Quanta: The math behind Wordle guesses

James Round for Quanta Magazine
James Round for Quanta Magazine

by Pradeep Mutalik, Puzzle Columnist for Quanta Magazine
December 2, 2022

In the simple game of Wordle, players have to guess a secret five-letter word in six or fewer turns based on clues about the presence and location of letters revealed by their previous guesses. While somewhat similar games have appeared in the past, everyone who plays Wordle on a particular day has to discover the same secret word, making it easy to share your attempts and discuss the game among your friends. The distinctive structure and presentation of the game inspired the questions in our latest Insights puzzle. The answers are discussed below.

One key to playing a good Wordle game is to choose a strong starting word. Computer analyses embodying information theory techniques suggest that starting words such as “slate” and “crane” enable you (or a computer algorithm, at any rate) to solve Wordles in the least number of turns on average. However, many human solvers feel more comfortable choosing a vowel-rich word such as “adieu,” “audio” or “raise.” This feeling has both an intuitive and a rational basis. First, placed vowels enable you to find a vowel “backbone” that can restrict the number of consonants you need to search for. For instance, if you know the word looks like _­AI_E after you play “raise,” there are just a few possible words left: “naïve,” “waive” and “maize.” Second, vowels maximize a quantity that can be called “coverage” — between just the five vowels and Y, we can get at least one positive letter in every one of the 2,309 answers. To get this kind of perfect coverage with consonants, you would have to try all 20 of them, which would require at least five turns.

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