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Words With Letters Medium

Words With Letters Medium – Every time I play Wordle, I am reminded of this quote from Arthur Conan Doyle’s character Sherlock Holmes, when he is solving another mysterious mystery:

How often have I told you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? – Sherlock Holmes

Words With Letters Medium

Wordle is a new daily word game that challenges you to guess a five-letter word in six rounds or less. After you enter a guess, the game will tell you which letters are correct and in the right place with a green highlight. Letters that are correct but not in the right place are indicated in yellow. Using this information, you can eliminate large numbers of words and guess better each round to solve the game. And like Sherlock Holmes, you are also in a race against the clock because you have to solve the game in six rounds.

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In this post, I will use computational methods to explore ways to solve this game efficiently. While I don’t directly use these methods to solve the daily Wordle game myself (where would the fun be in that?!), this analysis gave me insight into how the game works “behind the scenes”. Because of this, I feel that these insights help me play the game better and give me better outcomes.

Each round of Wordle gives you feedback on your guess, so you can deduce more and more information about the solution. Specifically, you learn the following information:

To explain the last item on this list: If you guess the word “RADAR” and Wordle responds with one “A” highlighted in green or yellow, then you know the solution has only one “A” in it. It’s a good strategy to start your first guess with a word that has five different letters, to cover as much ground as possible. But solution words often contain repeated letters, so at some point during the game you have to switch to guessing a likely repeated letter.

There are many ways to solve Wordle. Most people I know start with a random word and go from there based on the clues you get. Other people use the same two initial guesses, for example “RAISE” and “CLOUD” contain many letters that are very common in the English language.

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To start with words that have letters that are common in the English language, rely on tables of letter frequencies. For example, the word “RATES” contains more common letters than the word “EPOXY”. Starting with a word that has common letters is a good approach because it lets you eliminate many words quickly:

If you choose a word with the most common letters and one or more of those common letters are not in the solution, then you have identified a large number of impossible solutions. This significantly reduces the number of possible solutions, making the selection of the correct solution in the next round more likely.

Letter frequency tables exist for arbitrary words in the English language (or any other language for that matter), but here we only need to consider a very specific subset of the English dictionary: five letter words that Wordle accepts.

Words With Letters Medium

Wordle has its own unique five-letter word dictionary containing exactly 12,972 words. You can conveniently access this list of words using the Wolfram Language by loading the data from the following Data Repository object:

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There are some obvious English words like “lower”, “muggy”, “broth” and “brags”, but this Wordle list also contains some very obscure words like “maare” and “alary”. I don’t know the source of this Wordle word list, so we’ll just take this list at face value and use it to play the game. I should point out that there is a subset of this list that contains the actual solution words for the next few years. In this post I choose to ignore that subset of solution words as I consider them too much of a “cheat”.

Using the Wolfram Language, we can get the overall letter distribution of the Wordle words by concatenating all the words into a single very long string of 64,860 letters (12,972 words of 5 letters each) and using the CharacterCounts function to see how often each letter occurs:

So: StringJoin merges all the 12,972 words into a single string and then CharacterCounts counts how many times each letter appears. The KeySort function then sorts the result alphabetically.

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We can go a step further and see how common each letter is in a specific position in a five-letter word. For example, how common is the letter “E” in the third position? Here is the Wolfram Language code that achieves this:

This code looks a little more complicated, but it just counts how many times each letter appears in a specific place in a five-letter word. This gives the following output in a notepad:

So, for example, given the Wordle list of words, the number of times the letter “E” occurs in the third position is 882 times. Here is a more visually appealing version of this table:

Now that we have this table of values, we can assign a score to each five-letter word based on how common each letter is in a particular position. For example, the word “RAISE” has a score of 628+2263+1051+516+1522=5980. It is calculated by looking up the letter in the corresponding row of the table. For example, the fourth letter in the word, S, has a value of 516 in the fourth row of the table.

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Assigning a score to each word gives you a way to rank words. High-scoring words have common letters in places where they occur most often.

We can do this “score” calculation for every word in any five-letter word list, and the following Wolfram Language code does just that. It looks a bit more complicated, but all it does is look at each word in a list you pass to the function and calculate its score:

The word with the highest score is “SERE” which makes sense because all the letters are very common and in very common places (for example many words end with the letter S). In this article, I do not filter out words with repeated letters, such as “SERE”. Some people prefer five different letters in their initial guesses, which is only a slight variation on the approach I’m taking here. The highest scoring word with no repeating letters here is “CARE”.

The conclusion we can draw from the spelling feedback is that the letters S, R and E do not appear in the word, but the letter O does appear and it appears in the second position. We can write a small piece of code to remove all the impossible solutions and get a list of 520 words that can still be a solution:

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There are no new “good” letters, but now we can additionally eliminate the letters C, L and Y. We also now know that there is only one O in the word:

We have therefore further reduced the list of possible words to 117. The best guesses are now:

The additional information is that the letters G and A do not appear in the word. The letter N appears in the fourth position, and it is also the only letter N in the word. That leaves us with only 13 possible words:

We now have three known letters (OUN) and two letters we can rule out (P and D). This narrows the possibilities down to two words:

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If “MOUNT” was wrong, we would only be left with “FOUNT” as the final possibility, so either way, we would have solved the game in six innings!

To summarize this strategy: We loaded the list of allowed Wordle words into the Wolfram Language. We then calculated a table of values representing how often a specific letter occurs in a specific position of a five-letter word. Using that table, we then calculated a list of high-scoring words, hoping to optimize our chances of guessing the word (or nailing down as many letters as possible).

At each round of the game, we removed “impossible” words based on the game’s feedback mechanism, and then generated a new “most likely” word based on the remaining word list. Using this approach, we systematically reduced the possibilities from 12972, to 520, to 117, to 13, and finally to 2 equal scores. It is interesting to see how important the first guess is. This eliminated more than 95% of the possibilities. Even when you have a disappointing first guess (no matching letters), you still eliminate many, many words.

While there is no guarantee that it will always yield success in six rounds of the game, it does a good job of