Repetition in Permutation Wordle

TL;DR

This work analyzes a permutation Wordle variant on $[n]$, where a player guesses a secret permutation and learns which positions are correct. By formalizing strategies as $S=[s_1,s_2,\dots,s_n]$ and focusing on the cyclic shift strategy $\mathop{\\mathrm{CS}}$, the authors study information flow and repetition using displacement vectors and their left-variant $d_\ell$, constructing offending permutations that force repeated incorrect information for any non-$\\mathop{\\mathrm{CS}}$ strategy. They provide concrete algorithms for generating offending permutations when $2$ appears in the displacement vector and when it does not, then extend the analysis to derangements and cyclic-shifting components, including $\\mathop{\\mathrm{CSL}}$/\\mathop{\\mathrm{CSR}}$ cases. The paper culminates in a general framework proving that $\\mathop{\\mathrm{CS}}$ is optimal in avoiding information repetition across all strategies, with Maple implementations and open questions on counting offending permutations and extending the results to broader strategy classes.

Abstract

In a game of permutation wordle, a player attempts to guess a secret permutation in the fewest number of guesses possible. Previously, Samuel Kutin and Lawren Smithline (arXiv:2408.00903) introduced this game and proposed a strategy called cyclic shift, which they conjecture performs optimally. We continue our investigation of this conjecture by considering how information is obtained and, at times, repeated during a game of permutation wordle using an arbitrary strategy. This analysis includes several algorithms to construct a secret permutation which prompts inefficient repetition according to the player's strategy, as well as proofs of their efficacy.

Theorems & Definitions (11)

• Lemma 1
• proof
• Theorem 2
• Definition 3
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof