Five Letters, Six Guesses, Infinite Strategies

Tears. Crate. Adieu. Five letter words like these, seemingly unrelated, have taken the world by storm in the new New York Times trending word game — Wordle.

Wordle was created by Josh Wardle, a software engineer from Brooklyn, NY, as a romantic gift for his partner. Wardle’s game has a simple design, with a designated five letter “word of the day” that a player has six guesses to get right—but its simplicity is what makes it so novel, and now, over 300,000 people play Wardle’s game daily. 

Wordle 253, a Wordle example. When a player makes a guess, letters that are highlighted in green are letters that are in the correct place, and letters in yellow are letters that appear in the final word but in a different position. Letters in gray are not in the final word.

Since Wordle’s release, many have begun to publicize their strategies — looking for the most optimal words to start with to minimize the number of tries taken to guess the word — and “celebrity” scientists have begun to chime in as well. Mathematician Grant Sanderson, who runs the math Youtube channel 3Blue1Brown, recently released his algorithmic explanation of Wordle strategy , receiving over 7 million views within just three weeks of releasing this video.

A Wordle Algorithm

Sanderson’s video is centered on the creation of an algorithm designed to optimize Wordle guesses based on information theory, which deals with mathematically representing information. The wordle-bot attempts to generate the guess that will give the player the most information, essentially quantifying how good a guess is. The basic unit of information Sanderson uses is the bit, or binary digit, which is often used to represent values in computing — in Wordle terms, if your guess gives you one bit of information, it cuts the “uncertainty,” or the number of remaining possible answers, in half.

Professor Bill Wilson, who teaches discrete mathematics at De Anza College, explains that the “bits” of information you get from a guess is inversely correlated with how probable it is to obtain that information — essentially, uncertainty is most reduced when you see an unlikely pattern, like a word without any vowels, or one that starts with a ‘Q’.

“You can only get one of the three colors in each of five slots — yellow, green or gray — so there are three to the fifth possible results you can see,” Wilson said. “The results which are least likely to happen are the ones that give you the most information, because if there’s some pattern that would correspond to only one possible word, it’s not likely you’re gonna see that pattern. But if you do, you can celebrate, because the next time you got the word for sure, 100%.”

Senior Lydia Lu, who plays Wordle daily, has her own strategies to get the most “bits” of information possible from a given guess — without, of course, the precision of a computer algorithm. 

“My first words will always consist of the same three or four words recycled: adieu, crane and radio,” Lu said. “Through word of mouth, and just talking to other people I know who play the game, I know those are good words to start off with because they either use a certain amount of vowels, or they use certain letters that are the most common letters in English words.”

Sanderson’s algorithm adds refinement to the strategies that most players, like Lu, use by weighing not only which letters and words are most common, but also where letters are most commonly placed in words. Using logic similar to Wilson’s, Sanderson’s algorithm calculates an “expected information” value for each guess by multiplying the probability of a certain amount of information being gained from the guess times the bits of information gained.

A diagram of Sanderson’s algorithm in action. Here, the guess is “weary,” and the information gained if the ‘A’ and ‘Y’ are in the correct place is calculated by log2(1/p), where p is the probability of ‘A’ and ‘Y’ being in the correct place.

Sanderson’s equation then sums this product of the probability and the information from the potential outcomes of each guess to calculate an “expected information” for each guess. The word with the highest expected information is then designated as the best guess. 

Wordle in Practice

In theory, algorithms like Sandersons would guarantee optimal guesses for every Wordle game. However, enjoying Worlde requires no such precision, but rather intuition and a positive attitude. 

“On the first guess, the most important thing is to have common letters, and to try as many letters as possible,” Wilson said. “Then, if I feel like I’ve got a good picture of what it might be, I might try to ace it and get it in two or three guesses, but if there’s not so much there, then I’ll pick a word that has completely different letters, so I can find out more about what’s going on. That’s more intuitive than using anything quantitative.”

Wilson and Lu both believe that a “good” Wordle player can guess a word in three guesses, or less than four on average. Sanderson’s bot takes about 3.43 guesses, indicating that the precision of the algorithm is pretty much on par with regular human intuition. And, this intuition surrounding Wordle is what makes the game so appealing. 

“It gives people something to look forward to because you can only play it once every 24 hours,” Lu said. “It’s just a fun thing that you can do to start off your day, and because you can only play once, it takes a while for an individual to get bored from playing it.”

With Wordle as part of his daily routine, Wilson concurs with Lu. “I try to do it in four or fewer guesses — and I feel like it’s gonna be a good day when I get it in three. It’s a good start to my day.”

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