How to Solve Wordle With Information Theory

TL;DR

Choose Wordle guesses by maximizing the expected information revealed across all possible color patterns, rather than merely selecting common letters. Information is measured in bits as the negative base-two logarithm of an outcome’s probability, so rare patterns provide more information, while entropy combines every pattern’s information with its probability to score a guess.

Transcript

The game Wordle has gone pretty viral in the last month or two, and never one to overlook an opportunity for a math lesson, it occurs to me that this game makes for a very good central example in a lesson about information theory, and in particular a topic known as entropy. You see, like a lot of people I got kind of sucked into the puzzle, and lik... Read More

Key Insights

• Wordle is a guessing game in which a mystery five-letter word must be identified within six attempts. Gray feedback marks an absent letter, yellow marks a present letter in the wrong position, and green marks a letter in its correct position.
• A Wordle guess is valuable because its complete color pattern partitions the remaining candidate words. The pattern can sharply reduce the possibility space even when several guessed letters turn gray, since confirmed absences eliminate every candidate containing those letters.
• Common-letter strategies are useful but incomplete because they ignore letter positions and the probability distribution of feedback patterns. Comparing words such as "nails" and "snail" requires considering where letters appear, not merely whether the guesses contain the same letters.
• The valid-guess lexicon contains about 13,000 words, including many uncommon entries, while the curated list of possible answers contains around 2,300 words. The proposed challenge avoids relying directly on that answer list and instead seeks a more general strategy.
• An informative outcome is inherently unlikely because it leaves very few compatible possibilities. For the guess "weary," one illustrated feedback pattern leaves only 58 matching words, but under an equal-likelihood assumption its probability is only 58 divided by roughly 13,000.
• The most probable feedback patterns tend to provide the least information. For "weary," one illustrated pattern leaves about 1,400 matches and has roughly an 11 percent probability, while the all-gray pattern is described as occurring about 14 percent of the time.
• One bit of information corresponds to an observation that cuts the possibility space in half. Two bits reduce it by a factor of four, three bits by a factor of eight, four bits by a factor of sixteen, and five bits by a factor of thirty-two.
• Information is calculated as the base-two logarithm of one divided by an outcome’s probability, equivalently the negative base-two logarithm of that probability. Expected information evaluates a guess by combining each possible pattern’s probability with the information that pattern would reveal.

Questions & Answers

Q: How can information theory help solve Wordle?

Information theory provides a quantitative way to rank guesses by the expected amount of information their feedback patterns reveal. For every possible gray, yellow, and green pattern, a solver determines its probability and how much it would reduce the candidate space. Multiplying each pattern’s probability by its information value and considering the full distribution produces an objective score for comparing guesses.

Q: What does entropy mean in a Wordle-solving strategy?

Entropy is used to measure the expected information produced by a potential guess. A guess can lead to many possible color patterns, each having its own probability and information value. The strategy evaluates the complete distribution rather than focusing only on a favorable outcome. This balances rare, highly informative patterns against common patterns that eliminate fewer possible words.

Q: How is information measured in bits for Wordle?

A bit measures how strongly an observation reduces the current possibility space. If feedback cuts the candidates in half, it provides one bit. A reduction by a factor of four provides two bits, and a reduction by a factor of eight provides three bits. In general, information equals the negative base-two logarithm of the observed pattern’s probability.

Q: Why are rare Wordle feedback patterns more informative?

Rare feedback patterns are more informative because they correspond to a smaller portion of the possible answers. One illustrated pattern for "weary" leaves only 58 matching words out of roughly 13,000, creating a major reduction. Its rarity and its information are two sides of the same fact: an outcome that identifies a small candidate group has a low probability of occurring.

Q: Why is choosing common letters not enough for Wordle?

Choosing common letters can expose many likely matches or useful absences, but it does not account for letter positions or the full range of possible feedback. Two guesses can contain similar letters while arranging them differently, as with "nails" and "snail." A systematic ranking must evaluate how each arrangement divides the candidate words among all possible color patterns.

Q: Can gray letters provide useful information in Wordle?

Gray letters can provide substantial information because they rule out every compatible candidate containing those letters. The suggested pair "other" followed by "nails" covers many frequent letters, so even an all-gray result would be informative because words avoiding all those letters are rare. Information comes from reducing possibilities, not only from receiving yellow or green matches.

Q: Why avoid using Wordle’s curated answer list?

The proposed solver avoids depending on the curated list because the goal is to build a strategy that can play against more than the official site. The roughly 2,300-word answer list excludes some reasonably common five-letter words, and its source-code ordering reveals answers by day. General word-frequency data offers a broader way to prefer common answers without relying directly on that list.

Q: How should a Wordle solver rank an opening guess?

A solver should enumerate the possible feedback patterns for a proposed opening guess, determine how many candidate words match each pattern, estimate each pattern’s probability, and calculate its information value. The guess can then be scored by its expected information across the entire distribution. This approach evaluates both likely modest reductions and unlikely dramatic reductions instead of favoring either alone.

Summary & Key Takeaways

• Wordle asks players to identify a mystery five-letter word within six guesses. Every submitted word produces gray, yellow, or green feedback about absent letters, misplaced letters, and correctly positioned letters. A solver can use each resulting color pattern to eliminate incompatible candidates and progressively narrow the remaining possibilities.

• A systematic strategy must evaluate both how informative each possible feedback pattern would be and how likely that pattern is to occur. Rare outcomes can eliminate nearly every candidate, but common outcomes usually reveal less. Expected information accounts for this tradeoff by weighting the information from every possible pattern by its probability.

• Information is measured in bits according to how strongly an observation reduces the possibility space. Cutting the candidates in half provides one bit, reducing them by a factor of four provides two bits, and reducing them by a factor of eight provides three bits. The general formula is negative log base two of probability.