论文标题
无限词和策划人数
Infinite Wordle and the Mastermind numbers
论文作者
论文摘要
我认为游戏Wordle和Mastermind的自然无限差异，以及他们的游戏理论变化荒谬和Madstermind，考虑到这些游戏中具有无限长的单词和无限的颜色序列的游戏，并允许跨越游戏。对于每个游戏，秘密代码字都是隐藏的，代码破解者试图通过做出一系列猜测并收到有关其准确性的反馈来发现的。用$ n $字母的有限字母（包括无限的单词甚至无数单词）中的任何大小的单词词，代码破解者总是可以以$ n $ spest的速度获胜。同时，策划人数（被定义为无限的策划中最小的获胜猜测，对于长度$ω$而无需重复的颜色，$ω$的序列是不可容纳的，但确切的价值证明是独立于ZFC，因为它证明是完全等于最终的不同数字$ \ frak \ frak \ frak {d}（d iS ass y neq {理想$ \ text {cov}（\ Mathcal {M}）$。因此，我将所有用于游戏自然变化的策划人数放在连续体的基本特征的层次结构中。
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 $ω$ 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.
