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：本文研究了Wordle和Mastermind游戏的无限变体，即单词长度与颜色序列均可为无限情形，并允许超限游戏进程。对于每个游戏，存在一个隐藏的密码词，破译员通过一系列猜测并获取其准确程度反馈来尝试发现该密码词。在Wordle游戏中，若字母表有限（含n个字母），则无论单词长度为有限、可数无限甚至不可数，破译员总能保证在n步内获胜。与此同时，对于可数颜色集上长度为ω的无重复序列，无限Mastermind的最小获胜猜测集（即mastermind数）是不可数的，但其精确值独立于ZFC——该值可证明等于最终不同数𝔡(≠*)，这与贫集覆盖数cov(ℳ)相等。由此，我将游戏各种自然变体对应的广义mastermind数纳入连续统基数特征层级体系。