Permutation Wordle

TL;DR

The paper studies permutation Wordle, where the secret is a permutation $\sigma\in S_n$ and feedback reveals fixed points of $\sigma^{-1}\gamma$. It introduces the CircularShift strategy, proving that it solves exactly $A(n,k)$ permutations in $k+1$ rounds, with the average number of rounds for a random $\sigma$ equal to $(n+1)/2$, and conjectures its optimality. It then extends the problem to multicolor permutation Wordle, deriving analogous Eulerian-type counts via generating functions and recurrences, including the total round count $1+\mathrm{exc}(\sigma)+\sum_i \kappa(i)$ for colorings. The work connects Wordle-type permutation guessing to Eulerian numbers, higher-order Eulerian numbers, and $z$-excedances, and discusses links to Mastermind literature, highlighting average-case advantages and several open questions. Together, the results provide a natural combinatorial framework for permutation- and color-structured guessing games with precise enumerative characterizations.

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.

Figures (7)

• Figure 1: Permutation Wordle when $n=3$. We solve $1$ permutation in one round, $4$ in two rounds, and $1$ in three rounds.
• Figure 2: Permutation Wordle when $n=4$. We solve $1$ permutation in one round, $11$ in two rounds, $11$ in three rounds, and $1$ in four rounds.
• Figure 3: The deterministic strategy CircularShift guessing the permutation $\sigma = (7, 2, 4, 8, 5, 3, 1, 6, 9)$. The strategy succeeds in four rounds.
• Figure 4: We start with the sequence whose enumerator is $\sum_j (j+1)^n t^j$ (for $n = 3$) and apply finite differences (that is, multiply by $1-t$) four times.
• Figure 5: The deterministic strategy CircularShift guessing the permutation $\sigma = (7, 2, 4, 8, 5, 3, 1, 6, 9)$. We highlight the values in $R_k$ for each $k$; that is, those values that $\gamma_k$ places to the right of $\sigma$.

Theorems & Definitions (47)

• Definition 2.1: CircularShift
• Conjecture 2.2
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• Theorem 3.5
• Corollary 3.6
• Conjecture 3.7