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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

Fano

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:

  1. 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.
  2. 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...)

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  • $\begingroup$ Damn, love your puzzles (even if I rarely get anywhere before someone else does ha). Look forward to the next one. $\endgroup$ – MrSethward Jul 9 at 12:47
  • $\begingroup$ Look for the next one soon...think it'll be right up your alley! Thanks for the feedback! $\endgroup$ – Jeremy Dover Jul 9 at 12:56
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Okay, here's an attempt that I believe meets all the rules:

enter image description here

CHILE, TOGO, BELIZE - UN member states;

CHILE, INDIGO, BANDANA - Contain 2 consecutive consonants;

CHILE, PARANÁ, BABAHOYO - South American geographical locations;

BELIZE, INDIGO, PARANÁ - 6 letters long;

BABAHOYO, INDIGO, TOGO - End in 'O';

BELIZE, BANDANA, BABAHOYO - Begin with 'B';

TOGO, PARANÁ, BANDANA - Contain only one type of vowel (A,E,I,O,U).

If you don't like PARANÁ for this last one because of the accent, then I can also offer you SAJAMA or RAMADA as alternative 6-letter, one-vowel, South American geographical locations which have no consecutive consonants, do not begin with 'B' or end in 'O', and are not UN member states... I just prefer PARANÁ as the one which more people will have heard of, being one of the states of Brazil.

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  • $\begingroup$ Nice! This is similar to the solution I had. Interestingly, four of my categories were the same, though my three additional words were all different. Also, re: Paraná, I used Babahoyo so I cannot criticize :-) $\endgroup$ – Jeremy Dover Jul 9 at 12:40
  • $\begingroup$ @JeremyDover The two starters I found most difficult to pair up were CHILE and INDIGO. I figured these would have to rely on something linguistic, since very little else seems to connect them... Re the other linguistic ones, I tried to make these as non-trivial as possible, relying on word length and starting/ending letters specifically. I'd be interested to see your solution once you feel the puzzle has run its natural course... $\endgroup$ – Stiv Jul 9 at 13:09
  • $\begingroup$ @Stiv...will do. But as a preview, I tied CHILE and INDIGO with rot13(ORETNZBG, jvgu pngrtbel "ntevphygheny cebqhpgf") $\endgroup$ – Jeremy Dover Jul 9 at 13:12
  • $\begingroup$ @JeremyDover The problem with that is that CHILE would not qualify. CHILI would, but "CHILE" is unambiguously the country, not the pepper, at least in English. (It's a common mistake.) $\endgroup$ – Darrel Hoffman Jul 9 at 16:39
  • $\begingroup$ @DarrelHoffman I thought about that ambiguity myself but there's a lot of places online which do actually suggest that 'CHILE' is used for that purpose in some parts of America as well as in Spain... News to me! :) $\endgroup$ – Stiv Jul 9 at 16:54
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Here's a solution using only language structure:

CHILE, BELIZE, INDIGO - Contains the letter I

CHILE, RAZOR, BLUNT - Has 5 Letters

CHILE, BABAHOYO, HAZELNUT - Contains the letter H

BELIZE, BABAHOYO, BLUNT - Contains the letter B (Starts with it even)

BELIZE, HAZELNUT, RAZOR - Contains the letter Z

INDIGO, HAZELNUT, BLUNT - Contains the letter N

INDIGO, BABAHOYO, RAZOR - Contains the letter O

Fano Plane

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    $\begingroup$ On issue is that "no three of the four given words are in the same category." $\endgroup$ – hexomino Jul 9 at 10:59
  • $\begingroup$ @hexomino: I wavered over whether or not to make that a requirement, but ultimately decided to leave it as a recommendation. Since this is new, I want to see what folks come up with. I don't think this solution is bad, but if I were to write another I would probably put stronger limits on language categories, at least. Definitely interested in other puzzlers' feedback too. $\endgroup$ – Jeremy Dover Jul 9 at 12:01
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    $\begingroup$ Ah, I overlooked that. Maybe I'll change it and submit another answer later... $\endgroup$ – PelicanFive Jul 9 at 12:11
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Per @Stiv's request, here is the solution I put together:

Fano
INDIGO, BELIZE, AMAZON: All have exactly three vowels, two of which are the same
INDIGO, CHILE, BERGAMOT: All are agricultural products (per comment above, I've always used "chile" for the pepper, and "chili" only for the soup/stew which frequently uses chile peppers; perhaps this is a difference from different sides of the pond?)
INDIGO, BABAHOYO, MEXICO: All end in O
BELIZE, CHILE, MEXICO: All are countries
BELIZE, BABAHOYO, BERGAMOT: All start with B
CHILE, BABAHOYO, AMAZON: All are names of geographical features in South America
AMAZON, BERGAMOT, MEXICO: All contain the letter M
I adjudicate @Stiv's solution to be superior. I award myself no points, and may God have mercy on my soul.

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