Infinite Wordle and the Mastermind numbers
| Autor: | Hamkins, Joel David |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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, 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. Comment: 22 pages, 5 figures. Comments can be made at http://jdh.hamkins.org/infinite-wordle-mastermind. v2 has minor updates and corrections. v3 makes a minor correction to theorem 11 statement 2 and a few other minor ambiguities |
| Databáze: | arXiv |
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