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.

翻译：我考虑了Wordle和Mastermind游戏的自然无穷变体，及其博弈论变体Absurdle和Madstermind，将这些游戏拓展至无限长单词、无限颜色序列，并允许超限游戏进程。每局游戏中隐藏一个秘密代码词，破译者通过一系列猜测并接收准确度反馈来试图发现它。在字母表有$n$个字母的有限字母表Wordle游戏中，无论单词长度如何（包括无限长甚至不可数长单词），破译者始终能在$n$步内获胜。与此同时，Mastermind数——定义为在可数颜色集上、无重复的$\omega$长序列的无限Mastermind游戏中最小的优胜猜测集——是不可数的，但其确切值被证明独立于ZFC，因为它等于最终不同数$\frak{d}({\neq^*})$，该数又等于贫集理想$\text{cov}(\mathcal{M})$的覆盖数。由此，我将针对该游戏自然变体定义的各种Mastermind数置于连续统基数特征层级之中。