More fun with Wordle – Second guess analysis

Update March 26, 2022: Added some runts-to-win analysis and updated the “top-25” words again based on that data, see the update section below for details.

Recently I published some analysis for that holy grail of Wordle, the best starting words: Fun with Wordle.

That analysis looked at thousands of guesses for how many words those guesses eliminate on average for all the Wordle answers, as well as millions of simulations using those starting words to get the average turns-to-win playing hard-mode and easy-mode. It used those results to suggest the top starting words.

Since doing that analysis, and even while doing that analysis, I was thinking about taking things to the next step, looking beyond the first guess, to see how things look when considering also the second guess. I noticed in the data, that sometimes a guess that measures as a worse first-guess by that analysis might actually come out better when considering the results after a second guess. This got me thinking about doing some deeper analysis.

Doing the same kind of brute-force simulation analysis on first+second guesses for all the possible guesses would be over 160 million simulations, which I estimated to be about 18 months of compute time on my available hardware. So instead of looking at all the possible guess words, I took the top 100 from the previous first guess analysis and ran those. That took about five days of compute time to run all those simulations. Due to these compute requirements, I didn’t run simulations for both hard-mode and easy-mode. All the results here were run in easy-mode.

Based on the previous analysis, I expected the difference between the best and worst starting words to be minimal, and that certainly turned out to be the case. The best first+second simulation gives an average of 1.017237 possible guesses remaining, while the 100th rated result gives an average of 1.06955 possible guesses remaining. It’s important to keep that in mind when reviewing this data – any of these starting words are good and will give you a good chance to solve the Wordle puzzles. The point being, as noted in the previous report: don’t get fixated on “the best” starting word. Nevertheless, the data is still interesting (at least to me).

Here’s a table of the raw results:

        
1
rankwordmean resultfirst rank
first mean result
first worst case
differencetop 25
2
1LEAST1.0172372081.07228-19STAY
3
2DEALT1.0180166494.17266-62IN
4
3STOLE1.0193423987.7232-36IN
5
4LEANT1.0201821980.68220-15STAY
6
5TRAIL1.0205722382.63232-18STAY
7
6ROAST1.0207543285.84272-26IN
8
7SLATE1.0218231074.06228-3STAY
9
8SOLAR1.0224253385.84242-25IN
10
9STONE1.0226948097.65263-71IN
11
10RAISE1.023222162.541739STAY
12
11STALE1.0234471578.18228-4STAY
13
12STAIR1.0235954689.81268-34IN
14
13TRIAL1.0235993486.18232-21IN
15
14SONAR1.0244355391.09270-39IN
16
15IRATE1.025149365.9820012STAY
17
16LEAPT1.02540999100.36265-83IN
18
17TRAIN1.0256265792.27277-40IN
19
18AROSE1.025834468.1518714STAY
20
19ALONE1.0258731880.391931STAY
21
20ATONE1.0264012181.72196-1STAY
22
21CLOSE1.0264059099.34256-69IN
23
22YEAST1.0276058498.16266-62IN
24
23SCALE1.0280685190.71262-28IN
25
24REACT1.0283632482.832570STAY
26
25ALIEN1.0287833586.56229-10IN
27
26SLICE1.0292079499.76301-68
28
27STEAL1.0298617995.91246-52
29
28RENAL1.0303772984.34224-1
30
29ALERT1.0304331274.7720717OUT
31
30STARE1.030494873.5123422OUT
32
31ARSON1.031334100100.5270-69
33
32AISLE1.0314381678.8620216OUT
34
33SNARL1.0324048297.97249-49
35
34TRADE1.0325642583.32769OUT
36
35CANOE1.032754990.05223-14
37
36TREAD1.0328458197.8276-45
38
37SAUTE1.0330583786.72280
39
38TRICE1.0331534489.08286-6
40
39ARISE1.033222265.6217337OUT
41
40ISLET1.0333179599.91280-55
42
41NOISE1.0341276594.21268-24
43
42STORE1.0343222883.7725114
44
43PLATE1.0344356092.73265-17
45
44CRANE1.0347082282.227622OUT
46
45TRACE1.0347991477.0525731OUT
47
46CRATE1.0348811375.9125733OUT
48
47TENOR1.0362234789.842690
49
48ROUTE1.0366914889.882600
50
49CASTE1.037384288.052557
51
50RATIO1.0377316994.61197-19
52
51ALTER1.037765672.6620745OUT
53
52LATER1.038341772.7220745OUT
54
53LANCE1.0383676894.55284-15
55
54PARSE1.0384284589.452779
56
55SCARE1.0387222683.4424529
57
56SANER1.038748572.3222851OUT
58
57PEARL1.0387667094.75277-13
59
58PLANE1.0394028999.29297-31
60
59ADORE1.0394985491.472475
61
60RESIN1.039778798.74252-27
62
61SNARE1.039918973.6522852OUT
63
62SAUCE1.0403988698.29257-24
64
63RINSE1.0404114087.7325223
65
64CAUSE1.0411488899.18257-24
66
65RISEN1.0411527395.24252-8
67
66SCORE1.0419237194.89264-5
68
67LEARN1.0419581779.9422450OUT
69
68CLEAR1.042001559226113
70
69LATHE1.0422097295.16257-3
71
70CATER1.0423082783.5825743
72
71LASER1.0424251174.3921360OUT
73
72CADRE1.0426429199.55317-19
74
73SNORE1.0429583085.624443
75
74AIDER1.0432017895.9226-4
76
75GRATE1.04334187.7428634
77
76TAPER1.043387569227720
78
77SPARE1.0436216794.3527710
79
78SHALE1.0452016694.3324912
80
79CRONE1.0472028398.07297-4
81
80BLARE1.048458598.29303-5
82
81SHEAR1.0487357795.722474
83
82TEARY1.0493893887.0520744
84
83ROUSE1.0494336193.2325422
85
84HEART1.0497965992.6627325
86
85PALER1.0503596393.7627722
87
86LOSER1.0504943185.822455
88
87GREAT1.05069796100.06286-9
89
88SIREN1.0508493686.6125252
90
89HATER1.051429529127337
91
90TAMER1.0538337595.3728215
92
91SHARE1.0541144388.2824748
93
92LARGE1.0546736293.5228230
94
93SHIRE1.0557259399.722800
95
94URINE1.0582989299.582632
96
95GLARE1.058745892.3728237
97
96YEARN1.060167695.6622520
98
97LAGER1.06201498100.23282-1
99
98LINER1.0649337495.3523624
100
99EARLY1.06582597100.172232
101
100LAYER1.069555090.722250

In the previous analysis, we suggested the “best” starting words as LEAST or TRACE for hard-mode or SLATE for easy-mode. In this first+second analysis, our “best” starting word of LEAST holds up. It comes out on top here again. Here however, TRACE gets dropped from the top-25 entirely. SLATE holds on to a top-10 ranking but drops from “best” to number 7 (still very good).

Some big movers in this are LEAPT which moves from 99th in the first guess results, all the way to 16th in the first+second simulation results; STONE moving from 80th to 9th; CLOSE from 90th to 21st; DEALT moving from 64th all the way to number 2.

Words that would move into the top-25 after this analysis are:

wordnew rankold rankspots moved
LEAPT169983
STONE98071
CLOSE219069
DEALT26462
YEAST228462
TRAIN175740
SONAR145339
STOLE33936
STAIR124634
SCALE235128
ROAST63225
SOLAR83325
TRAIL133421
ALIEN253510
words moving into the top-25 starting words

We also have some big movers moving out of the top-25, including SNARE which was ranked at 9 and is now 61st; ARISE was our number 2 word and is now 39th; SANER was 5th and is now 56th.

Words that move out of the top-25 are:

wordnew rankold rankspots moved
LASER711160
SNARE61952
SANER56551
LEARN671750
LATER52745
ALTER51645
ARISE39237
CRATE46237
TRACE451431
STARE30822
CRANE442222
ALERT291217
AISLE211616
TRADE34259
words moving out of the top-25 starting words

These results suggest that there are likely guess-words outside the top-100 first-guess rankings that would be top-100 starting words, perhaps even better, perhaps even top-25. However, as noted, I don’t know how many more CPU resources I want to devote to this analysis. I did, however, try one of the favorite starting words, ADIEU, which is not in the top-100 first-guess rankings. It ranked 315th in the first guess simulation results. In running first+second simulations, ADIEU does move up significantly, all the way to 98th, but not into the top-25 starting words, at 1.061061 average remaining possibilities after two guesses.

I’ll be changing my guesser to pick from these new top-25 starting words. We’ll see if it has much impact on the overall turns-to-win of the guesser.

It takes about an hour to run all the simulations required for this first+second analysis for any given starting word. If you have one you’d like me to try, let me know and I can see how it does.

Update March 26, 2022:

I ran simulations on the new top-25 words shown above and made a new top-25 based on turns-to-win as well as failures (for hard-mode), including the popular favorite ADIEU. Here’s that data:

    
1
wordturns to winfailuressucccess
2
SLATE3.228101399.87%
3
LEAST3.230069199.96%
4
TRACE3.233102199.96%
5
REACT3.247615399.87%
6
CRATE3.249674699.74%
7
STALE3.256288399.87%
8
STARE3.258568499.83%
9
LEANT3.258778299.91%
10
TRAIL3.259436499.83%
11
TRIAL3.260624399.87%
12
RAISE3.262587599.78%
13
ALTER3.264731199.96%
14
LATER3.272451499.83%
15
SANER3.272767399.87%
16
CRANE3.273397199.96%
17
ROAST3.273515299.91%
18
TRAIN3.27383199.96%
19
IRATE3.274935399.87%
20
AROSE3.275249299.91%
21
DEALT3.276116299.91%
22
SNARE3.277778599.78%
23
LEARN3.277971399.87%
24
ALERT3.2802080100.00%
25
SCALE3.281629199.96%
26
CLOSE3.283362199.96%
27
ARISE3.284165499.83%
28
ALIEN3.284288599.78%
29
TRADE3.287695199.96%
30
ALONE3.288069499.83%
31
STOLE3.2893030100.00%
32
STAIR3.297051399.87%
33
STONE3.297227199.96%
34
LASER3.298352399.87%
35
AISLE3.30052399.87%
36
SONAR3.301821399.87%
37
SOARE3.302255399.87%
38
ATONE3.303726199.96%
39
YEAST3.313666499.83%
40
SOLAR3.318162299.91%
41
LEAPT3.321057199.96%
42
ADIEU3.439809899.65%

After this new adjustment, the words we added to the top-25 from our original top-25 are: CLOSE DEALT ROAST SCALE TRAIN TRIAL

And the words we removed from the top-25 are: AISLE ALONE ARISE ATONE LASER TRADE

In the end, LEAST holds up as, if not the best, certainly one of the best starting words. Likewise, ADIEU remains outside the top results.

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