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. 🙂

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