Wordle is NP-hard

TL;DR

This paper proves that the game Wordle is NP-hard to solve optimally, even with fixed word length, and explores its computational complexity and related open problems.

Contribution

It establishes the NP-hardness of deciding optimal strategies in Wordle and analyzes its parameterized complexity, providing a formal computational perspective.

Findings

Wordle is NP-hard to solve optimally.

Hardness persists even for fixed word length of 5.

Approximation of minimum guesses is NP-hard.

Abstract

Wordle is a single-player word-guessing game where the goal is to discover a secret word $w$ that has been chosen from a dictionary $D$. In order to discover $w$, the player can make at most $\ell$ guesses, which must also be words from $D$, all words in $D$ having the same length $k$. After each guess, the player is notified of the positions in which their guess matches the secret word, as well as letters in the guess that appear in the secret word in a different position. We study the game of Wordle from a complexity perspective, proving NP-hardness of its natural formalization: to decide given a dictionary $D$ and an integer $\ell$ if the player can guarantee to discover the secret word within $\ell$ guesses. Moreover, we prove that hardness holds even over instances where words have length $k = 5$, and that even in this case it is NP-hard to approximate the minimum number of guesses required to guarantee discovering the secret word. We also present results regarding its parameterized complexity and offer some related open problems.