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| title | date | math | tags |
|---|---|---|---|
| Cheating at Word Games: Part 2 | 2021-12-29T08:53:32-08:00 | true | [wordle] |
This is a sequel to my [previous post]({{< ref "/posts/cheating-at-word-games" >}}), where I laid out a Information Theoretical approach to algorithmically solving Wordle puzzles.
In that post, I considered whether there might be a strategy that takes a broader-view - rather than optimizing for maximizing information at every step, it might be profitable to make a less-optimal guess1 if the combination (guess1+guess2), in combination are "better". One approach that intuitively makes sense is to select a pair of 5-letter words which, combined, comprise the ten most-common letters in the corpus. The two guesses combined will give a clear indication of which of those ten letters are present - and, hopefully, indicate a couple of their positions, too.
These pairs were pretty easy to generate - but, far from giving a single best option, there are 447 of them! Interestingly, my previous suggested first guess - "roate" ("the cumulative net earnings after taxes available to common shareholders", apparently) - was not in there. Apparently there is no word that is an anagram of the remaining 5 most-common letters, "sincl".
The next step would be to find the pair whose partitioning1 gives the greatest information. As a heuristic before doing that computation, it would be sensible to start with the pair which contains the single best guess from the previous approach. (I suspect that this is equivalent to "the guess which is most likely to have correct letters (rather than present ones)", since correct letters are likely to be "rarer" and so to provide more information - but I haven't proven that yet). This might not be the best pair overall, if the second guess is significantly worse-than-average - but it's a good starting point!
That shakes out to recommending the pair of (soare, clint), which has quite a pleasing poetic image of Hawkeye in flight :) now that I have two strategies described ("always-locally-optimal" vs. "guess soare, then clint, then 🤷"), I'm looking forward to finding a way to pit them against one another against an automated implementation of the game. I suspect that they'll both reliably "win" in the same number of turns, so I'll either need to score them on their information/entropy properties (and probably have to dig out a textbook to make sure I'm doing that right), or construct some larger dataset for them to compete on.
(Check out the third post in this series [here]({{< ref "/posts/cheating-at-word-games-part-3" >}}))
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As described in the [previous post]({{< ref "/posts/cheating-at-word-games" >}}), each guess partitions the set of possible solutions into 125 subsets - one for each of the $5^3$ possibilities of
[first letter correct | first letter present | first letter absent] X [second letter correct | second letter present | .... ↩︎