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# Wordle Is NP-Hard

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LIPIcs.FUN.2022.19.pdf
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## Acknowledgements

We thank anonymous reviewers for helpful suggestions, Mark Polyakov for pointing out a minor bug in a previous proof of Theorem 1, and Wassim Bouaziz for pointing out some imprecisions in a previous version of this paper. The second author thanks Jérémy Barbay for his insightful conversation, and his awesome friends: Saranya Vijayakumar for proofreading the English in an earlier draft of this paper, and Bailey Miller for providing him a place (and tea) to work on this article.

## Cite As

Daniel Lokshtanov and Bernardo Subercaseaux. Wordle Is NP-Hard. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 19:1-19:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FUN.2022.19

## 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 𝓁 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 𝓁 if the player can guarantee to discover the secret word within 𝓁 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
• wordle
• np-hardness
• complexity

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## References

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