The best first word in Wordle…according to maths

Can you calculate the best opening word?

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Wordle may seem like a word game, but really it’s a mathematical one. Rather than debate the best opening word, why not calculate it with information theory?

Many thanks to Tran Nguyen for all her help with this video…in particular, for inspiring the graphs showing the entropy of different opening words. And also many thanks to Jason Liszka for open-sourcing his Wordle-solving code, which I hacked mercilessly to do the calculations for this video!

You can find the table of the optimal opening words according to this method below:

The best opening words

 easy modehard mode
all wordscommon wordsall wordscommon words
rankwordturns to winwordturns to winwordturns to winwordturns to win
1.REAST3.604TRACE3.625TRAPE3.805TRACE3.840
2.TRACE3.608CRANE3.635PRATE3.808SLATE3.841
3.CRANE3.610ROAST3.638SALET3.811LEAST3.846
4.CARLE3.611SLATE3.640LEANT3.812LEANT3.850
5.TORSE3.613TRICE3.641REAST3.814TRAIN3.851
6.RANCE3.614CRATE3.642TRACE3.815PRATE3.852
7.SALET3.614SLANT3.648CRANE3.816SLANT3.852
8.SLATE3.616STARE3.648LEAST3.818CRATE3.854
9.SNARE3.617LEAST3.651SLATE3.827CRANE3.855
10.CRATE3.620PRATE3.653CRATE3.829TALES3.857
11.LEAST3.620REACT3.654CARET3.829TRIPE3.862
12.TRONE3.620CORSE3.655CARLE3.829SPATE3.863
13.CARET3.621CADRE3.655TRICE3.829PLATE3.864
14.ROAST3.621TRADE3.656PRASE3.830TRICE3.864
15.CARTE3.621TRAIN3.657SLANE3.832DEALT3.866
16.ROSET3.621DEALT3.657PEART3.832TRIES3.866
17.PRASE3.622SNARE3.657ARTEL3.833TRAIL3.867
18.TRINE3.622LANCE3.657RESAT3.834STALE3.868
19.TOILE3.623CARET3.658TRADE3.835CRAPE3.870
20.TRICE3.623STALE3.660PARSE3.835TRADE3.870
21.SLART3.625SLICE3.660REIST3.836LANCE3.870
22.PARSE3.625STORE3.660CARTE3.838CLEAT3.871
23.TRADE3.626SCALE3.661REACT3.838CARET3.873
24.SOREL3.627TRIPE3.662CARSE3.838TRIAL3.873
25.STARE3.627STILE3.664SETAL3.839CREST3.875
26.RAINE3.628TRAIL3.664TORSE3.839SCALE3.875
27.SLANE3.628LEANT3.664TRINE3.840PLANE3.876
28.REACT3.631TEARS3.666SLART3.841REACT3.877
29.CARSE3.633SAINT3.667TEALS3.841LEARN3.883
30.PEART3.634SPATE3.668TRIES3.841NACRE3.884
31.REIST3.634RAISE3.669EARNT3.844PEARL3.884
32.TRAPE3.634PARSE3.670STALE3.844CADRE3.885
33.HEART3.635CATER3.670TALES3.844STORE3.885
34.ARTEL3.635CRAPE3.670TERAS3.844CASTE3.886
35.EARST3.635PLATE3.671ALERT3.845TREAD3.886
36.RONTE3.635PLANE3.671RANCE3.847ROAST3.887
37.STALE3.635SNORE3.671CATER3.848ALERT3.888
38.PRATE3.636AROSE3.672STANE3.849HEART3.888
39.RAILE3.636NITRE3.672ALTER3.849STARE3.888
40.TASER3.636SIREN3.673TEARS3.850CORSE3.889
41.SOARE3.637TARGE3.673ROATE3.850PARSE3.889
42.STANE3.637ALTER3.674SANER3.851TAPER3.891
43.TEARS3.637CREST3.674LEARN3.852SNARE3.891
44.ANTRE3.638TRIED3.674TRONE3.852TRIED3.891
45.LEANT3.638CLEAT3.675ROAST3.853TEARS3.892
46.ROATE3.638TRIAL3.676STARE3.853NEARS3.892
47.TARES3.640SCARE3.677HEART3.854LITRE3.892
48.AROSE3.641TREAD3.678ORATE3.854STEAL3.892
49.EARNT3.641SHALE3.678TASER3.854CATER3.893
50.RAISE3.641ARISE3.679LATER3.854STILE3.894
51.PARLE3.642GRATE3.679EARST3.854TENOR3.895
52.RESAT3.642NACRE3.679TARES3.855RINSE3.896
53.SLIER3.643TRIES3.679ROSET3.855SAINT3.896
54.STORE3.643CLEAR3.680SNARE3.856SLICE3.896
55.STIRE3.645PEARL3.680STORE3.857SORTA3.897
56.PAIRE3.648ALONE3.682TOILE3.857ALTER3.897
57.CATER3.648STAIR3.682SCARE3.857STAIR3.897
58.ORATE3.649ALIEN3.682PARLE3.858SIREN3.898
59.SANER3.649HEART3.682ANTRE3.859RATES3.899
60.SETAL3.649LATER3.682SERAL3.859SCARE3.899
61.ARIEL3.650SPARE3.682STRAE3.859SHALE3.899
62.ALERT3.650CASTE3.683RATEL3.860SNORE3.900
63.SERAL3.651ALERT3.683ALINE3.860SPARE3.901
64.ALTER3.651TALES3.684ARETS3.862LATER3.901
65.SCARE3.651AISLE3.685LITRE3.862NITRE3.901
66.LIANE3.652CARES3.685SOREL3.863GRATE3.902
67.RATEL3.652RINSE3.686TALER3.865IRATE3.902
68.COATE3.653RATES3.686ALIEN3.867CLEAR3.903
69.STRAE3.653LEARN3.687RONTE3.868PATER3.904
70.TALER3.654IRATE3.689RATES3.869ALIEN3.906
71.AISLE3.655SAUTE3.690ARIEL3.870SATED3.910
72.ALINE3.656TENOR3.690REALS3.870TARGE3.911
73.ALONE3.656TAPER3.690LIANE3.872ASTER3.912
74.SAINE3.656SOLAR3.691STEAR3.872CANOE3.912
75.ARISE3.657GLARE3.692PAIRE3.873NARES3.913
76.LATER3.657SORTA3.693STIRE3.874RENAL3.913
77.IRATE3.657PATER3.694RAINE3.875BLARE3.914
78.TRIES3.658STEAL3.695IRATE3.876LITER3.914
79.ALIEN3.659LITRE3.695SAUTE3.878CARES3.916
80.TEALS3.660SATED3.696ALONE3.880ALONE3.921
81.TERAS3.660CANOE3.697RAILE3.880THANE3.923
82.SAICE3.661SUITE3.698SHARE3.883SAUTE3.923
83.SAUTE3.662THANE3.698LARES3.884LASER3.924
84.RATES3.662BLARE3.698RALES3.885SOLAR3.925
85.SHARE3.663NARES3.700URATE3.885HATER3.926
86.TOISE3.663SABRE3.701SLIER3.887ARISE3.927
87.LITRE3.663SHARE3.702COATE3.888AISLE3.927
88.ARETS3.666LASER3.702SOARE3.888RAISE3.928
89.TALES3.666HATER3.703RAISE3.889GLARE3.928
90.LEARN3.667ASTER3.704LASER3.889ANISE3.930
91.URATE3.672LOSER3.705TOISE3.892LOSER3.932
92.ARLES3.674ANISE3.708SAINE3.894ATONE3.934
93.REALO3.674ATONE3.710ARLES3.895SABRE3.937
94.STEAR3.674RENAL3.714AROSE3.896SHARE3.940
95.REALS3.675ORIEL3.716OATER3.898SUITE3.943
96.LARES3.678SHEAR3.717AISLE3.899ORIEL3.951
97.RALES3.679NEARS3.721AESIR3.902AROSE3.961
98.LASER3.683ROUTE3.724ARISE3.902SHEAR3.965
99.OATER3.684LITER3.725SAICE3.908ROUTE3.974
100.AESIR3.686ROUSE3.726REALO3.909ROUSE3.978

Comments

  1. Please see

    https://medium.com/@tglaiel/the-mathematically-optimal-first-guess-in-wordle-cbcb03c19b0a

    Tyler Glaiel
    (Tyler and Glair are both words)
    Dec 30, 2021

    With RAISE as the first guess, you will average 3.49546 guesses to solve but have that sweet, sweet chance of landing that hole-in-one (1:2315)

    This version of the bot finds a solution in 3.49417 guesses on average, and has a worst-case of 5 guesses (meaning, it *always* finds a solution). The ideal first guess using this method is ROATE.

    Please review the algorithm you “hacked mercilessly”, you should be able to reproduce or refute the above. And have a look at why you have RAISE in both feasible and unfeasible test words.

    easy mode
    ROATE 3.638 – not solution set
    RAISE 3.641 – not solution set
    RAISE 3.669 – potential solution

    hard mode
    ROATE 3.850 – not solution set
    RAISE 3.889 – not solution set
    RAISE 3.928 – potential solution

    1. Hi Nick! Thanks for your comment. I think the difference here is that Tyler’s algorithm works slightly differently to this one—the first version scores words based on yellow and green squares, and the second looks just at the size of the solution set and not the entropy, hence they give slightly different results. 🙂

    2. Interesting. Raise has been my typical opening wordle for a while now, and I was unclear why it wasn’t in the top 100. Now I see that these are two different methods. Which one is best? 3.49 does sound marginally better than 3.6something for the various versions of the entropy bassed model. Guess I’ll stick with Raise…

  2. Nick presents ROATE. I’ve been starting with ORATE and doing pretty well…mostly 3’s and 4’s. Same letters. Different order. Does order make a difference?
    Also… although you mention S as one of the most commonly used letters, I haven’t seen it come up that frequently in a winning word. Could it be that its membership in most common letters is connected to its use creating plurals? So far it seems that winning words don’t appear to be plurals.

    1. Order does make a bit of difference because of the patterns created, but without looking into specific cases it’s a bit hard to work out exactly why! However, it could well be that R is a more common starting letter than O, meaning finding out whether there is an R in first place or not narrows down the search space more rapidly.

      The reason for the lack of plurals is that as I understand it the creator of Wordle excluded them when he designed the game!

  3. I’m surprised haole didn’t make it on the top 100 list? I’m not a math wizard or computer programmer, tho. 🙂

  4. I usually use “TRASH” and have found this works well most times. I was surprised this was not in your listing. Where would this rank?

  5. I’m a little surprised by the paucity of d’s and h’s in the list. These are very common as far as consonants go. I have had good luck with “death” as an opener, but have also used others such as “crate” or “stare.”

  6. If the goal is guessing in the least amount of times, don’t you also have factor in not playing a word that has been played. I was under the impression they are not repeating words.

      1. One problem with Steve’s idea is remembering all the answers. Writing them down would absolutely be cheating, since we’re doing this not, as he claims, to guess the answer in the least number of times, but rather to improve/maintain our mental alertness.
        Second problem, and even worse, possibly existentially so, is that at number 605 today, we’re well past one quarter of the possible words. What would we do without Wordle in 1710 days? Perhaps your computer could calculate the rate of anxiety increment as the last day approaches.
        Finally, I certainly don’t remember all the answers, usually can’t even remember today’s, but I’m pretty sure there have been repetitions. So there may yet be hope for us after the 2315 mark.

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