The Fano plane is the smallest example of a finite projective plane, with 7 points and 7 lines, which has the property that every pair of points lie together on a unique line, and every pair of lines meet in a unique point. If you're a nerd like I am, and you treat the points as words, and lines as categories, you can think of this as a really tangly Connect Wall. It's probably easiest to see with an example.
Let the words be RED, ORANGE, YELLOW, GREEN, BLUE, VIOLET, and WHITE. Then we can create lines as follows:
RED, ORANGE, VIOLET - primary and secondary colors to RED
RED, YELLOW, GREEN - colors on a stoplight
RED, BLUE, WHITE - colors on the United States flag (among many others)
ORANGE, YELLOW, BLUE - sum of letters (using A1Z26) is even
ORANGE, GREEN, WHITE - colors on the flag of Ireland (among others)
YELLOW, VIOLET, WHITE - colors on the uniform of the Minnesota Vikings
GREEN, BLUE, VIOLET - primary and secondary colors to BLUE
Now it's your turn! If I give you the four words: INDIGO, BELIZE, CHILE, BABAHOYO, can you find three more words, and seven categories, such that the result is a Connect Fano Plane? As an initial hint, in the solution I have no three of the four given words are in the same category. Let me suggest some ground rules:
- The categories can be based on wordplay, language structure, or definition, but simpler is better. For example, "word contains letter K" is fair, but "word contains either B or C" seems a little stretchy. "Word is one of RED, ORANGE or VIOLET" would result in rotten produce being thrown.
- To be a real Fano plane, not only must the words in a line meet the category requirement, but the words not in the line must NOT meet the category requirement. Otherwise, most of the color categories above could be "colors of the spectrum".
UPDATE: Having looked at the answers, I wonder if the "right" way to do this might be more like a traditional Connect Wall, where all seven words are given, and the seven categories must be determined such that all seven categories describe some final answer. That shouldn't be too hard to create. (Sobs softly into hands...)