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. For those in the former group, who want to refine their approach, there have been some initial discussions of strategy and an exploration of Wordle-eligible words, but there’s yet to be a thorough examination of decision-making in the game. As we dig deeply, there’s a lot more to the game than one’s choice of a starting word. We can also learn about how words work, and the English language, and perhaps even a bit about why the game is fun.
Where Wordle Comes From
The roots of Wordle are in the old board game Mastermind. In that game, one player (the code-maker) sets up a hidden pattern of colored pegs, and the other player (the code-breaker) has to guess that pattern. At each turn, the code-breaker lays out a series of pegs, and the code-maker responds: one kind of mark if the color chosen is in the pattern at all, and another mark if the color chosen is already in the right position.
If you’ve played Wordle, you see where this is going.
Mastermind has a legacy as more than just a game. Mathematicians, logicians, and others have long enjoyed Mastermind as a thought experiment to explore algorithms and problem-solving. Decades after its invention by Israeli postmaster Mordechai Meirovitz, and after selling 50 million copies in the 1970s, it could help us refine various theories about Wordle.
The original game of Mastermind had four slots and six colors. So: 6 x 6 x 6 x 6 = 1,296 possible arrangements. A harder version of playing Mastermind with home rules could leave one slot blank, meaning there were really seven colors available; that increased the possible answers to 2,401. In either the traditional or harder methods of playing, guessing the right combination of pegs in fewer than the 10 moves available required some strategy.
The optimal strategy in Mastermind is, at first, to guess as many colors as possible, to first find out which colors are actually in the solution. Then, once one knows the colors, it’s simply a matter of unscrambling one’s guesses until they’re in the right order.
Is Wordle Harder Than Mastermind?
At first glance, the designers of Wordle have taken the difficulty of Mastermind and massively increased it. Instead of four slots, there are five. Instead of six peg colors, there are 26 letters of the alphabet. In theory, that could mean there are nearly 12 million possible solutions to Wordle. And to make it even more difficult, you only get six guesses at Wordle, compared with the 10 in Mastermind.
However, the rules of the English language quickly reduce that number by a great amount. Wordle requires that all words tried be actual words; “ZZZZZ” isn’t a possible solution (or a possible guess), and dolphin Wordle isn’t the game we’re playing.
To apply the rules of English, Wordle keeps a stored list of five-letter words embedded in its code. The Wordle dictionary was revealed to the world by software engineer Bertrand Fan. I performed a series of analyses on that dictionary, alongside others, with packages available in Python.
The very first finding is that the list of words you can guess is around 1/10,000th of what would be possible if the rules of English weren’t a constraint: Wordle has only 12,972 words to choose from for possible guesses. The solution set is smaller—only 2,315 words—but that doesn’t necessarily make the game easier, since plenty of words in common usage, that you might guess, are simply not included as solutions. However, there are quirks to the words chosen as solutions that differentiate them from the rest of the English language, as we’ll explore.
There’s an inherent difference in how we’ll approach optimizing a solution to Wordle, compared with Mastermind. In Mastermind, you “know” all possible guesses and answers, since they’re just different arrangements of colors that are available. But in Wordle, your own language skills, memory, and the Wordle dictionary mean that no human (with the possible exception of an eccentric memory champ) could correctly identify 12,972 eligible words in as many guesses.
Because of this, it’s also impossible to do what the Mastermind mathematicians did and calculate how much closer to the solution you get when you learn of a letter that’s in it, since the value derived from that learning is different for every person.
After observing a number of people play the game, I’ve come to lean on a highly anecdotal line to draw between pure guessing and actually working on solving the puzzle: Knowing three letters in the solution, plus something about their positions, puts you on a path to informed and reasonable guesses. So we’ll be using that goal, of identifying at least three letters, to judge the quality of our solution.
For example, if you know that the letters O, P, and R are all in a solution, the possible words you can guess next are narrowed down to just 135 options, none of which contain J, Q, and Z. If you learn there’s an O, you’ve narrowed down 12,972 possible guesses to 3,911. When there’s also a P, the possible guesses are now 566. So as opposed to an ideal Mastermind algorithm that goes all the way to the solution, our goal here will be to get ourselves to what we’re assuming is a “winnable position” of knowing at least three letters in the word.
What solving in Mastermind and solving in Wordle do share is a basic strategy. As in all guessing games, the path to winning is found by reducing “entropy,” which in plain English is basically the amount we don’t know about an answer. Our problem when we start both games is an abundance of choice: Getting the right solution on our first guess would be pure luck. But by applying what we learn in our first guess to our second guess, and so on, we can reduce the number of answers still in consideration and get closer to solving the puzzle using our knowledge and reason.
To reduce entropy, we want to learn as much as we can on each turn. In Mastermind’s early turns, this means not repeating colors, since that reduces the opportunity to learn whether other colors are in the solution or not. In Wordle, an optimal strategy means avoiding repeating letters as much as possible. Instead of rushing to try to guess the actual solution on your second try, an ideal second word would repeat no letters from your first guess. That way, your first and second guesses test as many letters as possible, bringing you closer to that informed guessing point of three letters known.
With that in mind, we can find a path to getting as close to a solution as possible, as quickly as possible. We’ll identify the strongest starting words for our strategy, using what we know of English—and of Wordle—to get there.
All Letters Aren’t Created Equal
We’ve already seen how requiring arrangements of letters to be English words drastically reduces their potential randomness by well more than 99 percent. English words follow additional nonrandom principles that help us quickly funnel down the list of possible solutions and the best guesses to find them.
For one thing, all letters aren’t used with the same frequency. Here’s a breakdown of how many words each letter appears in, in the Wordle dictionary:
Nine letters—A, E, I, L, N, O, R, S, T—stand out far above the rest. The letter E is as close to an ideal initial guess as we can get, cutting the dictionary nearly in half: An E in our first guess tells us we’re dealing with either one of the 1,056 words containing it, or with one of the 1,259 that don’t have an E. But we want to avoid leaning heavily on other vowels in our early answers.
This is a key area where we’ll depart from a path many have chosen: It’s common for people to place a lot of vowels in their opening guesses (using words like arose). But we’re going to go in the opposite direction.
That’s because another way that English rules reduce randomness is how vowels work. There are only five vowels, and they’re needed in almost every word. Only 13 words in Wordle’s solutions have no A, E, I, O, or U, and those 13 are of the “sometimes Y” variety (Among the words available as guesses in Wordle, but not as solutions, there are five words that don’t have any vowels or the letter Y: crwth, cwtch, grrlls, grrrl, phpht.
This means that essentially every word you guess will have at least one vowel. If you use only one vowel per guess and choose a different vowel each time, you’ll have tested every possible vowel after the fifth turn.
Compare that with the consonants: There are 21 of them, and if you can guess four different ones at every turn (leaving space in each guess for a single vowel), you still won’t have guessed all possible consonants by the end of five turns.
Since our goal is to repeat as few letters as possible early on, we want to keep vowels to a minimum, as we’re going to likely end up repeating vowels to create words anyway. For example, let’s say we guess arose with our first word, which has five of the most popular letters; then we guess a word with the other five letters from the 10 most popular: unlit. We’ve now run out of new vowels to use in guesses three and beyond, and will be almost guaranteed to repeat letters as a part (and sometimes a large part) of our remaining guesses, since we need to use vowels to form words. If we’re repeating certain letters, we’re not guessing as many other letters as possible, and we’re not reducing the entropy as much as we otherwise could. Instead, what we want to make sure to do is to guess as many different consonants as possible early on, prioritizing the most popular ones. This means we learn less than is absolutely possible in the earliest guess or two, by skipping some vowels, in exchange for preserving the opportunity to learn more in total (including consonants) from several guesses than we otherwise would.
The Wordle dictionary contains 2,692 single-vowel words. Now we need to find the ones that make the best starting guesses.
The First Three Words We Should Guess
What makes an ideal first guess word? We’ve already learned that it should contain only a single vowel. It should also contain no repeating letters (sissy would be a very bad first guess), to give us more opportunities to test different letters of the alphabet. Removing words with repeating letters from our list of single-vowel words, we now have 1,783 choices.
When we score our shortened list by how many other words contain the letters of the potential guess words, here’s the resulting top 10 list:
The first four words all have the same five letters—E, N, R, S, T—in different arrangements, so one of those is clearly our best bet. Which one should we choose? Among the solutions, I’ve found that an “s” is far more likely to be the first letter than in any other position, so we’ll go with stern.
After that first guess, we’d already have identified three or more letters from 374 solution words, more than 16 percent of all possible game-winners. If that day’s word is one of the 16 percent, we’re now on the path to making well-informed guesses and should consider breaking away from our generic algorithm; if it’s not, we still need a system to arrive at our goal of knowing three letters in the solution:
Our second guess should repeat none of the letters in stern. There are only 151 single-vowel, no-repeating-letter words in the entire Wordle dictionary that don’t overlap with [e, n, r, s, t] at all. Of those, these are the ones with letters that match the most solution words:
You probably haven’t heard of yclad (an archaic word meaning “clothe”), haply (also archaic, meaning “by chance”), or glady (meaning “resembling a glade”), but they’re all in the Scrabble dictionary, too.
Phyla, clamp, and madly are all pretty close in score, if it feels more sporting to use a word you already know. But let’s be real: Next time you play Scrabble, you’re definitely going to use yclad, so you might as well play it here.
How are we doing so far, two guesses in? We’ve guessed three or more letters in 1,469 solution words—a filibuster-proof majority of all possible winners, in just two turns. If the day’s solution is one of those, now is the time to consider whether we want to make guesses that rely on what we’ve learned, and break away from the generic algorithm (by guessing as many nonrepeating letters as possible that seem likely to be in the solution, given what we already know). But if the day’s solution is one of the 36 percent or so that don’t have three letters in our guesses so far, we’ll still need to optimize for trying as many new letters as possible.
If we want to continue with the algorithm: After using terns and yclad, there are only three single-vowel words left in the Wordle dictionary that don’t overlap in letters with our first two choices, and they share many of the same four consonants: whomp, bumph, whump. All are in the Scrabble dictionary. Since the letter O is significantly more popular than U, we’ll go with whomp.
We now have guessed three or more letters in 2,060 solution words and have three guesses to build the rest of the word.
The last 255 words in the Wordle solution set are necessarily difficult to solve for—they involve rarer letters—and on the days when they’re the solution to the puzzle, we’ve now got three guesses left to try to find them.
There are nine consonants we haven’t guessed yet; in descending order of how often they appear in words where we haven’t guessed three or more letters, they are G, B, F, K, V, Q, Z, X, and J. Only two words contain at least three of them: befog and vibex. If we go with befog, there will be 92 Wordle solutions left where we’ve only guessed three letters; if we go with vibex (which isn’t in the Scrabble dictionary, so we might consider it unsporting to use it), there are 70 such words left.
After adding befog, we’ve guessed more than three letters for more than 96 percent of all puzzles we might face in Wordle, in four guesses.
However, if a given day’s word is one of the 92 solutions left where we haven’t found three letters, we still have some work to do. There are three groups within those solutions: the words either repeat letters we’ve already guessed, or they contain one or more of the letter I, or they have a QU in them.
With the eight letters (six consonants) we haven’t yet guessed, we can construct six different words in the Wordle dictionary that contain four: azuki, juvie, mujik, squiz, quirk, quick. We don’t need to guess the Q words, since all such solution words also have the letter U. That makes juvie our best guess.
We’ve now found at least three letters in all solutions used on 99.8 percent of days.
The Three Hardest Words in Wordle
We’ve now made five guesses, and there are still four words that could be solutions where we’ve only guessed two of the letters: queue, kappa, kayak, mamma.
Mamma, with its all-repeating letter lineup (it’s the only solution word composed of just two different letters), might be the single hardest solution to discover in the game. Every day that the solution contains a repeating letter brings complaints from many that the day’s puzzle was too difficult to solve. Overall, 749 solution words have repeating letters, nearly one-third of the total, and 57 repeat a letter twice, but mamma stands alone.
If our guesses have been [stern, yclad, whomp, befog, juvie], there are still four letters we haven’t used, and that could make it hard to get the solution in one last guess.
What this shows is that, in one way, Wordle can be quite difficult: Even five statistically optimized guesses would still make it a bit of a challenge to guess three of the words that could be solutions, in just six guesses.
But then again, that’s only three words out of the more than four years’ worth of words Wordle has lined up. On all but a few of those days, a sound strategy sets one up for a win.
Exploring the Wordle Dictionary Further
Wordle tries to make things a bit easier for us by expanding the usual English word gaming dictionary and giving us those 12,972 words eligible for guesses. A version of the Scrabble dictionary that I downloaded from a GitHub repository had only 8,636 five-letter words, far less room to guess than Wordle. We’ve already seen how Wordle includes some words that don’t really count as English, even in a very expansive definition. There’s a sense in which one can jam one’s fingers on the keyboard semirandomly and have a decent chance of landing a valid guess, like phpht (an exclamation in British English that is “an expression of mild irritation or reluctance” according to the Collins dictionary), crwth, (Welsh for an ancient Celtic instrument similar to a violin), or vozhd (a Russian Church Slavonic loanword meaning “leader”).
At the same time, the game curates a far more restrictive list of potential solutions: There are 2,315 words queued up, and all of them can be found in the Scrabble dictionary.
How are the solutions different from the rest of English? For one, they’re almost always really five-letter words. In the Scrabble dictionary, nearly one-third of five-letter words end in an S, meaning they’re likely plural versions of four-letter words; in Wordle solutions, fewer than 2 percent end in an S, and not one of them is a plural word.
The solutions also have a different structure than less common words, with vowels far more often in the middle of the word. In these graphs, I show which positions of the word—first letter through fifth letter—have more and fewer vowels, across three different sets of words: solution words, all Wordle words, and all Scrabble words.
When we look at an aggregation of the most popular 333,333 words found on the internet at one point and compare it with the Wordle dictionary, we find that Wordle’s words are far more popular.
In the following graphs, we look at how many five-letter words there are among these 333,333 most popular. When we look at all five-letter words, they are pretty evenly distributed in English. There are about as many five-letter words among the least popular (toward the right side) as there are among the most popular (toward the left) and in the middle.
But when we look at the 10,657 words included as guesses but not solutions in Wordle, we find they’re a fair amount more popular than the words selected from anywhere on the internet (their average ranking is closer to the No. 1 position, on the left).
This makes sense, because a great many words scraped from the internet aren’t in any way English at all, but some kind of transliteration or code, or something else (the lowest-frequency five-letter word is golgw, which appeared 1,271 times, and the most popular is about, which appeared 1,226,734,006 times).
And when we look at the words chosen as solutions, we see that they’re far more popular than the words available as guesses; the distribution is even heavier toward the left (the least popular solution word is still very clearly a word used in common English: elate).
What this all suggests is that the words we’re trying to solve for are very likely to be words we know. Even as we get frustrated while guessing, thinking the solution must be some very obscure word, the reality is different; the Wordle solutions are hand-picked to be quite a lot more popular than average. And perhaps that’s one reason the game is fun: We already know what we’re looking for, but we get to take a difficult journey to find it.