Infinite Wordle and the mastermind numbers

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 ω over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of $\mathrm{ZFC}$, for it is provably equal to the eventually different number $d\left({\ne }^{\ast }\right)$, which is the same as the covering number of the meager ideal $\text{cov}\left(M\right)$. 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.