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If you play Wordle in hard mode, with unlimited guesses, then you must solve any puzzle eventually. This is true because there are only a finite number of valid five-letter words, and it is not possible to repeat an incorrect guess in hard mode.

What is the longest possible game of Wordle, if unlimited guesses are allowed, but every guess has to be consistent with the hints from previous guesses?

Here is a valid 19-turn game in hard mode. Is this the longest possible?

  1. CILLS
  2. DILLS
  3. FILLS
  4. GILLS
  5. HILLS
  6. JILLS
  7. KILLS
  8. LILLS
  9. MILLS
  10. NILLS
  11. PILLS
  12. RILLS
  13. SILLS
  14. TILLS
  15. VILLS
  16. WILLS
  17. YILLS
  18. ZILLS
  19. BILLS
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    $\begingroup$ It's fine to not know the answer, but you do have to be careful about true open-ended puzzles because they're off -topic. It must be theoretically possible to firmly prove optimality. Here, with a limited word list and computers allowed, I think you're fine. $\endgroup$
    – bobble
    Commented Apr 10, 2022 at 16:35
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    $\begingroup$ You could prepend that list with AXLES $\endgroup$ Commented Apr 10, 2022 at 16:43
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    $\begingroup$ Here are some of my thoughts: 1) interpret each word as a vertex in a graph 2) draw an oriented edge from word A to word B if B is a valid guess after guessing A 3) the original problem is now to find the longest path in this graph, such that each word in the path has edges to all subsequent words in that path 4) notice that we're looking for maximum transitive relation 5) which is an NP-hard problem :( 6) this article describes maximal relation algorithm $\endgroup$
    – DL33
    Commented Apr 11, 2022 at 17:39
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    $\begingroup$ You could prepend by QUEUE followed by AXIAL $\endgroup$ Commented Apr 14, 2022 at 6:47
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    $\begingroup$ @DL33 although BBBBB after AAAAA is allowed but BBBBB after BCCCC and then AAAAA is not allowed (if the target is XXXXX) $\endgroup$
    – Sny
    Commented Apr 25 at 11:26

3 Answers 3

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I think the puzzle description is ill-formed: You say "Here is a valid 19-turn game in hard mode," and then show a game ending in BILLS. But BILLS is not a possible target word! So this is not a valid hard-mode game β€” no more than a game ending in AALII would be.


Major edit: My original answer had said, "I think anything allowed by hard-mode Wordle should be allowed by this challenge." But it turns out that NYT's Wordle actually permits all kinds of dumb guesses! For example, if your first guess is "🟨🟨⬜🟨⬜ ROOTS", a human would deduce that (1) the first letter is not R, (2) there is exactly one O, and (3) the target doesn't contain S. But Wordle happily accepts the second guess "🟨🟨🟩⬜⬜ ROTOS" which violates all of those deductions. In fact, Wordle happily accepts six guesses of "🟨🟨⬜🟨⬜ ROOTS" in a row! So, "what Wordle actually accepts" turns out to be an awful guideline for this puzzle.

My new suggested guideline is "Any word you guess must be logically consistent with all previous clues."


Another question is whether we should then limit the player to guessing only common words, e.g. guessing AALII or BILLS would be disallowed, since those aren't possible targets. But allowing uncommon words is more fun! So guessing AALII or BILLS is perfectly allowable, even though the human player might know those can't possibly be the target; because we're not trying to guess the target; we're just trying to play the longest possible game.

Still, restricting the search to just the common words does prune the dictionary and speed up the search. My best game so far which happens to use all common words (even though I think that shouldn't be strictly required) is length 9.

🟩⬜⬜⬜⬜ SASSY
🟩⬜⬜⬜⬜ SOOTH
🟩⬜⬜🟨⬜ SQUIB
🟩🟨⬜⬜🟩 SIEVE
🟩⬜🟩⬜🟩 SMILE
🟩🟩🟩⬜🟩 SPINE
🟩🟩🟩⬜🟩 SPIRE
🟩🟩🟩⬜🟩 SPICE
🟩🟩🟩🟩🟩 SPIKE

Using uncommon words (but legitimate target word SIGHT), I have length 18:

⬜⬜🟨⬜⬜ JUICY
⬜🟩⬜⬜🟩 ZIZIT
⬜🟩🟩⬜🟩 GIGOT
⬜🟩🟩🟩🟩 AIGHT, BIGHT, DIGHT, EIGHT, FIGHT
⬜🟩🟩🟩🟩 HIGHT, KIGHT, LIGHT, MIGHT, NIGHT
⬜🟩🟩🟩🟩 PIGHT, RIGHT, TIGHT, WIGHT
🟩🟩🟩🟩🟩 SIGHT

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    $\begingroup$ If you add SWIPE between SEIZE and SPINE, you can get another one in there. $\endgroup$ Commented Apr 25 at 18:46
  • $\begingroup$ I'm not sure what you mean by "common" words, because BILLS is quite a common words (ducks have bills, companies send bills for their services), and SOOTH and SQUIB are not particularly common. Are you talking about only using the target words? $\endgroup$ Commented Apr 25 at 18:48
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    $\begingroup$ The question does specify that "every guess has to be consistent with the hints from previous guesses", though. $\endgroup$
    – Jafe
    Commented Apr 26 at 1:10
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    $\begingroup$ Having programmed this up over the weekend, I expect (but cannot prove) that this answer is best case. I did get 2 other identical length answers, but they differ by so little as to not be noteworthy. $\endgroup$ Commented Apr 29 at 22:52
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    $\begingroup$ Turns out I was wrong. I still cannot prove maximum, but I have a 19. $\endgroup$ Commented May 7 at 3:14
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Following the logic from Quuxplusone's answer:

  • Target word must be a valid Wordle answer
  • All words must be valid Wordle guesses (using the list provided by OP)
  • All guesses must be logically consistent with all previous clues

I now have a length 19.
RIVER:

⬜⬜🟨🟨⬜ QUERY
🟨⬜🟨🟨⬜ EERIE
🟨⬜⬜🟩🟩 ICIER
⬜🟩⬜🟩🟩 TITER
⬜🟩⬜🟩🟩 PIPER
⬜🟩⬜🟩🟩 NINER
⬜🟩⬜🟩🟩 MIMER
⬜🟩⬜🟩🟩 BIKER
⬜🟩🟩🟩🟩 WIVER
⬜🟩🟩🟩🟩 VIVER
⬜🟩🟩🟩🟩 SIVER
⬜🟩🟩🟩🟩 LIVER
⬜🟩🟩🟩🟩 JIVER
⬜🟩🟩🟩🟩 HIVER
⬜🟩🟩🟩🟩 GIVER
⬜🟩🟩🟩🟩 FIVER
⬜🟩🟩🟩🟩 DIVER
⬜🟩🟩🟩🟩 AIVER
🟩🟩🟩🟩🟩 RIVER

3 unused letters: O, X, Z

369 unique answers (potentially incomplete)

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    $\begingroup$ Very nice! Looks like answers with two of the same wrong letter turn out to be extra powerful. $\endgroup$
    – Jafe
    Commented May 7 at 6:24
  • $\begingroup$ Yes. I was wondering how a word with less words "all missing the same 1 letter" could work and the answer is "miss 2 letters, but only use 1 new letter" $\endgroup$ Commented May 7 at 11:23
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    $\begingroup$ Hmm, the second solution has 🟨🟨🟨⬜⬜ AREAE followed by QUARE. I think the gray "E" logically forbids QUARE, because if the word were actually QUARE then we'd have had 🟨🟨⬜⬜🟩 AREAE instead. But I see nothing wrong with the RIVER solution! $\endgroup$ Commented May 7 at 18:12
  • $\begingroup$ @Quuxplusone - Thank you. Bug in my code. That is absolutely not allowed. The double-letter/triple-letter stuff is where I found all my bugs. $\endgroup$ Commented May 7 at 18:43
  • $\begingroup$ OK, bug fixed. Specific to double letters where one appears in target word. River still has lots of answers, but potentially not as many as stated in my answer. Running for a while to see. $\endgroup$ Commented May 7 at 22:53
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Building on the example in the question, and the comments and answers afterward, I managed to get a list of 22 words:

    AQUAS  _ _ _ _ β– 
    LEXES  β–‘ _ _ _ β– 
    OLIOS  _ β–‘ β–‘ _ β– 
    CILLS  _ β–  β–  β–  β– 
    DILLS  _ β–  β–  β–  β– 
    FILLS  _ β–  β–  β–  β– 
    GILLS  _ β–  β–  β–  β– 
    HILLS  _ β–  β–  β–  β– 
    JILLS  _ β–  β–  β–  β– 
    KILLS  _ β–  β–  β–  β– 
    LILLS  _ β–  β–  β–  β– 
    MILLS  _ β–  β–  β–  β– 
    NILLS  _ β–  β–  β–  β– 
    PILLS  _ β–  β–  β–  β– 
    RILLS  _ β–  β–  β–  β– 
    SILLS  _ β–  β–  β–  β– 
    TILLS  _ β–  β–  β–  β– 
    VILLS  _ β–  β–  β–  β– 
    WILLS  _ β–  β–  β–  β– 
    YILLS  _ β–  β–  β–  β– 
    ZILLS  _ β–  β–  β–  β– 
    BILLS  β–  β–  β–  β–  β– 
 

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