Information Theory in Word Games: Shannon Entropy and Guess Choice

Measuring uncertainty in bits

In 1948 Claude Shannon defined information as a reduction in uncertainty, measured in bits. One bit is the information you get from a fair yes-or-no question: it halves the possibilities. Two bits quarter them, three bits leave an eighth, and so on.

A Wordle puzzle starts with a fixed number of possible answers. Smotix Wordle draws its answers from a curated list of 731 common five-letter words, so before your first guess the uncertainty is log2(731), about 9.5 bits. Solving the puzzle means collecting 9.5 bits of information, and every guess is a question that returns some of them.

This is why counting bits is useful. It gives every possible guess a single score, the average amount it tells you, and lets you compare words as different as CRATE and ADIEU on the same scale.

A guess is a question with 243 possible answers

After each guess, every one of the five tiles turns green, yellow or grey. That gives 3 × 3 × 3 × 3 × 3 = 243 possible colour patterns. Each pattern is consistent with a different group of remaining answers.

A guess is informative when it splits the answer list into many small groups. In the extreme case, a guess whose pattern was the same for every answer would tell you nothing. A guess that put each answer in its own group would solve the puzzle immediately.

Shannon entropy measures exactly this. If a guess sorts the 731 answers into groups, its entropy is the average number of bits you learn from the pattern you see, weighted by how likely each pattern is. Large, lopsided groups lower the score; many small groups raise it.

• CRATE sorts the 731 answers into 111 different colour patterns, the largest of which holds 44 words.
• ADIEU produces only 65 distinct patterns, and its largest group holds 78 words.
• FUZZY, the weakest opener we tested, sends 477 of the 731 answers to the same pattern: all grey.

The best first guesses for Smotix Wordle

We computed the entropy of every one of the 4,517 words the game accepts as a guess, against all 731 possible answers, using the same colouring rules as the game (including repeated letters). The table shows the strongest openers, a few popular choices for comparison, and the weakest word on the list.

The top of the list is remarkably flat: the first six words are within 0.02 bits of each other. In practice any of them is an excellent start, and you can pick whichever you find easiest to follow up.

Why ADIEU and AUDIO disappoint

Vowel-heavy openers are popular because it feels useful to learn which vowels are in the word. The numbers say otherwise. ADIEU leaves 38 candidates on average and AUDIO leaves 61, compared with about 18 for CRATE.

There are two reasons. First, almost every answer contains A, E or O, so learning that one of them is present tells you little. In our answer list, E appears in 49% of words and A in 41%. Second, U and I are less common, so testing them uses a slot that usually comes back grey. Meanwhile the most useful consonants, R, T, L, S and C, go untested.

Position matters too. CRATE and TRACE put R, A and E in the places those letters most often occupy. In our list, R is the most common second letter, A the most common third letter, and E the most common last letter (163 of 731 answers end in E).

Average versus worst case

Entropy optimises the average. If you care more about avoiding a bad day than about your average score, look at the worst case instead: the size of the largest group a guess can leave you with.

On that measure IRATE (39) and ARISE (40) beat SLATE (54) and CRANE (63), even though SLATE scores slightly higher on average. If you play Hard Mode, where a bad early pattern is harder to escape, the smaller worst case is worth the tiny loss in average information.

ARISE has a second advantage: it is itself one of the 731 possible answers, so it can occasionally win in one guess. Of the strongest openers, ARISE, AROSE, ALERT, RAISE and STALE are all on the answer list.

Applying the idea after the first guess

The same reasoning applies to every guess, not just the first. After CRATE comes back mostly grey, there are 38 answers that share none of its letters. A good second guess is one that splits those 38 well, even if it cannot be the answer itself.

A practical rule: while more than about five candidates remain, choose the guess that tests the most common untested letters among them. Once only two or three remain, switch to guessing actual candidates, because a guess that might win outright is worth more than a few extra bits.

How we calculated these numbers

Answers come from the Smotix Wordle answer list (731 words). Allowed guesses come from the five-letter dictionary the game uses to check guesses (4,517 words). For each guess we computed the colour pattern against every answer, using the standard rule that a repeated letter is only marked yellow as many times as it appears in the answer.

Entropy is the sum of p × log2(1/p) over all patterns, where p is the share of answers that produce that pattern. Average answers left is the expected size of the group you land in. Because our answer list differs from the New York Times list, these rankings are specific to Smotix and will not exactly match published rankings for other Wordle games.