The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (2023)

In Wordle, players try to find a five-letter word by guessing. After each guess, you learn which letters are in the right place, and which other letters are part of the word but not placed correctly. Then you guess again – but be careful, you only have six guesses.

The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (1)

Variations of this game go back decades. Jotto is a paper version that doesn’t give the information about exact matches. Lingo is a game show that aired from the 1980s through the 2010s, with the same overall rules as Wordle.

What Wordle gets very right is in its community. Everyone gets the same word every day, and those words come from a curated list. Players can share their results, with the coveted goal of finding the correct word quickly.

Using mathematical analysis of the Wordle data, we share some advice that can help you win more quickly!

Use Letter Distribution

Which letter do you think is the most typical last letter in a five-letter word?

The list of words that are eligible for Wordle’s word of the day can be analyzed for patterns, including the most frequent letters in each position. Importantly for our analysis purposes, the list of words eligible for Wordle’s word of the day is hand-picked. The word list can be found here. The full list of 5-letter words you can use as guesses is here.

Over 15% of Wordle's words of the day start with S. Only six other starting letters appear in more than 5% of Wordle words.In order of frequency, they are C, B, T, P, A, and F. These starting letters might seem pretty surprising, but they are close to the order of general five-letter words.

(Video) Winning Wordle: The Best Starting Words To Use and Why | Mashable

In order of frequency, A, O, R, E, I, L, U, H appear in more than 5% of Wordle words as the second letter. There is a steep drop-off after that.

In both the second and third positions, vowels dominate. The following letters appear in more than 5% of Wordle words as the third letter:A, I, O, E, U, R, N, in order of frequency.

The following letters appear in more than 5% of Wordle words as the fourth letter: E, N, S, A, L, I, R, C, T, O, in order of frequency. By a wide margin, the most common fourth letter is E, which is the fourth letter in 1 in 7 Wordle words.

The last letter is the most interesting: E, Y, T, R, L, H, N, D all appear as the last letter in more than 5% of Wordle words. Compare this to the last-letter frequency of general five-letter words: S, E, Y, D, T, A, and R appear in more than 5% of the full set of five-letter words.

Tip alert!: Wordle words rarely end in S – it is the 15th most popular last letter. Wordle words end in S less than 2% of the time, compared to 31% of the full set of five-letter words. This is the result of the Wordle list being made from hand-picking words — and the contrast is surprising.

Interested in courses that explore topics like this?Check out AoPS Online's Introduction to Programming. The work in this article was done through computer analysis.

Use All Five Letters to Eliminate Options

When you start a game of Wordle, you’ve got a blank slate and can enter any five-letter word.

Imagine you’ve played your first word, CHAIR. The game responds by telling you the H is correct AND correctly placed, while all the other letter guesses are not in the word at all. This is actually great progress: There are now only 33 possible target words, including GHOST, SHEEP, and WHOSE. Which word would you play next?

Surprisingly, one of the best words to play next is STOLE, which does not have an H as the second letter. By playing STOLE, we can distinguish between GHOST and SHEEP and WHOSE, all of which will have different responses. In fact, after playing STOLE as the second guess, the worst case scenario is having to pick between three possible words on the third try. (OK, you could also play SWOLE with the same results.)

(Video) The best Wordle strategy – according to science

The important detail here is that playing the H as the second letter is worse than playing another common word, even though it adds the small chance of nailing the word on the next try. One of the best options that keeps the H in place is THOSE, but it’s important to see that this is less useful than playing STOLE. By playing STOLE, we get information about the letter L, which THOSE does not give. The only advantage of playing THOSE is that it has a 1 in 33 chance of being the correct word, while STOLE does not.

In Wordle’s “Hard Mode,” there’s a reason it forces you to play the H where it was found: It is harder. This especially applies when you have 3 or 4 letters placed, but there are many options for the final letters.

Tip alert! If you aren’t playing Hard Mode, rather than guessing one letter at a time, pick a word that cannot be correct, but tells you which of the many options to use next.

What Word to Pick First?

One of the fun things about Wordle is there isn’t one single answer to this question. Some people have a go-to starting word they always play, while others switch it up. Here are some suggestions.

The Best Average

A key measure of the quality of a Wordle word is the information you get from it. Let’s say your first guess is KUZDU. A whole lot of the time, you won’t get any information except that K, U, D, and Z aren’t in the target word. That makes this a terrible choice, and it can be measured.On average, after playing KUDZU, you still have 870 words that could all be correct. A better word would have a lower average number of words remaining.

Many of the best words are not common English words. These are the best 10 options alongside the average number of words remaining in the decision pool IF you choose it as your first word:

The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (2)

Every one of these options uses three vowels and two common consonants.

(Video) Oh, wait, actually the best Wordle opener is not “crane”…

Avoiding Worst-Case Scenarios

Sometimes, your first guess is bad. But you can pick words that minimize the worst-case options. These words narrow things down to the fewest possible remaining choices. They are especially good choices in Absurdle, an adversarial version of Wordle where the target word can change but still fits the selections made.

These are the best 10 options in terms of the number of lowest maximum possible number of words remaining:

The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (3)

The first five are all anagrams, and their worst-case scenario is that none of the letters are present. 93% of Wordle words have at least one of the letters R, A, I, S, or E.

Going for Green

Earning a green letter (right letter, right position) is very helpful on the first turn. These are the best 10 options for earning green letters, ranked by the average number of green letters found:

The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (4)

While these words can help, they tend to have repeated letters, and don’t generally do well at narrowing down the options if they miss.

Going for a Quick Win

While it’s proven that any Wordle can be won in 5 guesses or less, it’s prestigious to win faster! One measure of this is how likely a guess can narrow the options to fewer than 10 remaining options. For example, if you pick RAISE and you find that the R, S, E are all in the correct spots and the I is correct but in the wrong place, you instantly know the correct word is RINSE.

(Video) Solving Wordle using information theory

These are the best 10 options that give you the chance of narrowing things to under 10 options in one shot:

The Math of Winning Wordle: From Letter Distribution to First-Word Strategies (5)

All of them use three consonants, and two of three vowels each (among A, E, and I).

Final tip alert!There’s a lot to think about here, but note that the best first words always use five distinct letters and use multiple vowels. After the first word, take into account what the possibilities are, and find a word that distinguishes between the remaining possibilities. And, if you’re stuck, use the word list to figure out what your options are.

What’s your starting word? Tag us at @artofproblemsolving with #wordle.

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Written by Bowen Kerins, Senior Math Curriculum Developer, Art of Problem Solving

FAQs

What is the mathematical probability of Wordle? ›

Since there are 2,315 possible target words in Wordle, the probability that you will guess the target in exactly one try is 1/2315 = 0.000432. In fact, it doesn't matter which of the 2,315 words in the dictionary you choose to use as your initial guess since the probability of success is always 1/2315.

What is the most common first word guess in Wordle? ›

A pair of MIT researchers recently set out to find the optimal starting word for the popular online puzzle, discovering that the statistically superior first guess is SALET, which is a 15th century helmet.

What is the optimal Wordle strategy? ›

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.

What is the average Wordle distribution? ›

On average 95% of target words can be solved within 6 guesses with average game length is approximately 4 rounds. By the time the third guess has been made, a good player will have located about 4 target letters with 2 or 3 in their correct positions.

Is there an algorithm for Wordle? ›

The various Wordle algorithms all take slightly different approaches in how they solve the puzzle. Some start with knowledge of the solution list; others do not. Some allow any of the almost 13,000 five-letter English words as valid guesses; others (like WordleBot) use a smaller set.

What is a good Wordle distribution? ›

Canberra, Australia is the global city with the best Wordle average: 3.58 guesses. Sweden is the world's best country at Wordle, with an average of 3.72. The US is ranked #18 in the world for Wordle, with a national average of 3.92. The American state with the best Wordle average is North Dakota (3.65).

What are the 3 best words to start with in Wordle? ›

If, on the other hand, you're simply trying to win within the allotted six guesses, the top three words to play are “adept,” “clamp” and “plaid.” Using any of these three words will yield an average success rate in winning the game of 98.79 percent, 98.75 percent, and 98.75 percent, respectively, if you're playing the ...

What is the heuristic for Wordle? ›

The heuristic assumes that the underlying set of Wordle solutions roughly follows the list of most frequently-occurring 5-letter words in English, and with this heuristic we can get a 96.4% success rate on the algorithm from a test set of common words.

What is the average score on Wordle on hard? ›

If you continue always picking the best word, this will result in an average of 3.4212 guesses in normal mode and 3.5084 guesses in hard mode. Note: to be able to work this out, it's necessary to use an exact definition of the rules of the game.

What percentage of people win Wordle every day? ›

While most puzzles have a 99% solve rate, and even tough puzzles have a solve rate in the 80% range, today's Wordle has a solve rate of only 45%.

What percentage of players solve Wordle? ›

The same survey found that only 14% admit to doing so. Wordle players have up to six tries to guess a five-letter word of the day, with the game offering feedback after every try.

What are the odds of getting Wordle on the first try? ›

And the first result that popped up from Real Statistics Using Excel (which seemed credible) said: “Since there are 2,315 possible target words in Wordle, the probability that you will guess the target in exactly one try is 1/2315 = 0.000432.

What are the 10 most common letters in Wordle? ›

Many statistically-minded people like to make the absolute most out of their starting two guesses by squeezing in all of the top 10 most common letters. That means you need two words that include all of E, T, A, O, I, N, S, R, H, and L.

Has anyone solved Wordle in one try? ›

The odds get marginally better every subsequent game (e.g., 1 in 2,314; 1 in 2,313) assuming the correct answer isn't a repeat of the previous days' solutions. With that in mind, it's highly unlikely that 1% of all players are guessing correctly on the first try — and much more likely that some folks are cheating.

Does the average person solve Wordle? ›

Human Statistics Playing Wordle. On average, Wordle players can guess the correct word on their first try in only 0.02% of games.

Why is Crane the best Wordle starter? ›

On average, the bot now expects to solve a Wordle in 0.0009 fewer guesses on average when it starts with CRANE rather than SLATE. (If you play Wordle every day, you can expect to save one guess over the next three years by opening with CRANE instead of SLATE.)

What is the best word to use after Crane in Wordle? ›

The New York Times' Wordlebot originally recommended Wordle users on regular mode start with the word CRANE. But, as of August 2022, Wordlebot now recommends the word SLATE. If you're on hard mode, the recommended word has gone from LEAST to DEALT.

What is the average Wordle winning percentage? ›

While most puzzles have a 99% solve rate, and even tough puzzles have a solve rate in the 80% range, today's Wordle has a solve rate of only 45%. In fact, WordleBot claims that today's puzzle has an average solve rate of 6.3 answers....which means that most players aren't getting the puzzle correct.

What's a good average score for Wordle? ›

Most people should, on average, get it in at least 4, even on days where it is harder. Especially if you aren't making risky moves, four guesses should supply you with enough information to make a correct guess.

What are the odds of winning Wordle? ›

The true chances will therefore depend on how many five-letter words you know and how often you play Wordle. However, it's somewhere between 0.007% and 0.05%.

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