial Teaser: Too Many Answers

So my most recent puzzle attempt was pretty much a flop, unless there are lots of readers out there still trying to solve it. I started with what I thought (and still think) was a cool solution. But I didn’t spend much time trying to rule out other possible solutions, and it turns out there are an awful lot of them. And even if I’d tried to rule out them all out, I wouldn’t have succeeded without imposing some incredibly arbitrary conditions (like Gil’s suggestion, “The layout must look like my solution”).

Anyway, here’s the solution I had in mind to start with. I used A/a, B/b, and C/c to refer to the three different demographic distinctions.

[Shortly after I composed this post, Brian sent me a description of this solution; he was the only person to do so.] But here’s a much simpler solution, a version of which was first submitted by Chris Fulmer:

This solution, like many others (including those suggested by Patri Friedman and Chris Hibbert), relies on an up/down, left/right, inside/outside principle for segregating the groups. It doesn’t meet the “landlocking” condition that I added later, but when I added that condition, Chris responded by just taking one of the outside territories and wrapping it around all the others:

And then I began wondering if there was any reasonable set of conditions that would make my solution unique. Blar, who had submitted a solution essentially the same as my own, but using triangles instead of circles...

...suggested adding a condition that no territory touches more than three other territories (not including corner meetings). But the second solution above already satisfies that criterion. The third solution doesn’t satisfy the border-only-three criterion, but it does satisfy landlocking. So maybe we could impose both border-only-three and landlocking? Nope. Blar sent the following variation on the third solution above that meets both additional criteria:

What’s the bottom line? I think Blar gets it right: “All of these solutions are identical in terms of what territories are touching - the only difference besides topologically irrelevant shape-changing is what's touching walls.” And it’s not hard to modify any given layout to meet whatever wall-touching (landlocking) condition you want. I conclude that there’s probably no way to force my solution without a giveaway condition like, “Must use overlapping circles.” Ah, well.

ALSO: The computing background of many readers shone through in their choice to represent demographic categories using binary notation (000 = black male young, 001 = black male old, etc.). But one reader, Ben, used colors instead:

Primary colors (blue, red, yellow) represent pure categories (black, female, old), and absences of those colors represent their opposites. All the other colors are category combinations – e.g., green = yellow + blue = old black male. I don't know if this approach aids understanding, but I like it anyway.