Oh, wait, actually the best Wordle opener is not “crane”…
Following up on the Worlde-solver ( • Solving Wordle using i... ), discussing a minor bug and more details about how the best first word was chosen.
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Contents:
0:00 - The Bug
3:31 - How the best first guess is chosen
8:54 - Does this ruin the game?
Nice post by Jonathan Olson on optimal wordle algorithms:
jonathanolson.net/experiments...
More on optimal strategies:
sonorouschocolate.com/notes/in...
Code for this video:
github.com/3b1b/videos/tree/m...
These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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It takes courage to admit a self-made error, and in this case, an extra effort to make a video explaining it. Thank you Grant, this says, directly and indirectly, a lot of good things about you.
Real scientists are not afraid to admit their mistakes. People who claim to be scientists but insist they never make mistakes even though they clearly have should be avoided.
@@CristiNeagu Yep. This is a perfect demonstration of that. It also further applies the principle (and the scientific method) to find other mistakes--or possibly "mistakes" that are really just improperly-defined nuances--that further refine the answer and *also* delve deeper into the math and process.
Also he makes a bunch of money by issuing a correction video, and increases his credibility.
The responsible thing to do would be to delete the old video and upload a fixed version, but where’s the money in that?
@@TimwiTerby But what would learn from that? How is it more responsible to cover up a mistake than to explain it?
I always start with, "ADIEU" since it has four vowels. My second guess is, "STOMP" since it has the fifth vowel along with S,T,M,P, which are letters that are more commonly used.
STORY would probably do better, hitting the somewhat common ending Y and getting the common letter R.
@@petrie911 That's a good one. I've been using GHOST as the word after ADIEU.
i do adieu and then storm lol
@@petrie911 You can also use, "PORNY".
personally, my second guess is usually MONTH, and if that doesnt work i usally use a plural word with the letters i havent used yet.
Admitting a mistake and correcting it is the pinnacle of studying math and/or coding. I love how "noble" this channel is.
Get over it
@@oldoddjobs i think they were being positive
people acting like no one has ever admitted to a mistake before.
@@yt-dman a lot of people dont
I used the word rocks when I started
just so y'all know: 1) six words have been removed from the official answer set so it's now down to 2,309 words. The words that were removed: AGORA, PUPAL, FIBRE, LYNCH, SLAVE, WENCH 2) no answer word ever repeats. So as time goes on ... that list of 2,309 is shrinking by one every day. Ideally if you wanted to play the optimal first word, you'd rerun your code on a daily basis!
In fact, since we've just played Wordle 315, there are now only 1,994 possible answer words. That's a pretty significant chunk of words that have been eliminated; I'd be curious to see the optimal first word to solve for one of those 1,994.
I can understand why lynch slave and wench were removed, but why fibre pupal and agora
@@where_is_saucelikely due to fiber vs fibre. Same word, different spelling. NYT is based in... New York so they left in the American spelling and took out fibre
Why do they remove words?
@@Kabslantivity2000 They don't. Answers can repeat.
My mom plays Wordle, and she opens with the word “audio” every time to get the vowels. She tries to use the letter “E” in her second word. She’s only ever lost once.
@@dynamicdave2647 audio has 4 vowels tho
i did that too, but i use the word "ouija"
She could try 'adieu'
Just don't forget that y can sometimes function as a vowel! Like in cynic on Monday.
I use FLUID then ADORE to get all of the vowels and some of the consonants. Edit: ORATE as the second word avoids the double D.
It's interesting that SOARE keeps showing up in your lists, because it's also the top word from a simpler and more arbitrary method by Bertrand Fan ("The Best Starting Word in WORDLE"). They just counted how many green and yellow tiles you get from each starting guess if you use that guess on the whole answer list, but gave greens twice as much weight as yellows.
That is surprising!
It makes sense as it has really common letters
Just use any word without more than two vowels. Then using elimination you can figure it out, Especially if you take out E in the start.
This actually sounds like a good hard mode strategy, where you're trying to get a letter that appears first, no matter the correct spot or not, and that informs your next guess. Interesting!
@@georginovalsan4715 AUDIO has four vowels
I started playing wordle really late, about 7 months ago, and on my own came up with a group of starting wordles that seemed intuitive to me based on my experiences watching Wheel of Fortune growing up. And my list is SLATE STALE STEAL LEAST TALES I was honestly super proud of myself for coming this close to the best answer just by thinking about how other word games work
its just a,e,l,s,t rearranged
@@joosh_rocks yep, that's the point. A group of words that combined three of the five most common consonants and the two most common vowels. Like I said, I was thinking of Wheel of Fortune. In the bonus round, before 1988 players had to guess 5 consonants and one vowel. But the fans realized there was an obvious set of letters to pick to increase your odds, and so eventually a LOT of players began selecting r,s,t,l,n,e, with very little deviation. That led to the modern bonus round, where the puzzles are larger but you automatically get those 6 letters for free, and have to guess 3 more consonants and 1 more vowel before solving. My backup word, if I strike out on guess 1, is ROUND, which knocks out the other two Wheel of Fortune letters and gets 2 more vowels.
I tend to start with either ARISE or RAISE, and for the second word I use CLOUD or COULD, then sometimes I'll use NYMPH for the third word.
@@Knowbody42adieu works
Same
An excellent follow up video. Came for an answer, but stuck all the way through and gained some insights.
Yeah, as a real human playing the game, I actually aim to find the most consonants, and ideally the first letter of the word, as this helps my brain think of possibilities more readily, instead of having just vowels and maybe a letter somewhere in the middle. So statistically "best" and "best for me to use when trying to solve the puzzle" are likely to be very different.
What's your top consonant-heavy opener? I usually go with SNORT for the best coverage of the top consonants, and I agree it's much more intuitively helpful than anything else I've got from a more technical script.
@@edbrims I always try to use a different word every time. Though I don't generally remember which words I've used before. It's more fun and interesting to think of new words each time.
@@edbrims I use TRANS or RANTS or RENTS
@@Mr.D.C. I also always use RENTS!
Yep. I like starting with STERN, followed by something like plaid or mulch.
There are two joys in solving a puzzle: The first is finding the solution; the second is exploring the process. Applying information theory to the process is a joy in itself, and for some a joy greater than finding an answer. Similar feats are done that result in Rubik’s cubes being solved by robots, or machines that can sink a basketball from anywhere on the half court. The mechanics of invention are not finding a solution, but in finding a new puzzle in the process that leads to a better solution. Farming is advanced not by figuring out how to make it rain or have longer days to work, but on solving problems of yield and disease. Transportation is advanced not by breading better horses, but making cars, trucks, and airplanes. This is true out-of-the-box thinking, looking at the problem with eyes that are larger or smaller than the problem in its original form requires. There is a joy here that needs no apology, because the exercise is in its own right a challenge to be met and the consequences of such exploration may yield innovations with far greater benefits than knowing the answer to today’s Wordle.
yes
I ain't reading all this but congratulations or sorry about that I guess
My engineering professor would love this paragraph.
I always go with “adieu” and “snort”. I missed one word “corer”. I guessed cover, cower, comer. Annoying but is what it is
Slow clapping for a whole hour at this post
Your esence of linear algebra has helped me so much honestly, please make esence of basic analysis🙏🙏🙏 The linear algebra series has made me from barley getting through exams to acing them with ease, as it so much easier now to study, mor fun, and it makes sense. Before I was just memorizing everything to get through and to pass the exam.
I always like to look around my location and use the first 5-letter word I find. If it contains common letters then it's possible to get more green and yellow letters. If it contains more uncommon letters like Q and X, if they're right, then it narrows down the guess opportunities.
cool!
Public corrections show a dedication to intellectual integrity. I am so grateful we have creators on KZhead who are dedicated to quality and honesty. Keep it up, Grant!
Absolutely! We all make mistakes. What matters is how we handle them. Kudos to Grant for his integrity!
Props to you for praising youtubers who admit mistakes, rather than harping on them even more for the mistake they made. People really need to learn to reward good behavior(admitting mistakes), rather than punishing good behavior.
Wish this would be more common. Everyone makes mistakes, but it's the way how you handle those. Stand up to it, admit it, correct it - thats the way it should be.
I always like to start with "Share", usually followed by "Point" if the first guess produces all greys. Seeing "soare" as such a highly rated first guess is a bit reassuring.
I also use point as a second guess often! I start with whatever word pops into my head though
microsoft sharepoint is not good though
STARE is my go-to
I also start with "share". If the target word has both A and E, I start looking for consonants (and maybe try to find the location of A or E if they are yellow). If not, my next word is "doubt" (probably not ideal for consonants, but covers two other vowels). If both first words have been mostly grey, I continue with "lying" for the last two vowels and a bit more consonants. After that, it's tailored by the info I have so far.
@@MagikMako Sure, but then you can never win in less than 5. I often solve in 3, occasionally even in 2, by not having a fixed starting word and by not fishing for letters unless I really need to. I only use a non-overlapping word when I am desperate and then usually just one.
My favourite starting word is 'adieu' as it shows four out of the five vowels immediately. You can tell a lot about a word by the placement of its vowels.
I always play "aisle" and usually follow with "round". That covers all 5 vowels and 5 common consonants.
I tend to do “aside”, then “clout”. But I really like adieu!
Audio and bytes gets every vowel including Y and a lot of common consonants
Excellent point towards the end about overfitting to the answer list not being a good strategy long term, since it's now got an editor who's theoretically going to be picking an answer each day, similar to how crosswords are human generated rather than procedural, rather than having an official answer list (I think the allowed guesses list is the same?) I think they're committing to keeping to the 'no add an s plurals' for what they pick as answers, mind. This also allows them to do thematic answers - Maybe on the 25th of December Carol will be more likely to be the answer than it would be around July, for example. (I rarely play Wordle. I do often find myself playing squardle, which is a grid of 6 criss-crossing wordle puzzles in a waffle shaped pattern, three vertical, three horizontal, where clues can give you help on other answers alongside the one you put it in - so an E will tell you if there's an E both in the clue you guessed it and the clues going the other direction that 'see' that letter and your answering one horizontal and one vertical each time where my opening is usually point, clear, mushy - which is also my opening for the weekly puzzle (which fills in the holes to make it 25 simultaneous, crisscrossing, wordle puzzles))
So far, the editor has not picked an answer that was not on the list of 2309 (changed from 2315 back in early 2022) words, but rather picked a different word from that list than the baseline scrambled list. Predictably, some people are mad about it, and the editor is "ruining Wordle", which is clearly not possible since 3blue1brown already did! ;)
Waffle is basically the same as squardle but simpler and I suppose slightly harder for the actual gameplay but in terms of processing information given to you, it is probably slightly easier. It also does themed versions on specific days.
That's the same reason I like playing sedecordle, it runs sixteen wordle games at once - each guess is applied to all sixteen, so playing CRANE will tell you the first word has a green R and yellow E, the second word has a yellow CR and a green N, etc. Lots of fun strategizing to maximize the information you get out of each guess, and you only get 21 guesses to solve 16 puzzles, so if you have six wrong guesses you can't win
@@TimMaddux The NYT editor is branching out from the word list now. There have been two answers in the last month: BALSA, and KAZOO which were not in the original word list. If you overfit your answer to the (former) answer list, it wouldn't be able to solve those problems.
Curious how things would differ if this system was applied to hard mode (i.e. always use all known info)
At the end of the last video, I showed a very brief clip for this. It actually doesn't change as much as I would have expected. Salet remains the best, with an average of 3.53, followed closely by "least" and "slate"
@@3blue1brown I love to use LEAST and it's iterations (e.g. STEAL). Nice to know I stumbled into a near optimal algorithm. Still takes me time to finish though, my word lookup skills are far less efficient than those in python!
You don’t have to use all known info. You can use letters you already know do not exist. It sounds pedantic but in the context of this video the word information means all information.
@@3blue1brown Consider all the words ?ight. I get these words. bight, dight, eight, fight, hight, light, might, night, right, sight, tight, wight. All you have is that you know ?ight. Now what tactic do you apply? You could try them one by one, but that means an average of 6 guesses. Far better to pick words you know are wrong. FHWBMDELNRST are the starting letters. For example, TENDS. You know its wrong, but depending on the matches, you get TIGHT, EIGHT, NIGHT, DIGHT or SIGHT in 2 guesses. If you don't get a match, try WHIRL That gets you HIGHT, LIGHT, RIGHT and WIGHT in 3, Then MAFIA will show FIGHT or MIGHT directly, and if you get no matches BIGHT. The last three in 4 guesses That's at most 4 guesses, average 2.833. The key observation is that a guess you know is wrong, can be the best strategy.
@@Nickle314 I like this. I often have the most luck when I pick a totally different word just to narrow down which letters aren't there. Even if your first two guesses are all grayed out, you've still gained a ton of information
i mean a lot of it just takes time to analyze, it's good that you were able to identify this
HELP MY!!! My muscles are too big! I am a big tall man and my muscles are even BIGGER! I use them to get views but they HURT so much!!! Because they are heavy. Do you have any advice, dear act
This was very interesting as an information theory thing. I have come up with Scart, being folky as my "optimal" starting strategy. Thanks and keep up the good work.
The algorithm gave this to me yesterday and today I am blessed.
08:55 it's like in chess. Computers have solved so many openings and movesets, but often the optimal move is totally not the best move, because you can't follow up with a random movement pattern thinking 8 rounds ahead
true although in really high levels of chess they do and have learned alot from the chess engines
Or in solving Rubik's Cubes. Humans need a method instead of being told by an algorithm what the optimal move is.
Computers haven't "solved" any of the openings yet, at least none that actually get used, because that would mean the game's outcome would already be known from the beginning. What they have solved is the end game, modern tablebases can look up the outcome of any legal board position with 7 or less pieces by going through 18.4 terabytes worth of data. Now they're working on a tablebase for 8 piece end games, which will require over 5 petabytes of disk space, a difference of two orders of magnitude for one extra piece, so it might take a while before we get to the full 32.
@@ultru3525 yeah totally! I meant it in a way where stockfish and AlphaZero will sometimes choose a virtually optimal move, which wouldn't be humanly optimal, as you can't see the opening or the follow ups
@@eftorq Sure, but most of the time, they’re still guessing based on relative piece value, piece activity, and other heuristics, not fundamentally different from how good chess players approach the game (at least for StockFish, AlphaZero is pretty much a black box). But it is true that you shouldn’t blindly follow an “optimal” line of moves if you have no idea why it’s deemed optimal, and the strength of your opponent also plays a role, e.g. Mikhail Tal was famous for sacrificing material in order to create attacks too complicated for mere grandmasters to defend against, but computers would never play them because they can spot that one line of 20+ moves countering it and would assume their opponent sees it too. Also, surprisingly the opposite can happen as well on occasion, where a human spots a great move that a computer may have skipped because it “only” looked 10 or so steps ahead and its evaluation algorithm didn’t spot any improvement, so it prunes that branch in favor of more “promising” lines. The most well known example of this is probably the Traxler Counterattack, there’s one line that completely blindsides StockFish, and allowed me, a chess amateur, to beat it at max difficulty.
I've been using "crane" since the first video came out, and it works alright for me. I use "pious" as my second guess if the first guess doesnt turn up much yellows or greens, cus it uses the remaining vowels and some common consonants.
Try using alien too
Eyyy another pious user. That's fun
I use crane then molts, works so far.
I use louis
i use hard mode
Hey Grant, Excellent video - thank you. I've enjoyed the information theory aspects and the comparison of different search strategies. One thing I'm surprised about is that no one has yet tried to tackle an analysis that involves "hard mode" - in which there are two further requirements: 1) any correct guesses (green) have to remain in future guesses, and 2) and correct letters (yellow) have to be included in future guesses. The question I've had is not which word gets the best overall score but which word has the fewest possible outcomes in which you cannot guess in 6 tries. For example, guessing vowel-heavy words (think ADIEU) is a bad idea - you easily get caught knowing the vowels and not the consonants and don't have enough remaining guesses to distinguish between all possible words.
After watching this video, I immediately played today's wordle on 4 July 2023, and opened with the word "CRATE". It just so happened that the last 4 letters - "RATE" were all Green !! A very well made and explained duo of wordle videos. Looking forward to more!
But was it grate or irate?
I've actually never played Wordle and just wanted to know how this clever man cracked the game with math....
@@xelnoc2100 It's crazy. I'm crap at games like Boggle and Scrabble, but for whatever reason I rock at Wordle. It's a fun challenge.
Trust a mathematician to try and suck the fun out of something.
@@RUBBER_BULLET lol if you cant appreciate what he did then dayum
this is pretty much the reason to watch any math video
it actually makes you smarter!
It is interesting that "CRATE" and "TRACE" deliver pretty much the same result, but "SALET" and "STALE" don't. That being said the positions of the letters matter and the odd case in this is "CRATE/TRACE" matching in score rather than "SALET/STALE" not matching in score. This calls for a follow-up video.
My bet is that it's not a coincidence most good first guesses have an E on the end. Lots of words end in E, and it's always worth knowing straight up if its on the end or not, it will always narrow down the word either way.
The way the English vocabulary is structured, some letters appear more often in specific positions. For example, there are plenty of words that start with S, but there are way fewer words with S as the second letter. You get more information out of S as the first letter than the second letter.
To be fair, C doesn't usually appear in many other spots and is usually together with his friends E or H
You could take a step back and stop restricting guesses to words. In that situation, you're really guessing a number of separate puzzles, but with some cross-information across them. You might consider "what is the most common first letter of all possible 6 letter word answers? And then guess that as the first letter because it is most likely to reduce your result set. This doesn't take into account the possibility of the letter being elsewhere in the word, which would further weight the value of guessing that letter in that position. That said, with this approach you could get pretty close to finding the results that these algorithms found without having to necessarily rely on (potentially) opaque algorithms. :)
@@Lachy474 Very good observation, i think it might be due to the wordlist that the algorithm works on being finite and very small. With a different set, same algorithm would probably yield different results.
There are very similar opening words used in both this presentation and how I make an opening choice. I use the official Scrabble (TM) dictionary and filter out all the 5 letter words (12972). Then count the frequency of each letter in all those words, the first 5 letters and frequency are {s, 6665}, {e, 6662}, {a, 5990}, {o, 4438}, {r, 4158} and of all the words in the dictionary that contain those 5 letters are {aeros, arose, soare}.
Oh wow, I've been using soare for a couple of weeks after reading an article listing a few good words. The amount of vowels but also very frequent consonants made me go with it, I half expected to see it here, that's fun
When this game came out, I immediately thought of its extremely similarity to cracking WW2 Enigma code, especially the part used to find "cribs" from ciphertext. The pre-computed lookup table is analogous to diagonal board used on the improved Bombe machine which basically serves as a 5*5 string comparision circuit.
Damn bruh I just had a stroke and ascended to heaven and was reborn again to write this comment
^ what this dude said
^ what this dude said
^ what this dude said
^ what this dude said
After seeing all those articles and social media posts about your last video, I am hugely impressed with your honesty and humility to call out how that is actually wrong. Didn't think my respect for you could go much higher. Loved the last video and love this one too - thank you :)
The idea of quantifying information was fantastic. It'd be awesome to have the long awaited Inverse Fourier Transform video sometime soon too :)
This kept showing up on my feed and last night I wanted a starting word so I chose CRANE. I now have a first round completion.
My initial approach was to do the "Wheel of Fortune" strategy and try guesses that use as much of R, S, T, L, N, and E as possible for my first 2 guesses, and I see those letters appearing a lot in the top suggested guesses from your bot. It may have been interesting to see how choosing guesses based on letter frequency (rather than word frequency) affected the results.
Likewise.
I often start with Learn, and follow up with Hoist if learn didn't hit many letters.
I like using Terns or Stern because it hits 5 of the 6 high frequency letters!
The problem is that the standard quoted letter frequencies was based on an analysis of normal English text, so is biased by the commonness of words. Not to mention including the extremely common short words "the", "in", etc, that can't appear in Wordle. You'd have build a new letter freq analysis around the Wordle target-list (not even the full Wordle dictionary.) (Aside: My preferred opener and second is based on that same (wrong) general English letter frequency analysis.)
GUILT FERNS is funny and effective
this makes sense, when i implemented it i got that list with soare, roate, and raise at the top so i was confused. Glad i didn’t just miss the point some how
ok
Where did you get the word frequencies from?? Hope there is is a dictionary of word frequencies lying somewhere on the internet, free to download.
@@raghavendrapotluri5861 I actually just used all possible guesses to find the best first guess entropy. Then to find the average solution score for a word: I used all possible guesses until there was only 4 or less (arbitrarily chosen) actual answers possible (then i only guessed possible answers) as a lazy side step.
the wordle accepted guess and possible answer lists are available in source code. And for the possible guesses be sure to add the possible answers bc the accepted guesses list doesnt contain the possible answers
I have been using RAISE as my first guess for a while ant it's pretty solid.
Many thanks for your great videos. And I suggest making video about non-integer base of numeration.
I usually shake things up with my starter words, but I was happy to see several of my starters on the list. Stare, arise, raise, roast for example
“Maybe it’s not about the optimal opener… but the Information Theory we learned along the way” 🧐 - 3Blue1Brown (paraphrased 😅)
I always open with "DAISY", and if that doesn't give much information, I follow with "OUNCE", which gives me all vowels including Y. It's not perfect by any means, but it gives me some comfort to know all the possible vowels are included so in the worst case I'll have at least one clue.
I think “Irate” is a solid choice
Try starting with "stern" and "audio", gives all vowels and all the most common consonants.
@@lukesteiner8934 bro that's what i do..
This is why I use "arose" and "unity" in a similar way. Nice spread of consonants, too.
@@lukesteiner8934 I use "suite" for that.
I absolutely never start with a consistent word. Everyday I stare at the puzzle and pull a credible word out of the sky. Hasn’t failed me yet. I’ve been playing for many months.
Great graphic design, great content!
I like starting with “young”, because it helps me with letters and patterns that trip me up if I DON’T get them, and I can infer more common letters more easily if the uncommon ones are already in place. I also play in hard more so I don’t have a standard second guess, but I will usually try to get as many vowels as possible in the second guess :)
Young then whale then brick
I think you have the right idea. I’ve learned much about information theory and I appreciate that you not only let us know about an error, but that you took the analysis further out of curiosity. I love your content - keep doing what you do!
This is so excellent thank you
I usually use audio since it has 4 vowels in it. Each vowel has a silent e 5 word that you can use to follow up. It works most of the time for me. (Shame,prune,shine,shone are just examples.)
I always respect when a content creator realizes a flaw and acts to correct it. I also entirely agree that these technically optimal words are rather irrelevant for human players. I always get the best results by eliminating most vowels early, so I was surprised by crane, unsurprised by soare and again surprised by salet. I'm incidentally a medieval enthusiast, so I do consider it a word. However, I do consider the word to be spelled with 2 "L"s, making it fake anyway. In my vowel elimination strategy, I am a bit disappointed I never thought of "audio" before seeing it at the end of this video.
i dislike the word audio for opening because if your goal is to use all vowels, most of the words with single e are with double letters. however if your side goal is to minimize the tries it takes then go for it. i usually play with elimination of letters in mind because it feels like scrabble that way.
I always start with “stare”. It gets out the s, t, and r which are probably the most used consonants and then the a and e for the vowels. Yeah, it’s boring but I almost always solve the word puzzle so it works for me :)
I don't do that but I guess Rinse (or an anagram of it since there are man Risen, Resin and Siren) then guess Octal as that eliminates 10 of the most common letters in the english language then if there is not enough information to get anywhere I would guess "Bumpy" or "Dumpy" as that can eliminate a further 5 letters eliminating 15 letters in total. The only time it doesn't work for me is when the answer is a really really obscure word I have never heard of so would not consider guessing over a more common one in the english language. I saw this strategy on the primarily Sudoku focused youtube channel "Cracking the Cryptic" and the least number of guesses I have managed to get a word in was 2 since when I guessed "Rinse" or any one of its anagrams (I forgot what I guessed since I guess all of these from time to time) there was a green letter and like 3 yellow ones in the grid too and I managed to form the final answer in 2 guesses ALSO another thing to note when he mentioned algorithms changing due to new words being added or older ones being removed this strategy is not affected as all it does is eliminate as many letters on the board as possible and does not involve trying to follow algorithms
That's my starter, too! Unless I'm playing WordleUnlimited, and then I just use whatever the last answer was, just to mix it up.
This is my word too, plus putting the e at the end feels like a good idea
I use "tears"... Which has all the same letters! I usually follow it up with "pound"
I use stair
I enjoyed this explanation, and the previous one. I was looking at how WordleBot does it, and I did not understand the 'expected information' in its analysis. I was just using its same idea of [color] groups. I am amused and sympathetic with your issue of the combinations of yellow letters with duplicated letters, as that has been the main difficulty with my programming efforts. I think I finally have fixed those issues. As far as the 2315/12546 word lists go, that ship sailed long ago, with the 2309/14855 lists the NYT uses today, combined with the fact that it occasionally uses final answer words that are not in the 2309-word list. (Like on June 9) I started out this programming project (and I am not a serious programmer) with the idea of finding the 'best' word, but I am finding that the difference between a word with 3.402 average guesses being better than a word with 3.405 average guesses similar to the fact that the 13th of the month is more often a Friday- statistically true, but essentially meaningless. I do, however, test SALET every day, along with 20 or so other words. I am leaning towards LEAST for my computer, and CREST for my personal use (I like consonants, especially combining consonants), but am still finalizing my programming,
I often use crane as my second guess. Right after guild. Those give me a solid amount of info. Glad I came to a reasonably optimal answer on my own
In recognition for your brilliant first attempt, "crane" should henceforth be known as the Sanderson Opener (in reference to the Parker Square)!
The Sanderson Gambit
@@mikailvandartel Chessle
It's also nice to see it at #6 on the final corrected list, so it's not like it's a bad opener
@@mikailvandartel chessle
I bet discovering this mistake felt mortifying. We've all been there; most of us are fortunate that our mistakes aren't seen by millions of people. Thanks for publishing the correction so quickly and transparently!
I've been starting with SAINE for about a month now. I looked at letter frequency in each position for the 2315 Wordle answers (found on a web page somewhere) and this was the highest scoring word I could come up with. I think it is slightly more likely get some green which I think helps solve the puzzle.
I was starting with snare and pilot; but recently I've switched to slate and groin. Just looking at scrabble letter scoring can tell you the commonality of each letter and getting all the 1 point letters in two guesses is pretty efficient for human players.
Some of these comments with no context are WILDDD
I've worked with and have read a lot on information theory, yours was by far the best in explaining what bits are and what the formula for entropy means. Great video!
Information comes from information information hakusho. Only the moon can know if information exists or not. Magic? Hmm? Exactly.
I appreciate that you correct your mistake with more knowledge and exhaustive explanation. That's the correct way of doing science and I hope you do more in the future!
I appreciate the analysis from an information theory standpoint. There might be an interesting point to be made from the best theoretical analysis to develop a solid strategy in wordle. I believe the best theory would be one from a linguistic standpoint. For example, we know that (in the English language) nearly every word needs a vowel. So early words that eliminate all the vowels are handy - adieu, ouija, etc. For a 5 letter word, you can probably expect 2 vowels, frequently 1 vowel, but rarely 3 or more. Clear follow up words that eliminate any remaining vowels (if necessary) as well as picking up on letters that commonly form consonant blends are useful. In a 5 letter word with 1-2 vowels there will have to be a consonant blend or two. S, R, L, H, and rarely w are our letters commonly used to form consonant blends. You might decide that it's unnecessary to eliminate all the vowels in the first 2 guesses, and that it's better to eliminate the more common vowels and letters used in consonant blends on the first guess instead. Therefore, SHARE is actually a very useful word as a first guess. It picks up 3 of the most common letters used in consonant blends, as well as the common A vowel - and perhaps the most useful vowel, E. Of course, a curve ball can always come your way. You need to remember the use of double letters, and there are the consonant digraphs that are tricky such as KN, NG, or MB. A linguistic analysis may not get you the absolute best starting word, but it can give you the tools needed to beat the game better. It would be interesting to see whether information theory or linguistic knowledge gets better results overall.
Interesting. I'd been using Steal, which is an anagram of Salet, as my first guess to pretty good results.
I'm curious to know how the bot performs on Hard Mode, where every subsequent guess must include the letters you've found before. Since that restricts how much information you can gather on later guesses, can the bot still maintain an average 3.4 score?
As you say, Hard Mode limits the information you can gain, so presumably the optimal score (and the bot's score) would get worse, but I think something else interesting might happen, too. The Hard Mode rules, by constraining your guesses presumably drastically prune the search space. So, even though the optimal score is probably worse than in easy mode, the bot might be able to search more deeply and come closer to optimal than it could in easy mode. It's not impossible that it could find the actual optimal strategy as opposed to a heuristically close one, but the exponential explosion of possibilities probably mean it couldn't. But it should be able to search deeper which is cool.
@@doctorbobstone i think it would make it easier in that case, since it obviously knows every word unlike humans, and can obviously make up optimal paths after said words. the thing to keep in mind here too is, the game is also a robot, so it specifically makes it in a way that you cant just 'lose', as in literally being softlocked out of an answer (if you're a robot ofc) so BECAUSE of that, i think it would actually end up favouring the algorithm by quite a bit since it gives a LOT more information to work with by extension
I find I actually solve the Wordle in less guesses when I play on hard mode. If you play "easy" mode, you are intentionally playing words that you KNOW are wrong, since they don't contain letters that you've been told are already in the answer. If I get 1 gold square in my first guess, I will sometimes play easy mode. But those usually end up being 4/6 or 5/6. (my avg is 3.8/6 out of 42 games and I've gotten three 2/6)
@@High-Tech-Geek I agree. When I realized there was a hard mode, it didn't matter because I had been playing easy mode as if it was hard mode, using the letter in the word and I have to say, I think that is easier. When you guess a word that you know is wrong just to eliminate one more possible letter, you're wasting a turn. I have been averaging 4.1/6 out of 37 games played with 24 of those games being 4/6.
@@Charizard24682 if you're saying that the bot could potentially play better than the humans, I agree. Similarly if you're saying it could get closer to the optimal score, I agree. If you're saying that it could literally be easier in hard mode (as in better score for the same word), I think that if that's true I'm missing something about hard mode. You could (AIUI) play easy mode using hard mode restrictions for your strategy if that was the optimal choice, so the flexibility of easy mode should give you strictly more information and the ability to reduce the search space faster (or, in the worst case, no slower) than playing in hard mode would. Is there something about hard mode in missing?
Love the ending! So interesting that the computer algorithms are optimized for words with lots of vowels. My personal strategy is to start with words with lots of consonants, since I will likely "accidentally" guess the right vowels eventually if I use a different one each time. Having a couple consonants is more helpful for me to guess the word than, for example, knowing that "e" is the fourth letter. Very interesting.
I agree, and think it's easier to guess the vowels from the consonants than the other way around. At least for human use.
@@bobsassaman564 Yeah, it definitely is.
I loved how similar the theory behind the wordle probability is to poker theory! Maybe an idea for a video..?😬
ABOUT followed by STICK and then RHYME< is the best for me. just because it covers all the vowels and the top 3 constants, i think. S, T and H
A very interesting 3-word opening sequence that a friend taught me was: Ideal, Snort, Chump. Works like a charm, but it's not the best to get the answer in as few guesses as possible.
I find the best 3 word combo is: ROUTE, CLASH, PINKY
I used to use Ready, Climb, Shout! But I've found I get the most satisfaction getting it in three guesses, so I've been using Biome as a start for a while now. The 3-word combinations are amazing at getting it in four guesses
@@blankpyrosis yeah for me 3 is great, 4 is ok, 5 or 6 are trash and not getting means I need to go back to bed
UNITY BREAD CHOPS is mine :)
i used arose and tulip and decide which letters i want for my third guess
This was such an interesting video! I learned about overfitting, heuristics and entropy in uni earlier this year, and being able to place all of that here felt so satisfying :)
That sounds so satisfying! May I ask what you're majoring in?
Route, snail was my go to. Great game with many great strategies.
Just found this video, and I must say I find it both amusing and vindicating. I’ve been using LEAST as my opener almost since day one. Sometimes I’ll mix it up with STALE or another anagram, but the letters are the same. Must be a good one, because I’ve never lost. Maybe I’ll throw SALET in as another of the anagrams.
This second video is so great, Something that is missing in the age of online self-presentation: admitting errors and failure and then fixing it. Brilliant video! Really good!
The double "E" problem happened to me last semester in high school coding! I didn't know how to solve it but and I asked my teacher, but she said to not worry about it. This helped though!
It’s too bad she didn’t help you figure it out! If you’re still interested, one way to approach the problem would be to keep track of which letters have been “matched” (i.e. assigned a color); this allows you to prevent matching the same letter twice.
Congrats that it helped you!
I'm so glad you put out this video! I feel validated. I have a few friends that kept telling me "crane" was the best opener, but I was like "eh really?". I picked a word that I liked that I felt gave me more information with each word including common letter patterns. This algorithm made sense, but I felt like the thing this missed was the information you can gain from grey letters and common word patterns. For example, I like using the word 'Horse' because words that have a H, typically start with H or it is part of a pairing (TH, SH, CH, PH). If the H is yellow I will use a word that has all the rest of the pairing letters. Also, it doesn't account for the way each person thinks. Thank you so much for updating!
i usually start with trace or stale, so I now feel very validated
I have such a deep respect for this channel… I’ve always hated math but ever since I started watching this channel about 2 years ago I can definitely begin to see its beauty… the Fourier transform series is just amazing . Really paired nicely with Varitasium’s latest video on calculating tides and nondigital computers. Your animations are awesome and your voice is lol very soothing.
No way, he taught me Fourier transform yesterday, and it took me until literally now to realize he also made the wordle video! huge respect to them!
Even when you're wrong, it's still great to watch and learn from you! Way to go Grant! Keep up the good work!
Crane is the best starting word today!
Soare being the best one-step-deep opener pleasantly surprised me seeing as my prefered opener is "share"; just one letter off. I quite like "share" as an opener because of the fact that eliminating the H early on means that i can also eliminate clusters such as TH, CH, PH and so on… great video as always!
yeah, that H can be really tricky when trying to limit the possibilities on later guesses as a human - I'd bet SHARE, SHIRE, or SHORE is one of the best human-viable openings, especially if you have a planned 2nd word when you don't get a lot of info (I run with STORE -> PLAIN, but SHORE -> PAINT might be better)
I love this take on videos - wanting to get a cool fundamental point across but also "knowing how the internet works..." and having to address the "technically correct" question even when it's totally irrelevant. I think I saw someone in your comments on the first video pull an "Um actually..." saying that they'd run the entire word list at full depth and "wordle was a solved game" - but that doesn't teach you anything! I don't even remember what word they said was the best opener, but I remember your information theory arguments, which are so much more interesting and useful. I can't tell you how many comments I got on a video reply to Veritasium that said "Well I just got to the end and I'm not sure If Derek was wrong. Was Darek wrong??" and I want to reply to all of them "IT DOESN'T MATTER who's technically correct! Isn't electricity cool???"
It's really wild to see speed-of-light delays in action. It's just so instantaneous to us that intuition often fails, not to mention that the forces are all invisible! A student of mine was inspired by your Gerymandering video to create a similar project on his own, and we've had a lot of discussions about annealing, stochastic simulations, and emergent complexity.
Heyy love your videos also Derek was technically wrong
Great comment & great channel!
In the 'speed of electricity' video, Derek being right or wrong is actually pedagogically relevant because the "technically correct" clickbait teaser reflects how a person's mental model of electricity works. If people come away from the video thinking Derek is correct then they have imprinted the less correct model in their mind, less correctly corresponding to how the real universe works.
Ummm actually Why do people have to care about "learning value for other people" of a method they use to uncover the truth? That sounds like a very ineffective way of research, besides where's the fun in making a subpar guess just to satisfy people who aren't even that interested So yeah, educational videos should be educational, but I don't see why everyone should work like that or why would that be a better method at all
I did an analysis of letter frequency, and scored words based on occurrence of those letters. I didn't use the official Wordle list, I garbed the Linux English dictionary. Then I sort the words in that list by most frequent letter occurrence (I drop duplicates as those don't reveal as much information) and came up with my go-to starting word of "arose" ... then as you play you fill in your grey/green/yellow results, and it builds regex patterns that it then uses to eliminate words from the list, and then sorts the remaining words by letter frequency (of the remaining set) again. That started making Wordle boring, so I switched to Quordle and have never failed to solve it so far. Not as deep a dive as you took to run the game fully for each word, but good enough to give me a likely list of words to pick from.
My strategy with wordle is to start with either Adieu or Audio (whichever works), then to base the second guess on a hopeful vowel catch. There are a lot of words, using the most frequent consonants and just one vowel so it's easy to 1-2 punch a lot of information out. For example, start with ADIEU, then (assuming there is an E) follow with STERN - two guesses with no letter overlap, and 1-4, 6-8, and 12 most frequent letters in the English language. It almost always gives enough information to make a very accurate guess 3rd - with both a lot of letters ruled in/out and placement of the vowel narrowed down considerably. Then it's just about using your knowledge of the structure of English words to guess intelligently - Y is 19th most frequent, which is almost the lower 3rd of the alphabet, however is commonly used as a stand-in vowel, so if you know there's a single vowel at the start or end of a word, it's highly likely that the word either has a repeat vowel and/or a Y in it.
As more of a game theory guy, I'd be curious about applying different goals than just "lowest average score". "Don't Fail" is a logical goal in order to maintain success streaks, and then "Best score without the possibility of failure". Things along those lines.
I'd expect that people who try to play Wordle well lose very seldom. I've only played it 20 times, but I've only had one six and never lost. (My six was because S_ILL can be completed with H, K, P, T and W; I'd eliminated T already, didn't notice the possibility of P and guessed H and W before getting SKILL.)
My small group plays "quickest response" - any screenshot sent to the group is timestamped, and if quick there's no time to lookup any published correct answer
I was having exactly this discussion with someone. MIN MAX and LOWEST AVERAGE are not as fun as highest chance to get in 2 or 3 (i.e. a 4 5 or 6 are meh results, what you want is the buzz of the quick result, maximizing for the chance for that is quite a different idea)
Also, personal goals for the fun of the thing. I like to see how well I can do creating a different starting word every day using a couple of simple criteria. I think the game would lose charm for me if I just followed a rote formula each morning.
Lowest avg score will invariably lead to not failing though. That's the whole premise of the video. Quantify the most likely options and reassess with the limited word pool you create
9:20 I can confirm that NYT already changed the list, I used to open with ABOUT, PIERS, and LYNCH depending but now lynch is not in the word list which makes it an unacceptable guess (and completely screwed me over on CYNIC day, when I found this out lol). Could do an updated video with the new word list, if available somewhere!
ahhhhh so that's why I could no longer use the word b*tch
@@ebergelefsen nor slave or whore
'cynic' day was NOT a nice day!!
@@ViKBiTViT rly? i got that one quite easily! and I used crane haha
cynic was tough, first time it ever took me 6 guesses
I based my heuristic on letter frequency per position for five-letter words, which led me to highly value vowels. Word choices like ADIEU and AUDIO stuck out to me, but I had the sense that they overvalued vowels (vowels often bully each other for position, and it's not very common to have many vowels in a word anyway). After a fair bit of trial and error, I settled on AEROS as my primary starter. It was cool to see SOARE matching my starting letters, but arranged in a very different order. My other fun wildcard trick is that I've dedicated some memory to recall about twenty different words that contain four consonants and 'Y'.
NYTimes stated they got rid of words ending in s (at least, plurals of 4-letter words). Soare would then be a btter choice than aeros.
I’ve always had great success with “strep” as my opening word. E and S are two of the most common letters that appear in English, and T and R are near the top.
Oh, I don't even play the game. I'm here purely for the math, this stuff is interesting. Please do keep up the great work!
From my understanding "Sallet" actually pronounces the hard t at the end. There is a separate word in french for the helm. But the helm typically covers the top half of the face, with a gorget or... bevah? bever? Covering the lower half of the face. Sometimes they were attached, and other times separate.
It's spelled "bevor" usually, and yes, often sallets were used with them, but not always. Early sallets did not have visors and the helm's name comes from the italian "celata"
Yes, and the pronunciations are shown on the screen when he googles the word.
@@ARR0WMANC3R funnily enough, it's pronounced beaver. Sallets are my favorite helmet so I was aghast when he pronounced it so badly, even when the IPA was onscreen.
@@SirSkeleto nobody looks at an ipa and thinks "ah, the prounciation". 🙂
@@pmcgee003 some of the symbols are pretty arcane, i will admit, but that's definitely a T at the end there
I like GHOST as a first answer because it at least has the potential to tell you if any of the common digraphs GH-, SH-, or TH- could be in the word. I usually guess something with the vowels A and E next because there are fewer vowels than consonants. Bear in mind that I have never tested this, but I have an average score under 4 so it seems to work.
I’m glad I saw this, because it’s done me well, 487 games, 366 day streak and (spoiler) my first 1 solve, with the only interruption being a single day that I did Pzazz.
I don't think knowing the "optimal" word ruins the game. What's optimal for a computer is definitely not what's optimal for me, because my brain doesn't work like a computer. I start with "adieu" since it gives me a clue as to which vowels are in the word, then I build off that with which consonants are fairly common. Crane/Salet simple don't work well enough for the process I have to go through to get the word.
ADIEU is great for solving the word within 6 tries. But it's not a great word if you are trying to solve it in the least amount of tries. Knowing the vowels still leaves you with plenty of consonants that use those vowels.
ayyyy, ADIEU club!
before learing about these videos I used stale everytime, and it's amazing how this word is similar to crane and salet lol
exact reason I use "audio"
Seems boring to always start with the same word. I try to come up with new words every time. Keep it fresh.
Personally, I really like using the word AUDIO as of late, it just gets past nearly all of the vowels for me, and I get through likely consonants based off what I get. If I feel like I need to determine if there's an E, I will of course go for it in my second or third guess. My average score is 3 or 4! I used to use RAISE because it got rid of pesky letters to me. If that was a full blank, I love to RAISE DOUBT on wordle! But that was before I started winning more often with 2 or 3 using audio, it feels good having a strategy that works for me that clearly isn't very orthodox
adieu is a pretty good one
buoys! BUOYS!!!!
Louie and Smart are my go to
There’s no need to get all the vowels
@@Pondredia ouija is also ok
I don’t really have any desire to discuss Wordle or watch a video on it, but the prospect of an ultimate starter word is pretty cool, and also I wanted to share that for whatever reason I almost always start with “weird”
Thanks for this! It has probably been asked before, but I am curious: Is there a difference between hard mode and normal mode, as to what word would be better to start with? I have a feeling it would.
A year too late, but definitely still curious on the answer. I’ll keep scrolling to find the answer, but it’s frustrating to see everyone commenting “I’ll guess ADIEU then STORY”. Yeah, obviously that’s the more efficient. But that’s why hard mode is harder.. you can’t just guess two random words with a collection of the most popular Wheel of Fortune letters. Would love to see another video about hard mode, not to find the “best guess”, but I think it fundamentally changes second guesses and the game in general.
I usually start with "steal", trying to knock out some of the most common letters, then "chomp" to try to group together some less commonly used letters that are hard to fit into others words
“Chino” is a better 2nd guess than “Chomp”
fellow steal starter here
I would use "stale" instead if I were you since it has the same letters but has the vowels in more common position, increasing your odds of getting greens.
@@MagikMako no its, flame shunt brick podgy Shunt is 2nd
I use AUDIO and STERN as my first 2 opening words. I have a tendency to over analyze and thus take too much time on word games. So these 2 words actually give me just enough info to get it on my 4th word 90% of the time. 9% chance on 3rd. 1% on 5+.
This is a great start. I start with SAUCE and JOINT and get it on the 4th most of the time. I think I might start with audio since that's a great word to get all the vowels
Dairy and Caulk
RAISE and DOUBT
@@guywithglasses5309 based on what you said, I assume you got todays then 😂
@@sgtpepprs3 Actually no I didnt lol after not getting it I am using it as a starter from now on out of spite
I have a topic suggestion you might be interested in: Homomorphic encryption and what mathematics theories there are that this works. I know it doesn't has anything to do with this video or PI, but I think you can explain this topic very well so that everyone is able to understand the basics.
Something I'd like to see evaluated (forgive me if this exists buried within the comment thread) is how the "Hard" vs Standard mode requirements impact expected score. The "Hard" mode requires players to utilize what is learned in previous guesses, vs. the standard mode which does not force guesses to map to previous information. I play using both models depending on what typically guess one yields. If I get say three unplaced common letters, guess two might be a word totally orthogonal to guess one. Most of my successes in three moves came from this. Curious if a brain clearly bigger than mine had insight on how the strategy would work in theory.
I use a two-word starter: ORATE + JUICY. None of the letters repeat, so I have the benefit of knocking out 10 letters (and every vowel) before I make a real guess. While this sacrifices the opportunity to win on the second guess, I find it significantly raises my (human) ability to win on the 3rd or 4th guess.
Yeah I do the same kind of thing. I rotate through a few different starting words, but i always try to get some of the "uncommon letters" out of the way first (they are more common in wordle by virtue of wordle words rarely being commonly used words, whereas the RSTNLE deal came from running the letters in a newspaper, aka lots of "the"s, "and"s, "tells"s, contractions, and other words that wont show up in a wordle.) and make sure to be able to make a second guess with all new letters if the first guess is a complete bust
ADIEU, FROST, GLYPH are very good starters. Surprisingly, that knocks out 15 different letters.
The most common letters in wordle answers are EAROTLISNCUY, so I often use "lysin" after "orate", especially if "orate" doesn't get any hits.
@@gmalivuk I came to that conclusion too, and I’ve been following up with INCUS.
@@Zerphses Now that I've gotten my own scoring algorithm fixed (my first go had the same problem as Grant's, though I knew it at the time and just went with it anyway because it's fast and gives only slightly inaccurate results), [[redacted because it was still broken]] LYSIN is a bit more informative than INCUS, against both the list of possible words and against the shorter list of possible answers.
This is great
I am so happy because it's good online apps
Excellent app
CRATE is the best one for a real mind to think, as it limits the most answer options on average for any word, as you have a very small chance of all greys (follow up with lions or solid) and when you do get a letter, most options with that letter are thrown away by the other letters being there (or not) For people that like combo starters, PARTY SLICE and HOUND/MOUND (M appears eight more times than H, but H is in about 20 more words)so pick which one you like more, it doesn't make much of a difference
personaly speaking i think the best words go in this order regardless of what letters u get right. fjord, waltz, nymph, quick, gybes. that leaves u with one guess but also leaves u with every letter in the English alphabet used accept for two x and v so its pretty easy to get the word after that
I keep it interesting by starting each day with the previous day's answer. I'll never get it in 1, but it keeps it challenging sometimes. Other times, it actually starts of incredibly strong. On the day when the answer was ultra and the previous day's answer was ulcer, there was no way for me to get it on anything but the second guess.
I like to just use a random word and hope that one day I solve the puzzle on the first guess
My first two guesses are almost always irate and soupy, because by the end of round 2, I know exactly which vowels make an appearance, and I've either included or excluded p, r, s, and t, which I find really reduces the possible answers.
I have fun using Adieu to figure out what vowels are in it then go from there
@@Arcavi0us ooo
@@madscurr I use tears and pound normally to start off
I would be interested in a statistic of how often it's better to choose a word that's already impossible because of the results of the previous guesses.
Probably very common on the second guess and pretty rare after that.
Unless my 1st or 2nd word makes it somewhat obvious, I usually go for 15 unique letters among my first 3 words. Something like "panes", "fight", "could".
hypothetical situation that might fit this: say you have everything but the first letter, and it ends in _ATES . let's also say you only had 2 guesses left. it could be GATES, FATES, RATES, DATES, HATES, MATES, etc. instead of guessing what the word is, your 2nd to last guess would be better served entering a word you know is wrong with multiple of those options covered like FRAUD or HARMS since that would elimnate more possible options. pretty specific i guess but that dynamic between finding out new letters vs what position the letters you already have are in is pretty interesting.
@@bogidrums If my first choice of REACH doesn't land anything, I run with MOIST.
I used CUNTS today which used U which I already knew wasn't in the word.
I like CREST and AUDIO for my first two guesses. Five vowels and most of the common consonants
While I always thought getting the vowels out was most important, my word of choice was Baste. A as second, ends in E, uses B and S... T I think is more common than K. Haven't played in a bit, though. I should try again with my word and yours to see the difference firsthand.
I always tried to maximize the possible digraphs in each guess, as to hopefully strike at two letters to match. STING = ST, SN, NT, NG for example.
ooh, nice! this is good because it helps you figure out where the vowels might be based on if there are digraphs
@@Cloiss_ But alternatively, digraphs by definition likely contains less information because they're basically 2 letters representing 1 sound. But like the video said, human heuristics differ so much from the game-theoretic play that it's totally possible this works!
@@lekhakaananta5864 I suspect digraphs actually provide more information, since Wordle doesn't care about sounds, and eliminating possible digraphs makes it easier to shrink the list of reasonable answer possibilities.