# Infinite Wordle and the Mastermind numbers

arXiv: https://arxiv.org/abs/2203.06804 · HTML: https://mlsanity.com/paper/2203.06804/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2022.

## Built on by (most cited first, 1 of 1)

- [Using Wordle for Learning to Design and Compare Strategies](https://mlsanity.com/paper/2205.11225/) (2022)

## Abstract (arXiv)

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 frakd(neq^*), which is the same as the covering number of the meager ideal cov(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.