Permutation Wordle

TL;DR

This paper introduces permutation Wordle, a game involving recovering hidden permutations through strategic guesses, and establishes connections between the game's solving process and Eulerian numbers, extending to colored and signed permutations.

Contribution

It presents an optimal guessing strategy for permutation Wordle, links the number of permutations solved to Eulerian numbers, and extends the framework to colored and signed permutations.

Findings

The number of permutations solved in k+1 rounds equals Eulerian number A(n,k).

The strategy extends to suited permutations with a recurrence relation.

Results relate to Eulerian numbers of type B for signed permutations.

Abstract

We introduce a guessing game, permutation Wordle, in which a guesser attempts to recover a hidden permutation in $S_n$. In each round, the guesser guesses a permutation (using information from previous rounds) and is told which entries of that permutation are correct. We describe a natural guessing strategy, which we believe to be optimal. We show that the number of permutations this strategy solves in $k+1$ rounds is the Eulerian number $A(n,k)$. We also describe an extension to suited permutations: the setter chooses a permutation in $S_n$ and also a coloring of $[n]$ using $s$ colors. We generalize our strategy, give a recurrence for the number of suited permutations solved in $k+1$ rounds, and relate these numbers to the Eulerian numbers. In the case of two suits, or signed permutations, we also relate these numbers to the Eulerian numbers of type B.