Solving Wordle using information theory

https://www.binghamton.edu/news/story/6327/s-m-a-r-t-these-researchers-used-math-to-crack-wordle

by hhs • 1 month ago

60 64 1 month ago

binghamton.edu

https://orb.binghamton.edu/nejcs/vol8/iss1/6/

ano-ther• 1 month ago

Paper: https://orb.binghamton.edu/nejcs/vol8/iss1/6/

jezzamon• 29 days ago

I thought this was old news? I remember people making videos about using information theory to solve Wordle back when it was particularly hyped. (After writing this I checked, there's even a 3 blue 1 brown video on this)

My favourite along those lines was solving wordle in 1 guess using the distribution of coloured squares on social media https://www.kaggle.com/code/benhamner/wordle-1-6

GL26• 29 days ago

Yes 3blue1brown made a series of videos which explains it : https://www.youtube.com/watch?v=v68zYyaEmEA ; https://www.youtube.com/watch?v=fRed0Xmc2Wg .

seirl• 28 days ago

Also same exact name as the paper?

csande17• 29 days ago

It's not clear how the strategy in the article differs from the one used by Wordle Bot, the analysis/feedback system that Wordle links to on the victory screen at the end of the game: https://www.nytimes.com/interactive/2022/upshot/wordle-bot.h...

The first page of the published paper ( https://orb.binghamton.edu/nejcs/vol8/iss1/6/ ) also claims that the game was developed by "Josh Wordle", so maybe it just isn't the highest-quality scholarship in the world.

wasabi991011• 29 days ago

also claims that the game was developed by "Josh Wordle",

To be fair, that is a 1 letter typo; the developer is in fact "Josh Wardle".

madcaptenor2• 28 days ago

I had wondered if getting wordle in 1 based on social media data was possible! Now I know.

kps• 29 days ago

To paraphrase von Neumann, isn't that how everyone does it?

_ache_• 29 days ago

Yeah, exactly. My girlfriend did it too, using information theory. In my solution, I did chose a word that minimized the size of the larger set of possible guesses. It's not strictly information theory, but it's a good approximation of finding 'the word that brings the most information [this turn]'.

I suspect that this researchers work involves formalizing and proving the optimality of their solution.

gkoberger2• 29 days ago

This isn't groundbreaking science, sure, but it is a great way to get people interested in a topic. After all, it wouldn't be on Hacker News if the word Wordle wasn't in it. I'm a huge fan of using things like this to teach science, math and engineering.

We started using "Solve 100 Wordles programmatically" as our technical interview, and people _love_ it. They get really into it and have fun. It's pretty easy to do inefficiently, and it's great to watch people build on it and try to improve their scores.

It has two benefits: 1/ everyone clearly understands the problem 2/ people see it as fun rather than a drag.

tialaramex• 29 days ago

Another small benefit: Everybody understands this isn't what the job is. Nobody is hiring you to beat Wordle, and you solving this is clearly not somehow on the path to their actual task, they are asking you if you can write software which solves a problem and you're demonstrating that you can do that, which makes sense.

I think "Solve 100 Wordles programmatically" sounds like a lot of work, so that'd probably be a "No" from me unless it was last hurdle for a job I was enthusiastic about but unlike "Write a program to solve this class of graph problem" I at least wouldn't be worried that you're trying to get me to do work for free.

Actually Wordle solver as Code Review task sounds like maybe a more interesting live interview than the one we do today. "Here's this mediocre Wordle solver, what is your feedback in review?" has the advantage that they've probably seen a Wordle puzzle before but it's not an example problem they've seen in fifty textbooks.

gkoberger2• 29 days ago

Well, the 100 Wordles is just "Solve one Wordle" in a for-loop. If you're an even somewhat decent engineer, it takes under 10 minutes to get it working (inefficiently).

Then we encourage people to do whatever they want next: improve their average score, build a frontend UI for it, solve on Hard Mode, etc.

In the past, we never did technical interview questions like this. We always asked people to bring their own project, and work the way they want to. However, with the addition of AI, we hit a wall: we want people to feel they can use AI in a way that mimics how they'd actually work day-to-day, BUT we also need a simple check to make sure they understood engineering basics.

doctorpangloss• 29 days ago

it's a drag... i mean if you're 22 it's fun. if you love games, or you're older, or mature, i don't know. the pretentious stuff Josh Wardle says is tough. (https://gdcvault.com/play/1027882/-Wordle-Doing-the-Opposite). he says all sorts of stuff that is wrong ("innate understanding of language" except if you're a kid, which is the average gamer...)

it's like when interviews asked people to implement 2048. it's not even the best version of that game, Threes! is.

plants• 29 days ago

plug to my blog post where I did something similar a few years back! https://willbeckman.com/wordle.html :). Not sure if this is an identical solution, but it was a fun little project.

dredmorbius3• 29 days ago

I'd quickly realised that a set of words which covered most of the alphabet (20 words, leaving b, g, j, q, v, and z excluded) allowed solving virtually all Wordle puzzles. The game quickly lost any challenge.

  wimpy
  crowd
  thank
  fuels

Altering order might give faster results. The order presented leaves the most common letters (e, t) for last. Z is quite uncommon, q is virtually always followed by u, similarly common pairs such as ch, sh, and th, as well as three- and four-letter combinations ing and tion, though those won't show frequently in five-letter words of default Wordle.

It would be possible to vary word choice based on revealed matches and hits, but if your goal is simply to solve (rather than minimise attempts), the above list works quite well.

magneticnorth3• 29 days ago

The game quickly lost any challenge.

I only play on hard mode for this reason. My next guess must always be a possible answer based on my current information, and that varies the puzzle enough from day to day that I still find it enjoyable to play occasionally.

drivers99• 29 days ago

Does that not lead you to situations where you have to guess the last letter on each round? Or do you choose words to avoid getting trapped in that situation in the first place?

dieselgate• 29 days ago

I haven't played wordle much since it was purchased but NYTimes ran a story in the last couple days saying how "Hard mode" is actually easier based on their research. Just throwing it out there as food for thought not in a "gotcha" sort of way.

https://www.nytimes.com/2026/06/18/upshot/wordle-hard-mode.h...

dofm• 29 days ago

Yeah, I play on hard mode. I have three words that seem quite unlikely but cover most frequent letters and common patterns, that I use to quickly work through possibilities in my head.

I haven't failed to solve in months and I usually play every day. I only picked it up again a few months ago in search of a little ritual for the morning. Wordle, Parseword and the NY Times Sudoku on Hard mode. Good for 25 minutes of break at some point in the morning.

shawn_w• 29 days ago

Until I read some of the comments here, I didn't realize anyone used any other approach with Wordle. A word that doesn't include the known letters is just unnecessarily increasing the number of guesses it takes to solve and doesn't make sense...

Brendinooo• 29 days ago

gotta see how your strategy holds up in sexaginta-quattuordle

lkbm• 29 days ago

This is how I played Quordle and Octordle, but with just three words (doubt, glyph, raise). Might not scale for all combinations.

mcphage2• 29 days ago

I'd quickly realised that a set of words which covered most of the alphabet (20 words, leaving b, g, j, q, v, and z excluded) allowed solving virtually all Wordle puzzles. The game quickly lost any challenge.

You can even go further—there's a set of 5 words which uses 25 out of 26 possible letters, leaving you one more word to enter the right answer.

But here's the thing: while that means you'll almost always win, your # of guesses will always be high.

but if your goal is simply to solve (rather than minimise attempts)

Pretty much nobody's goal is to simply solve. Once they've played it for a few days, everybody's goal is to minimize guesses. That's the flaw in having a long word list—you always do badly.

topaz0• 29 days ago

The only virtuous goal is to type some fun words imo.

dredmorbius3• 29 days ago

There are times that enough hits are revealed, or other patterns, that guessing the actual answer is possible in fewer than six attempts. I think three-try wins are not unheard of for me, though honestly it's been years since I've played.

starky• 29 days ago

In simulations, their approach solved 99% of Wordle puzzles, while the traditional method solved just 90%.

This seems wrong to me, getting a 98%+ solve rate for Wordle is pretty common.

kens• 29 days ago

What I'm interested in is the best starting word. Using Shannon entropy, the paper finds that it is "tares".

bee_rider• 29 days ago

I think it depends on whether you use a conventional dictionary as the population of possible words, the valid guess list, or the (known, at the time, for the originally version of the game at least) possible solution list. I guess you could also use the possible solution list minus the words that had already been guessed in previous iterations.

I mean, you had to give yourself an artificial constraint because IIRC the next solution was actually built into the page anyway, not obfuscated in any way.

devNoise• 29 days ago

A couple years ago, I wrote some js to sort a 5 letter word file. The value for a word was based on letter frequency in each letter location. Then I took the words that didn't have repeating letters. The I choose a word at the top that I liked and grep'd for some other words to use for the 2nd and 3rd try. In 3 tries, you have used over half the alphabet and all vowels. This usually lets me get the word by the 4th try. Here are some word sets you can start with.

- pares, moity, bundh|bunch - pares, monty, build|guild - pares, doily, bunch|munch

- pores, banty, guild|mucid - pores, canty, build|guild - pores, manty, build|guild

- bares, ponty, guild|mucid - bares, moity, punch|dunch - bares, monty, guild|child

- cares, ponty, build|guild - cares, moity, bundh|gulph - cares, monty, build|guild

futune• 28 days ago

I did this immediately when wordle came out, too. Here's what I got for top 5 starting words along with the entropy:

"tears" 6.00570508021708 "teras" 5.95925128791865 "teals" 5.90122552646963 "tares" 5.88711668336419 "lears" 5.86158270234831

Using another metric which I called consistency (I think it was just minimizing the biggest bucket possible afterwards, trying to beat absurdle fast), the best starting word was tied between "arise", "raise", "aesir", "reais" and "serai". All of these have a worst-case scenario of 168 valid words remaining.

I find it interesting that different implementations seem to come up with slightly different rankings. Maybe the dictionaries are different?

boothby3• 29 days ago

I crushed wordle within a few days of its popularity entering my sphere. It was pretty easy to brute-force a decision tree minimizing the average number of guesses using a lowly python script and a few days of qpu time.

Don't Wordle[1] is significantly more interesting; I've got a solver but the maximum score takes my lowly python script upwards of a day (per day) to solve using brute force. For now, I solve it with a heuristic that terminates in about 20 minutes. My old wordle solver was useful to find a good but suboptimal tree for identifying the answer in 5 undos or less.

Today:

  Don't Wordle 1491 - SURVIVED
  Hooray! I didn't Wordle today!
  ..... 8089
  ..... 4647
  ..... 2492
  ..... 1026
  .Y... 231
  ..G.. 100
  Undos used: 3

    100 words remaining
  x 10 unused letters
  = 1000 total score

My puzzle ethics are: you can and should download the dictionaries of valid answers and valid guesses, you're allowed to keep them separate, but you must not keep the list of answers in its original order.

[1] https://dontwordle.com/

cvoss3• 29 days ago

Another great variant is Unfair Wordle [1]. The opponent does not fix the answer upfront but instead evades the player's guesses as long as possible, providing you with the least information it legally can give (according to the usual rules) while still preserving a valid game completion path. The result is that your guesses end up looking extremely unlucky in retrospect.

[1] https://tweakimp.github.io/unfairwordle/

arcastroe• 29 days ago

Woah. I was surprised to win on the first try: cramp, ghost, blind, bulky, bevel, bezel

BoppreH• 29 days ago

I also found it surprisingly easy, starting from my triple "olden, party, music" (calculated years ago for state space coverage and memorability). Followed by "fight", then either "shark, brash" or "brash, shark".

Maybe it's harder if you're a careful player that doesn't waste the first three words.

madcaptenor2• 29 days ago

see also "absurdle": https://qntm.org/files/absurdle/absurdle.html

headz• 29 days ago

I "crushed it" when it came out with Right click -> View source -> Find the list of words. All the future words were in the code already.

bre1010• 29 days ago

I also noticed this. I was working on my solver but my dictionary kept suggesting options that wordle didn't consider to be words at all (what the heck is usr/share/dict/words anyway??). So I looked through the source for the dictionary wordle is using and then also found the list of all solutions which really took the fun out of that whole exercise :(

boothby3• 29 days ago

This is why I explicitly mentioned that in my puzzle ethics. I reject that as a valid solve.

(And amusingly I said "qpu time" when I meant CPU)

NooneAtAll3• 29 days ago

what's qpu?

boothby3• 29 days ago

That's a typo, I meant CPU! I work with quantum computers, where we do discuss QPU (quantum processing unit) time, but that's not how I did this.

trollbridge2• 29 days ago

Hacker News commenter uses grep -i ^.u...$ /usr/share/dict/words | grep -i c | grep -i -v '^..c..$' | grep -i -v '^...c.$' to crack today's Wordle

macintux• 29 days ago

I admit after a half hour of flailing a couple of days ago I got desperate enough to use /usr/share/dict/words. Turns out "emoji" isn't in my copy.

trollbridge2• 29 days ago

Those are my favourite Wordles. It becomes more of a challenge.

adamgordonbell• 29 days ago

It's trivial to determine the best guess at any point based on what options it cuts out.

But I ended up building an alphaWordle, using MCTS and a reinforcement loop just to get a feel for how AlphaGo approach to solving games works.

It's not a 'smart' way to solve it, but its pretty instructive and I could compare its moves to the theoretical best move to see it progress.

https://github.com/adamgordonbell/bitter-lesson-demos

senderista• 29 days ago

Isn’t “maximize information gain at each step” already the standard approach to such problems?

_whiteCaps_• 29 days ago

If you want a challenge, try https://lirdle.com/

One letter per line is a lie.

anigbrowl• 29 days ago

I think most people understand such problems are analytically tractable, but not everyone understands that the challenge and pleasure of a game is being able to do so unaided.

waffletower• 29 days ago

I made a Wordle solver in Clojure a while back which is backed by an English word frequency database. My family believes it is cheating to use it. Much more fun to write that code than to play the game though :D

Bratmon• 29 days ago

Didn't like 20 different YouTube channels do this exact thing back when Wordle was relevant?

aidenn0• 29 days ago

I wrote a wordle solver that was fairly straightforward; it just brute-forced all of the possible outputs, minimizing the maximum number of remaining possibilities.

The very first step was too slow to be interactive (IIRC it took about a minute), but fortunately can be precalculated. With a good first-guess, the number of remaining words is small enough that you can just brute-force.

Note that this is not theoretically optimal; you would want more than one level of lookahead for that, but it's good enough to solve almost the entire YAWL dictionary in 6 guesses or less.

whycome• 29 days ago

The NYT world bot that "reviews" your game also has entropy as one of the things it addresses I think.

I prefer to not have or use predetermined words. It's most fun to actually feel like you have a chance at solving it in one, and have the challenge of more or less information from first play. It's interesting to compare the solve distribution that the game lists when you're done. I wonder what information can be gleaned from that?

silentmafia• 29 days ago

Nobody mentions that this was already done by 3Blue1Brown?

https://youtu.be/v68zYyaEmEA

egberts1• 27 days ago

I have a tree.

I always started with 3 words

POINT LASED CRUMB

This eliminates not only the most frequent letters but least frequent letters (Bloom filter), leaving rest of infrequent letter set to guess by narrow deduction.

Never failed me.

Solving Wordle using information theory

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