Infinite Wordle and the Mastermind numbers

  • J. D. Hamkins, “Infinite Wordle and the mastermind numbers,” Mathematics arXiv, 2022.
    [Bibtex]
    @ARTICLE{Hamkins:Infinite-Wordle-and-the-mastermind-numbers,
    author = {Joel David Hamkins},
    title = {Infinite Wordle and the mastermind numbers},
    journal = {Mathematics arXiv},
    year = {2022},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {Under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    doi = {},
    eprint = {2203.06804},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://jdh.hamkins.org/infinite-wordle-mastermind},
    }

Download article at arXiv:2203.06804

Abstract. I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number 𝕞, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.

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