[HN Gopher] Square Theory
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Square Theory
Author : aaaronson
Score : 389 points
Date : 2025-05-27 15:33 UTC (7 hours ago)
(HTM) web link (aaronson.org)
(TXT) w3m dump (aaronson.org)
| lisper wrote:
| Aargh! I hate it when people quote text as images because it
| makes it impossible to cut and paste.
|
| > Jet black/Jet Blue ... catnap/dognap
|
| My favorite examples are how prepositions can change the meanings
| of idioms. For example, to be "down for" something and "down
| with" something mean the same thing, but to be "down on"
| something means the opposite. (And going down _to_ X means
| something very, very different from going down _on_ X. That last
| example is also interesting from a geeky HN point of view because
| the preposition imposes a type constraint on the binding of X,
| which is why I had to use "X" instead of "something" :-)
| robinhouston wrote:
| > I hate it when people quote text as images because it makes
| it impossible to cut and paste.
|
| That's much less true than it used to be! I don't know what
| device you're using, but on my iPhone I can seamlessly copy the
| text from that image.
| lisper wrote:
| I'm using Firefox on a Mac.
| layer8 wrote:
| https://support.mozilla.org/en-US/kb/text-recognition
|
| Doesn't this work?
| lisper wrote:
| Not for me. No idea why. Maybe this doesn't play well
| with NoScript or AdBlock.
| layer8 wrote:
| There's also TextSniper. I'm not a Mac user; Microsoft
| PowerToys provide that feature on Windows.
| bee_rider wrote:
| Down for lunch?
|
| Down with lunch!
|
| (Breakfast food is yummier).
| kps wrote:
| > I hate it when people quote text as images because it makes
| it impossible to cut and paste.
|
| Yes. In this case the text is in the ALT tag, which would help
| _if_ browsers exposed it.
| teach wrote:
| And "down for" something is very nearly synonymous with "up
| for" something. (:
| keeganpoppen wrote:
| what a lovely Baader-Meinhof for Ricki Heicklen's delightful
| "Unparalleled Misalignments", which showed up on HN not too long
| ago!
| lblume wrote:
| https://rickiheicklen.com/unparalleled-misalignments.html
| aleph_minus_one wrote:
| > what a lovely Baader-Meinhof
|
| Which attractive terrorists?
| kens wrote:
| The link to the HN discussion, by the way, is:
| https://news.ycombinator.com/item?id=43891128
| jakupovic wrote:
| This is pretty cool. I wonder what logic/math describes it?
| BoiledCabbage wrote:
| While it doesn't fully describe it, his category theory diagram
| reference seems relevant to me.
|
| The stricter of the squares seem to be a homomorphism. But the
| "looser" ones which don't "preserve structure" after the
| transformation but "find a new structure" are some of the more
| interesting ones.
| AaronAPU wrote:
| Semantic Bayesian hyper-graphs where each of the percepts have
| strong correlation between each other.
|
| I'd argue you could bind them tighter by giving the corners
| strong relationships to each other as well.
|
| We find these sorts of dense correlations pleasing because it's
| the natural way we discover meaning. Even though in this case
| the meaning is fairly superficial.
| oliwary wrote:
| > If you've ever tried to construct a crossword, you'll find that
| the framing of a crossword grid under square theory feels right.
| When you're nearing the end of the grid-filling process, finding
| valid crossings of words to fill that final corner of a grid,
| there's a satisfying "clicking" feeling--a sense of magic--when
| it all fits together, analogous to the wrapping-around feeling of
| completing the square.
|
| If you enjoy this feeling, I think you would like my word game
| https://spaceword.org. The goal is to arrange 21 letters in a
| square that is as tight as possible. No one has achieved a
| "perfect" pattern yet, but people are very close, often leaving
| only 3 spaces blank!
| JoshTriplett wrote:
| Do you have a version of spaceword that's not a "daily" game?
| I'd be interested in trying it if so.
| oliwary wrote:
| There is a weekly mode with 63 letters, thinking of adding a
| monthly mode as well. Or would you prefer puzzles that are
| always open?
| JoshTriplett wrote:
| Puzzles that are always open is exactly what I mean, yeah.
|
| Some "daily" games call this kind of generated puzzle a
| "practice" mode. But whenever I encounter a daily game, I
| go straight for that mode, which is what most games would
| just present as the game itself.
| oliwary wrote:
| Fair enough! :) I'll look into adding a practice mode.
| JoshTriplett wrote:
| Thank you very much! I really appreciate it and look
| forward to trying it!
| JoshuaDavid wrote:
| Fun game! Though I dispute that people are "very close" to
| achieving a perfect pattern.
|
| To get a "perfect" pattern you'd need to find three 7 letter
| words that can stack on rows adjacent to each other to form a 3
| letter word in each column. Such arrangements _do_ exist, for
| example: o p e r a t e a r r o w e d
| r e s e n d s
|
| but they are very rare - I estimate something on the order of
| 0.002% of combinations of three 7-letter words have any valid
| arrangements. Assuming that you're using standard ETAOIN letter
| frequencies, the typical bag of 21 letters will usually have
| just a handful of combinations of three 7-letter words so a
| given puzzle has a << 0.1% chance of having a perfect solution.
|
| But there are 12,000x more ways to rearrange 21 tiles within an
| 8x3 grid, and the word choices are more forgiving as well (if
| you draw 7 letters from the etaoin frequency distribution,
| those 7 letters in order are much more likely to form a 3
| letter word followed by a 4 letter word than they are to form a
| 7 letter word). Pretty much every puzzle should have at least
| some solutions fitting within an 8x3.
|
| Additional note: 3 blank spaces is the best non-perfect
| arrangement, since the grid is only 10 tiles wide. One blank
| space could only be achieved by a single 23-letter-long word,
| and two blank spaces could only be achieved by a 10 letter word
| next to an 11 letter word, and an 11 letter word would not fit
| inside the 10x10 grid.
| isodude wrote:
| My first initial thought when I saw the game: spaceword golf.
|
| Like any golf, you start with the smallest square possible and
| increase it with each level. You get less points for how
| perfect the the square is.
| kelseyfrog wrote:
| I'm a big fan of killing time on long drives with friendly word
| games. One of my favorites is a mix between rhyming and square
| theory. Here's how it works: one player picks two words that
| rhyme perfectly. Then, for each of those words, they choose a
| clue word, usually a synonym, but any kind of related word is
| fair game. They say those two clue words out loud, and the other
| players have to guess the original rhyming pair.
|
| What makes it fun is trying to reverse-engineer the original
| rhyme from the clues. It's like solving a little logic puzzle.
| It's easy to come up with new puzzles, but cracking them can be
| surprisingly tricky. Still, the structure gives just enough to
| keep it solvable most of the time.
|
| 1. Somewhat described here https://bestlifeonline.com/jeopardy-
| rhyme-time-opera-version... It's actually quite difficult to find
| a description of the category many of us are already familiar
| with.
| cooperaustinj wrote:
| Do you know any unfriendly word games I can try?
| kelseyfrog wrote:
| You could add the additional constraint that the words have
| to insult the guesser based on their unique psychological
| vulnerabilities. Hope that helps!
| stronglikedan wrote:
| Perhaps give the Argument Clinic a call.
| hypertexthero wrote:
| FYI https://en.wikipedia.org/wiki/Argument_Clinic
| Someone wrote:
| Reminds me of https://en.wikipedia.org/wiki/Rhyming_slang:
| _"The construction of rhyming slang involves replacing a common
| word with a phrase of two or more words, the last of which
| rhymes with the original word; then, in almost all cases,
| omitting, from the end of the phrase, the secondary rhyming
| word (which is thereafter implied)"_
| finnh wrote:
| My family calls that game "pink mink"!
| jacobolus wrote:
| As far as I know the most common name is "hink pink", if
| anyone wants to look this up (or sometimes "hinky pinky").
| Here's a 1981 book,
| https://archive.org/details/hinkpinkbookorwh00burn/ and
| here's a short description from the 50s, https://archive.org/
| details/realbookofgames0000unse/page/134... Searching
| further, apparently _Hink Pink_ was the name of an 18th
| century pirate ship; I 'm not sure if there's a relation to
| the game.
|
| According to this 1941 _Life Magazine_ issue, teenage girls
| in Atlanta were making up rhyming pairs like this at the time
| under the name "stinky pinky". https://archive.org/details/L
| ife-1941-01-27-Vol-10-No-4/mode... Webster's Dictionary from
| the 60s has the game listed under that name, https://archive.
| org/details/webstersthirdnew0000phil_l0b1/mo... and that name
| also seems to continue to today, e.g. by the radio show
| _Loveline_.
| kelseyfrog wrote:
| Thank you! What a fantastic find. This is exactly the kind
| of book I would have checked out at the library as a child.
|
| It's possible I found this decades ago and the origin of
| how I learned this game was lost to time :)
| jonny_eh wrote:
| We call it Awful Waffle, based on a Board Game called Brain
| Strain. They had "Awful Waffle" as an example.
|
| https://boardgamegeek.com/boardgame/8785/brain-strain
|
| I made a proof-of-concept daily game:
| https://awfulwaffle.jonabrams.com/
| gostsamo wrote:
| My personal recommendation is this game _1. Not for travel, but
| a very good in forcing interesting associations and making you
| mad at your partner, which is a certified sign of a good game.
|
| _ 1 https://boardgamegeek.com/boardgame/178900/codenames
| gms7777 wrote:
| If you like codenames, you might also enjoy decrypto [1], it
| scratches a very similar part of my brain. There's a set of
| secret words, and the codemaster needs to give clues that are
| specific enough that if you know the secret words, you can
| make the connection, but vague enough that you can't guess
| the secret words.
|
| [1] https://boardgamegeek.com/boardgame/225694/decrypto
| vunderba wrote:
| Our family plays "Match Three" during long drives where one
| person comes up with three words and whoever correctly answers
| with a word that can complete or precede any of them becomes
| "it" and chooses the next set.
|
| Homophones and proper nouns are considered acceptable.
|
| So for example: (Fox, Lone, Crossed)
|
| The answer would be: Star Star Fox - a well
| known rail shooter originally on the SNES Lone Starr
| - the only man who would dare give a raspberry to Dark Helmet
| Star Crossed - a Shakespearean reference to two people whose
| relationship is doomed
| kelseyfrog wrote:
| Love it! Does the person who comes up with three words have
| the connecting word in mind from the beginning, or no?
| joshhug wrote:
| The part about dad joke square theory got me thinking about this
| classic scarecrow joke, which feels like an example from some
| higher order version of square theory:
|
| "Why was the scarecrow given an award?"
|
| "He was out standing in his field."
|
| The fact that a scarecrow's job is to be "out standing in his
| field", and that excelling at one's job can be phrased as being
| "outstanding in his field" is an incredible linguistic
| coincidence.
| cdkmoose wrote:
| Also, My friend accidentally drank a bottle of invisible ink...
| she's at the hospital now, waiting to be seen.
|
| "Waiting to be seen" having slightly different meaning with
| respect to hospitals and invisible ink.
| simianparrot wrote:
| For some reason this old saying popped into my head reading
| that. I know it's not related but:
|
| Time flies like an arrow. Fruit flies like a banana.
| yojo wrote:
| My favorite part about that quote is the broken symmetry
| between the double meaning of the second sentence and the
| single meaning of the first.
|
| It begs one to consider the possibility of little "time
| flies" snacking on arrows. Which I guess completes the
| square?
| anyfoo wrote:
| Gosh, after all those years I've only just realized the
| double meaning of "fruit flies". Thanks!
|
| Before that, I just thought it was more of a non sequitur,
| but still amusing. There was just something inherently
| funny about imagining a banana hurling through the air in
| an awkward tumbling motion, right after the sagely abstract
| concept of time and its elegant arrow metaphor.
| j2kun wrote:
| The classic, "why did the chicken cross the road" also fits
| into this genre, but nobody seems to understand that "get to
| the other side" means "to cross over from life to death." Every
| time I explain this to someone they are shocked that they never
| knew this meaning.
| Upvoter33 wrote:
| That is funny. We finally figured out this double meaning a
| few years ago and I have been on the same quest since.
| gwd wrote:
| That's a failure of the joke not to set it up -- one of the
| "top corners" of the square is missing. Chickens normally
| don't make an effort to get to the "other side" (as far as
| we're aware anyway).
|
| To make the square you'd have to do something where the
| context of "the other side" means past life into death. e.g.,
| "Why did the spiritualist put his ear towards the road? To
| hear from the other side."
| wavemode wrote:
| My understanding is that that interpretation is an urban
| legend.
|
| Wikipedia attributes the joke to an 1847 article, which is
| phrased in a way that clearly isn't intended to have some
| deeper meaning: https://en.wikipedia.org/wiki/Why_did_the_chi
| cken_cross_the_...
| j2kun wrote:
| TIL, thanks!
| albedoa wrote:
| > Every time I explain this to someone they are shocked that
| they never knew this meaning.
|
| You might have taken this as a hint?
| notfed wrote:
| Is it a coincidence though? You could have started with the
| phrase "outstanding in his field", recognize the double
| entendre, and simply consider whether it's anyone's actual job
| to "stand in a field". Scarecrow is one of many possibilities.
| winwang wrote:
| https://en.wikipedia.org/wiki/Commutative_diagram
| noqc wrote:
| Commutative diagrams are to these squares what category theory
| is to analogy.
| gowld wrote:
| noqc, can you diagram your comment?
| ogogmad wrote:
| There's a mapping there, however it's not natural.
| Garlef wrote:
| I think Double Categories [1] would be a more appropriate
| setting: In a double category, the vertical and horizontal
| arrows are of different types. In usual commutative diagrams,
| they are of the same type.
|
| [1]: https://ncatlab.org/nlab/show/double+category
| ogogmad wrote:
| Why don't some phrases arrive at the same meaning? They don't
| commute.
| SamBam wrote:
| This is clever, and I want to spend some more time thinking about
| it. In a sense, I think this is basically saying that you can put
| the standard SAT-style analogy questions ("Lumen : Brightness =
| Inches : Length") in a square, and that most crossword clues
| could be represented as weird SAT analogies. Or maybe I'm
| stretching the analogy.
|
| But I think that the "Diagonal" that the author suggests for the
| connection between "Donkey" and "Elephant" and "Party" isn't
| quite correct. The key is that both the Donkey and the Elephant
| are a "Party Animal." You can't ignore the "Animal" part, it
| describes them: they are each the animal that represents their
| party, the "party animal."
|
| I'm not sure the correct way to represent this in "Square
| Theory," but it's not just linking "Party" to the animal in
| question.
| varjag wrote:
| Leibnitz once famously said, "Music is the pleasure the human
| mind experiences from counting without being aware that it is
| counting". Perhaps solving crosswords is the pleasure mind
| experiences from doing group theory.
| cshimmin wrote:
| My favorite example, which was an honest translation error from a
| non-native speaker friend: Hand job (he meant to say manual
| labor)
| lubujackson wrote:
| Heard from a non-native speaker watching a missed basket in a
| basketball game: "Another rim job!"
| zahlman wrote:
| Something similar once famously happened with a Japanese
| vtuber: https://www.youtube.com/watch?v=UeQ5K5DQiDI
| paulddraper wrote:
| There are a number of comedic sketches based on the linguistic
| similarity of "booty call" and "butt dial."
| munificent wrote:
| What a lovely post.
|
| The "Grubhub" square fits some other alternatives: "Grubclub",
| "Bitesite", or "Eatmeet" (but eww).
| dleeftink wrote:
| Kind of exciting to see Algirdas Greimas' semiotic square derived
| from (game) first principles!
|
| [0]: https://en.m.wikipedia.org/wiki/Semiotic_square
| gowld wrote:
| > However, there's nothing about the square structure that
| dictates the edges must represent phrases and synonyms.
|
| I got there and thought "Category theory", and, lo, that's the
| next paragraph.
|
| > Let's talk about Scrabble, one of the seven most important
| games [link to review of Seven Games book]
| https://www.nytimes.com/2022/01/25/books/review/seven-games-...
|
| Seven Games was mentioned in another HN discussion last week, by
| 'danvk talking about his Boggle solution:
| https://news.ycombinator.com/item?id=44084022
|
| HN hasn't yet taken an interest in that:
| https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...
| rossant wrote:
| Same. I couldn't help seeing a commutative diagram before he
| had a chance to say "category theory".
| mellosouls wrote:
| Nice article! It feels like there should be something AI-
| zeitgeist-related in there referencing word2vec or similar.
|
| OT: Going by the url, link here on HN and slightly adjacenty vibe
| I got to the bottom and signature before realizing this wasn't
| _Shtetl-Optimized_ finally made mobile-friendly.
| jacksnipe wrote:
| Wordcels discover category theory lol
| bitcurious wrote:
| I came up with a (lambda? Triangle?) loop a few days ago.
|
| Read until sleepy. Sleep until ready.
| underyx wrote:
| The author of this post would love the video game Lingo:
|
| https://store.steampowered.com/app/1814170/Lingo/
| astrolx wrote:
| How can someone hate "etui" so bad !?
| RupertEisenhart wrote:
| Great article, but I can't believe it didn't mention the sator
| square!
|
| SATOR
|
| AREPO
|
| TENET
|
| OPERA
|
| ROTAS
|
| (Very easy to commit to memory too since most of the letters are
| right there in the name!)
| scubbo wrote:
| GNU Terry Pratchett
| zem wrote:
| one of my favourite english curiosities follows this structure -
| "outgoing" and "retiring" are both perfect antonyms (enjoying or
| not enjoying socializing) and perfect synonyms (leaving a
| political office or job)
| nssnsjsjsjs wrote:
| Fun read. As they say in New York, Arigato!
| geodel wrote:
| Huge fun. Besides mixing/matching like this I mix across
| languages and that give an extra dimension to create such fun
| phrases.
| akoboldfrying wrote:
| GrubHub is certainly a good name, but if I came across an app
| actually named "Food Central-Place" I would have no choice but to
| install it on the spot. It just has a certain anti-ring to it.
| AStonesThrow wrote:
| Next time you're in NYC, try searching the web for "Thai Food
| Near Me"!
|
| https://www.thaifoodnearmenyc.com/
|
| There was a local food-truck operator named "Pho King" and
| eventually they established a storefront ... well, I see one
| closed, and another opened up. Formerly known as "Pho King
| Kitchen" and now there's the "Pho King Eggroll" place.
|
| Fred Armisen did an SNL bit about this, too.
|
| Not far from me, there is a ghost kitchen cluster. It's tucked
| away in a commercially-zoned neigborhood, and it serves all the
| food delivery services. Apparently, you can walk in too. I only
| accidentally patronized them once, when they had some great
| _larb_ on offer. I think the report says there 's 15 different
| menus and "virtual kitchens" in the building, just turning out
| food-to-go.
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(page generated 2025-05-27 23:00 UTC)