We've all been inundated with Wordle's little colored boxes, showing the path to winning (and the less frequently shared path to losing) that a person took on a given day. With the announcement of the "low seven figures" sale to the New York Times, we're sure to see a lot more of it in the coming months and years. As we explore Wordle, we can find out a lot: Software engineer Angie Jones joked that one's strategy could become part of a job application, and of course there's intramarital bragging rights and the opportunity to get out of studying for the SAT. We learn about how we think--and that too much thinking can generate disrespect. It can also generate cushions. Some players are methodical; others enjoy the game best when they're not thinking too hard.
There's no disputing that the Wordle craze has taken social media by storm, but the best way to play -- and the ultimate word to start with -- is a subject of contention. Looking for a scientific answer, US-based software developer Chris Wanek analysed the frequency of each letter among the 2,315 words the game can use as solutions. 'As a computer scientist, I want some proof -- what is the actual optimal starting word?' he asked in reply to linguist Danny Hieber's suggestion that'irate' is the best. 'After looking through the source file, I managed to find the entire word list. 'I made a quick script to load all of the words, calculate the frequencies of each letter, score each word based on the frequencies of the letters in the word, and then calculate the most optimal starting word, he explained.
We develop a new approach that computes approximate equilibrium strategies in Jotto, a popular word game. Jotto is an extremely large two-player game of imperfect information; its game tree has many orders of magnitude more states than games previously studied, including no-limit Texas hold 'em. To address the fact that the game is so large, we propose a novel strategy representation called oracular form, in which we do not explicitly represent a strategy, but rather appeal to an oracle that quickly outputs a sample move from the strategy's distribution. Our overall approach is based on an extension of the fictitious play algorithm to this oracular setting. We demonstrate the superiority of our computed strategies over the strategies computed by a benchmark algorithm, both in terms of head-to-head and worst-case performance.
Most mathematical questions one could have about Wordle are settled by now, and a few remain open. I summarize here what is known, as far as I can tell. First, let's clarify a few things about the game: Wordle comes with a dictionary of 12972 words that the player is allowed to use as guesses. They are essentially all 5-letter combinations one could reasonably argue are English words. The "secret" word that the player has to discover is also always in that dictionary.
Realistically, at this point, you don't need us to tell you about the viral puzzle game Wordle--five letters, six guesses, no shenanigans. Its virtues as a collective shared experience, a moment of escape amid various current miseries, have been eloquently lauded. Our Twitter timelines are now coated with green, gray and yellow squares, and that feels fine. Wordle may have also coated your timeline with heated opinions about how everyone else is playing it wrong. As people who spent our days thinking about how best to put letters in boxes long before last week--Nate is a professional crossword setter and official test solver for the New York Times daily puzzle--we have some humble suggestions for how to choose your Wordle guesses.