Wordle has fast become a modern classic played by millions. As a puzzle enthusiast, I feel as thrilled by it as an astronomer seeing a visible supernova. Only twice before have I felt and seen such widespread excitement about a puzzle: first, early in life, when I encountered Scrabble and crosswords, and later during the Rubik’s Cube craze of the 1980s. Today we’ll explore this modern supernova of the puzzle world.
Wordle is a simple game in which you have to find a five-letter word in six guesses. After each guess, you get clues about how close your word is to the target. Wordle is similar to a paper-and-pencil game called Jotto from the 1950s and also resembles the Mastermind game that uses colored tokens instead of words. But what makes Wordle easier and more appealing than its precursors is that it gives you visual feedback about each letter’s position in your guess: green if the letter is in the correct position, yellow if it’s part of the correct word but out of place, and gray if it’s not in the word.
While its delightful mix of luck and intuition is part of the game’s appeal, so too is its tie-in to information theory. Every Wordle move presents a chance to find a mathematically calculable best choice or choices, at least on average. This connection has been well explored in Patrick Honner’s Quantized Academy column and in Grant Sanderson’s excellent YouTube video. From the point of view of computer analysis, Wordle, being a far easier game than Chess or Go, has essentially been solved, as explained by the mathematician Alex Selby. All possible starting words have been ranked according to how many turns it would take on average to arrive at every word on the official Wordle list. With the best starting words, computers take an average of 3.41 to 3.42 turns to solve all possible Wordles.
But people are not computers. While the computer analysis recommends starting with words like “stare,” “slate” or “crane,” many of us prefer to first focus on the vowel structure of the word, starting with “adieu” or “audio.” Most humans also cannot calculate every possibility, relying instead on intuition, general principles and rules of thumb to do their best. Some tasks that are easy for computers are hard for humans, such as making a list of all possible outstanding words. This means that information theory solutions cannot be directly transferred to humans. We need to take human psychology, cognitive preferences and shortcomings into account in order to fashion good strategies.
In this column, I offer four puzzles and four questions. The puzzles are attempts to use objective techniques to address some interesting aspects of Wordle. The questions concern aspects of Wordle that have piqued my curiosity. I hope these open questions, with no clear-cut answers, make for good discussion. A computer search might help with some questions, but you can also just describe what you think are the best ways to figure them out.
According to The New York Times’ WordleBot, “adieu” is by far the most common starting word that people use (at about 7%). Two other vowel-rich words, “audio” (4%) and “raise” (3%), round out the top three. Computer favorites such as “stare,” “crane” and “slate” are less popular (2% to 3%). This is probably because most people think about word structure using vowels as the skeleton.
Assuming we want to focus on vowels as early as possible, let’s try to determine which of these three vowel-rich starting words is best. The following table gives the frequency with which the eight letters in the words “adieu,” “audio” and “raise” occur in each position over the entire Wordle answer list of 2,309 words.
Based on this table, determine how many greens and yellows you can expect to get over the entire Wordle answer list for each of the three vowel-rich starting words: “adieu,” “audio” and “raise.” What does this tell you about their expected performance as starting words?
This raises the question of the relative value of getting a green versus a yellow result for a letter. We humans love greens because they save us a lot of cognitive effort. Getting five greens solves the puzzle, but getting five yellows means having to try out various anagrams. So is a green worth 1.5, 2, or more yellows? How would you quantify this?
A) If you get all five yellows on your first turn, what’s the maximum number of turns it can take to find the answer, assuming best play?
B) Is it ever the case that having a letter in a certain position turn yellow is more valuable than seeing it turn green? If so, can you give an example and explain why this should be?
While “adieu” is not actually a Wordle answer, you can still use it as a guess. Beyond Wordle’s current answer list of 2,309 common words, you can guess any of about 15,000 words, some of which are highly obscure. For example, the first four of these possible guess words, in alphabetical order, are “aalii,” “aapas,” “aargh” and “aarti.” These kinds of words are likely to be known only to tournament Scrabble players.
Does a person with a good vocabulary of obscure Scrabble words have an advantage or disadvantage in playing Wordle?
Wordle makes it easy to share and compare your scores with friends. It helps that the scores have an almost perfect correspondence to golf’s. Good human players can solve most Wordles in four tries, so four can be treated as par. Therefore, three is a birdie, two is an eagle and the miraculous one would be a hole-in-one. A very good human player would average a little below par over the long term, just like a very good golf player.
This ease of sharing makes for good group fun, but it can also lead to angst and envy. In a survey of Wordle users conducted by Solitaired, about 10% of Wordle players admitted to cheating, most of them doing so strategically once or twice a week. It is very easy to cheat — spoilers abound on the internet, and you can solve the day’s Wordle first on another device or in private mode. Given all this, it is quite possible that the actual incidence of cheating is even higher.
However, the incidence of suspected cheating is likely even higher than that of actual cheating. It’s hard to estimate how often a good player can beat par, and it is easy to be suspicious when someone racks up a string of good scores. Obviously, if someone keeps getting a hole-in-one, they’re probably cheating, but it gets murkier for birdies and eagles. Before we accuse our friends, let’s try to figure this out objectively.
A traditional scientific criterion for investigating further is if the probability of an outcome occurring by chance (the alpha value) is less than 5% or less than 1%, depending on the goals of the researchers. The result is then deemed to be statistically significant at the 5% or 1% level. Since it is not nice to suspect people of cheating when they are not, let us choose the more conservative 1% level in this investigation.
Suppose you belong to a Wordle group of 10 players who have been sharing results with each other every day for 200 days. Assume that a very good human player can expect to get a birdie every 2.5 games, an eagle every 40 games, and a hole-in-one every 2,000 games (which are reasonable real-world estimates).
A) How many birdies in a row would be significant at the 1% level in your group during this time?
B) How many eagles in a row?
C) How many holes-in-one in a row?
It’s entirely possible that the frequency of good results in your group is significantly higher than the frequency predicted by chance, without anyone cheating. How would you explain this?
The End of Wordle
Wordle is a superbly designed game. For the individual solver, it is quick and easy to play and rewards skill and luck, reasoning and intuition. It does not require a specialized vocabulary, so most people can get good results often and easily, and with great visual feedback. Socially, it is the same puzzle for everyone and is easy to share and bond over. This explains its popularity.
However, one aspect of Wordle is poorly designed and will need to be changed soon to avoid compromising the game. The problem is the way that solutions are picked. A well-designed system would pick a random word from the list, with an equal probability of any word being picked on any day. The Wordle answers, however, come from a predetermined list set in a fixed order. The current list will run out sometime in October 2027.
This means that once a word is used, it will not appear again for five years! So when the words “theme” and “thyme” came up within a few days of each other, players who were just missing the middle letter in TH_ME in the later Wordle knew exactly what the answer was. And if you keep track of which words have been used, the list of potential future solutions will shrink with every passing day.
Consider a person with a perfect memory of past solutions. To such a person, the answer would be obvious on the last day of Wordle’s 2,309-word list. Can you quickly estimate how many holes-in-one this person would expect to get over the duration of the entire list, without doing the actual calculation? Then if you can, try and do the actual calculation.
Wordle was designed so that the solution word is already present “client-side” on your device the very first time each day that you connect to the website and download the code to solve the day’s puzzle. One advantage of using a predetermined list is that Wordle’s administrators don’t need to actively select a new random word every day — the solution changes automatically to the next word on the list at midnight based on the change in date. But this opens the door for hackers to find all of the solutions for the next five years.
How would you design Wordle so that it keeps the client-side design, ensuring that everyone gets the same solution word on a given day, but randomizes the answers in a sensible way without requiring a change in the code every day? (There’s no need to give coding details, just an overview of what techniques or algorithms could be used.)
Let’s hope The New York Times finds the solution to this question (or uses your answer) and makes the change before the end of the game is nigh.
Enjoy your daily Wordle. Happy puzzling!
Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt or one of the two Quanta books, Alice and Bob Meet the Wall of Fire or The Prime Number Conspiracy (winner’s choice). And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online, so solutions to the puzzle above should be submitted separately.)