Spoiler: Best words include: RAISE, IRATE, ARISE
Like many others, I’ve been playing around with Wordle to identify an optimum strategy. There are dozens of articles out there on the internet and people have many interesting approaches. However, nearly all of them disappoint me in one particular respect:
I stopped enjoying Scrabble a long time ago once I’d learned that the spirit of the game had been destroyed by a Scrabble Dictionary containing a vast number of words that no normal person would recognise or use. The game changed from a word game to a memory game. Now all the experts have a vocabulary of words that they never use except when playing Scrabble. That same malaise is indirectly threatening Wordle.
I say indirectly because it is clear that the Wordle solutions have been kept strictly within the bounds of a reasonable vocabulary; and this to the extent that it even seems that 4-letter words in their plural 5-letter form have not been candidates for solutions (although they are accepted as guesses). Bravo to the developer and the NYT for sticking to the spirit of a word-game.
However, Wordle uses two dictionaries: a solutions dictionary of 2,309 words that is used to set the wordle for each day and a validations dictionary of 10,657 words that are considered legitimate guesses in addition to the solutions. See footnote 1 for my reservations about the latter.
Other Research
Most researchers seem to be content to use both dictionaries. I disagree for two reasons:
- At the very least, the validation dictionary needs to be purged of all the nonsense which defies the spirit of a word-game; recommended words should not go to this bad place.
- More practically speaking, the validations dictionary never supplies the solution and it is not needed for a good strategy as I hope to demonstrate in this article.
My Analysis
Some commentators have used thought experiments (or common sense) to explain why their approach must be good. I think they overlook the fact that Wordle is, in fact, very easy (especially when not using Hard Mode) and when they get good results that seems to justify their analyses. Other researchers analyse reported games to reach their conclusions about what represents good play. Their conclusions must be tainted by selection bias since people show off good results (including lucky ones). Still others use computer simulations to learn what works best in terms of minimising guesses. I prefer these last approaches but I reject their reliance on the validations dictionary. All of them, as far as I can tell, play Wordle in easy mode; i.e. they allow the player to ignore letters revealed previously in order to widen their search.
My approach is closest to the last category but slightly different. My simulation works exclusively with the solutions dictionary, only guessing words which are in the dictionary to ascertain which guesses will pare down the solutions dictionary to as few words as possible in one guess. See footnote 2 for some remarks in this regard. I then use the highest performing words to guess every possible second word. I’m interested here, not in finding a second word (which, of course, depends on what has been learned from the first) but in measuring which initial guesses will lead, on average, to very small solution sets after two guesses. My simulations use Wordle’s Hard Mode; any letters revealed must be used in subsequent guesses. If a word has repeated letters, I never use it as a first guess since it makes no sense to not spread the net wide for information. (It surprised me to discover that as much as 31% of the dictionary was made up of words with repeated letters.)
Approach and Conclusions
We have 1 chance in 2,309 of winning in one guess (if we stick to the solutions dictionary). My simulations calculate that , if we played any random first word (again, from the solutions dictionary), we reduce the solution set, on average, to 176 words before our second guess. I’m looking for words that do much better.
My approach proceeded by taking each word in the solutions dictionary in turn as the first guess, and then evaluating it against every other word in the dictionary as if that word was the wordle being sought. Evaluation means using what is revealed to count how many words remain in the dictionary and finding the average for all words. On average the dictionary was reduced to 176 words by the first guesses, but some first guesses were much better than others. This was a very computational intensive procedure involving almost 4 million evaluations.
How many words are still in the vocabulary before the second guess?
Some words drove down the solutions dictionary size enormously. Here are some really good performers who reduced the set to less than 80 words before a second guess was tried:
- < 65 words: RAISE, ARISE, IRATE
- < 70 words: AROSE, ALTER, SANER, LATER
- < 75 words: SNARE, STARE, SLATE, ALERT, CRATE, TRACE, STALE
- < 80 words: 10 words including: AISLE, LEARN, LEANT, ALONE, LEAST
Note the number of anagrams or near-anagrams in the lists showing very clearly the letters that need to be investigated early. Importantly, I think you can probably recognise and define every word in this list, so that we remain within the spirit of a word-game.
Some words touted on the internet didn’t fare nearly as well, such as AUDIO (average 182)
In a similar vein, some words were (unsurprisingly) shown to be very poor first guesses. I note with a smile that CHUMP made this list, one of the words I chose (randomly) for a previous article. You need to assiduously avoid:
- > 500 words: JUMPY, JUMBO
- > 450 words: BUXOM, QUICK, MUCKY
- > 400 words: FUNKY, DUMPY, PUDGY, CHUMP, GUMBO, WIMPY
… and after the second guess?
Assuming we chose one of the best words for my first guess, I wondered how many words would be left, on average, after our second guess. Remember, we’re playing in Hard Mode, where we must use any previously revealed letters in our guesses. So we need to identify those words which are frequently available after the eliminations of the first guess and which produce very reduced dictionaries after the second guess. Using somewhat subjective cut-offs I identified 63 which were each still available to more than 15% of the full dictionary and which produced dictionaries of less than 6 words (on average):
BRUNT CAROL CLOSE COUNT COURT CRUST ELITE FIRST FOIST GIANT GRUNT HEIST HOIST INTRO ISLET MOIST NOISE NORTH OLDEN ONSET POINT POSIT PRINT ROOST SCONE SCORN SCOUT SHALT SHEET SHIRT SHUNT SINCE SINEW SINGE SKIRT SLEET SLEPT SLIDE SMITE SNIDE SNOUT SONIC SOUTH SPELT SPILT SPITE SPLIT SPURT STONY STORY SUITE SURLY TERSE THORN TONIC TORCH TORSO TORUS UNITE UNSET UTILE WORST WRIST
There is sufficient diversity in this list that some of the words from this set will be available to the player, whichever word they chose for the first guess (and despite the prevalence of letters frequently used {A, E, R, S, T, N}) and that these words will greatly reduce the dictionary.
So I ran a simulation (230,000 games) where I used the above methods to choose the first and second guesses and then played random choices from the remaining dictionary against every possible wordle and I got the following result:
| Out in | 1 | 2 | 3 | 4 | 5 | 6 | Fail |
|---|---|---|---|---|---|---|---|
| %age success | 0.03% | 5.72% | 35.84% | 38.23% | 14.97% | 3.93% | 1.28% |
Interestingly, the failures mainly arise from a particularly pathological type of rhyming word. They share the characteristic that they differ by only one letter in the same position and that these letters are among the less common and so are frequently found last. The following 7 suffixes account for six or more words and therefore you can fail even after getting the entire suffix by your second guess. There are 20 further suffixes with five and between them all they account for 38% of all failures :
- EIGHT – substitute F, L, M, N, R, S, T, W (not to mention BIGHT, not in solutions)
- BOUND – substitute F, H, M, P, R, S, W
- BATCH – substitute C, H, L, M, P, W
- BATTY – substitute C, F, P, R, T
- COWER – substitute L, M, P, R, S (not to mention DOWER, TOWER, not in solutions)
- DAUNT – substitute G, H, J, T, V
- BILLY – substitute D, F, H, S, W
If BOUND was the wordle and you chose POUND as your first guess, playing in hard mode you still have a 17% probability of failure!! It’s just bad luck. I remember WATCH coming up a while ago in the game but I was lucky enough to choose it early.
Spirit of the Game
So that’s my analysis. Use it if you like but please bear this in mind:
The research above was my response to intellectual curiosity inspired by the game. If I am truly honest though, I’d have to say the a strong adherence to the spirit of the game would mean ignoring the conclusions revealed above and choosing your words as the spirit moves you. It’s a game and some of the fun is to be found in toying with it. Personally, I like to start with any mad word (of five different letters) that comes into my mind, usually aiming for less popular letters in the hope that I don’t reveal good ones too soon and be obliged to use them in all subsequent guesses. I still rarely need to go to a fifth guess.
Footnote 1 – Validations Dictionary: While the solutions dictionary is an excellent collection of words that should be recognisable to the average english speaker, the validation dictionary is a travesty containing foreign words, technical terms, archaic words, abbreviations and complete nonsense words such as CRWTH, CWTCH, GRRLS, GRRRL and PHPHT. I find the inclusion of these words to be against the spirit of the game, especially when we know that the solutions have been chosen with care to suit an average english speaker. Recommendations from people who analysed play have included words like ARLES, LARES, RALES, RAILE, REAIS, SOARE and ROATE. I have never used any of these words in conversation nor could I define any of their meanings. Plurals have also been recommended when we know they will not be solutions and an anagram might do the job as well; e.g. STARE instead of RATES. Using a validation word in a guess is an immediate concession that you will not find the solution on this turn, even by chance. The validations dictionary would be perfectly acceptable if its vocabulary were chosen with the same care as the solutions dictionary. However, it has no place in my simulations since it will never change the count of words surviving evaluation.
Footnote 2 – Perfect Information: Clearly this approach is a perfect information approach in the sense that I’m assuming the player knows all the 5-letter words in the solutions dictionary … and no others. That is most unlikely. Although I imagine that most players will know the full dictionary, it is more than likely they will know words that are not in the dictionary, hence the need for the validations dictionary so they are not frustrated to have a legitimate word rejected by the app. This means that there are other possible guesses which will reduce the solution space. However:
1) Allowing words which are not in the solutions dictionary won’t change the averages I have calculated for words already in the solutions dictionary except to the extent that a player will occasionally lose a turn because they tried one of them; the words chosen remain the best (even if not as good as advertised) since all words carry this risk
2) Some of the new words may perform better at reducing the solutions dictionary words but I’m inclined to doubt this on the basis that it was a set of anagrams or near-anagrams that won the above contest and words that I consider legitimate validations dictionary words (i.e. recognised by and used by the average english speaker) will be structured very similarly to words in the solutions dictionary. Arguably some 4-letter plurals will be good as the letter S is important.
Asynchronists 