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:

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:

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:

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:

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.