(Update: my conjceture has been disproven, at least with current word lists.)

Happy New Years everyone! I hope everyone can enter 2022 with some optimism.

Everyone is playing Wordle. It is very cool and fun. It is basically Mastermind, the word guessing game, but with a cool interface and a software keyboard that makes it very addicting.

On top of this lots of people are now playing Absurdle and Evil Wordle. In these adversarial variants, the game doesn't pick a word immediately. Instead, every time you make a guess, it figures out what kind of clue it can give that keeps the most possible words in play.

So on each guess the game will partition all of possible answers in the word list based on what sort of clue you would get based on your guess. And then the game decides to give you the clue with the most possible variants!

Playing "normally" will quickly lead to needing 8+ guesses, but some words quickly allow convergence (the optimum seems to be 4). There's also some chatter about trying to find very long chains (which usually happen when you lock in some suffix or prefix and you end up having to just iterate over the dictionary).

But what about trying to make the answer a specific word? What if you want to play Adversarial Wordle (knowing how the algorithm chooses clues that leave the most words on the table) in a way to force the game to choose MOUSY, PLEAT, or GEESE?

My gut feeling is that for many words, it will simply not be possible:

The Adversarial Wordle Limiting Conjecture

Given an English-language word list, it is impossible for certain words in the list to be the final answer in an Adversarial Wordle session using the strategy of minimizing word eliminations when providing clues.

For example, it might not be possible to make guesses that lead to GEESE being the final answer, no matter how hard you try.

I believe this is true by considering that words tend to match categories, while rarely having 50% partitions. For example, many words are plurals of other words. But this is less than half of them are! So if you ask the Adversarial Wordle Machine "is this a plural" (by placing an S at the end of your guess), it just feels more likely that the game would choose to eliminate plurals rather than give you a good hint.

Because of this, and the fact that most words have properties that allow them to be grouped, I believe that certain words that have "too many properties" would always end up getting eliminated during the game. I have a strong feeling this is true for at least one word, if only because of the reality of things like anagrams. But my strong conjecture is that this is true for at least 20% of words in the word list.

Someone please prove me wrong, and at lesat find a way for us to get to a fun word like QUINE.

(Aside: A very good third party Minesweeper clone I remember playing 15+ years ago whose name escapes me, has a similar idea. It provides partial clues that are "solvable" without locking in the location of the mines. If you make a "random guess" when clues are available, the game will place a mine there and cause you to lose. They went through the trouble of fixing Minesweeper, and will punish you for not following it! Inversely the first click wouldnever hit a mine)