Learn & Understand

Information Theory and the Mathematics of the Optimal Wordle Guess

The companion calculator gives the raw odds of a random guess among the remaining possible Wordle answers, and notes that strategic play beats random guessing. The mathematics behind that strategy is genuinely beautiful: the best Wordle players are not trying to guess the answer at all, they are trying to gather information, a concept made precise by information theory. Understanding how information and entropy define the optimal guess, and why the mathematically best word early in the game is often one that cannot be the answer, turns a probability into an appreciation of the deep math behind a daily word game.

The Goal Is Information, Not the Answer

The key insight that transforms Wordle strategy is that a guess should be judged not by its chance of being correct, but by how much it narrows down the remaining possibilities. A guess that happens to be the answer wins immediately, but a guess that is not the answer can be far more valuable if it eliminates many candidates by revealing which letters are present, absent, or correctly placed. Good players think of each guess as a probe that partitions the remaining word list into groups based on the colored feedback, and the best probe is the one that splits the list most evenly, learning the most regardless of the outcome. This reframes the game entirely: you are conducting an experiment to extract information, not gambling on a hunch.

Entropy: Measuring Information

Information theory gives a precise way to measure how much a guess reveals, using the concept of entropy, roughly, the expected amount of information (in bits) a guess will yield.

A guess whose possible feedback patterns split the candidate words into many roughly equal groups gains the most information, because whatever the result, the remaining list shrinks dramatically. A guess that usually leaves most words still in play gains little. Entropy quantifies this expected reduction, and the optimal guess is the one that maximizes it. This is the same mathematics used to compress data and transmit messages efficiently, applied here to a word puzzle. Choosing the highest-entropy guess is the mathematically optimal strategy for gathering information fast.

Why the Best Guess Isn't the Answer

The most counterintuitive consequence is that, early in the game, the best guess is often a word you know cannot be the solution, because it contains common, untested letters that split the possibilities efficiently. If a word made of frequent letters would, whatever the feedback, slash the candidate list far more than a word that might be the answer, then playing the non-answer word is the smarter move, it sacrifices a tiny chance of a lucky immediate win for a large gain in information that improves your odds on subsequent guesses. This is why expert solvers and Wordle-solving algorithms open with information-rich words designed to test many common letters, rather than trying to hit the answer on the first try. Guessing to learn beats guessing to win, until you have learned enough. The random-guess odds the calculator shows are the floor that this information strategy improves upon.

Greedy Versus Optimal Play

There is a further layer of depth. Simply maximizing information on each single guess, a "greedy" strategy, is very good but not perfectly optimal, because the best play considers the whole sequence of guesses and the worst-case scenarios, not just the immediate step. Truly optimal Wordle solving, computed by algorithms, plans ahead to minimize the expected or worst-case number of guesses over the entire game, which occasionally differs from the greedy choice. This mirrors a general theme in decision-making: locally optimal choices are not always globally optimal. For a human, greedy information-maximizing is an excellent and practical approach, while the fully optimal strategy is a solved problem for computers. Either way, the guiding principle is information, not luck.

Playing Wordle Mathematically

Use the calculator to see the baseline odds of a random guess, and understand the strategy that beats it: the goal is to gather information, not to guess the answer, information theory's entropy measures how much a guess reveals, the optimal guess maximizes that information even if it cannot be the answer, and truly optimal play plans the whole sequence rather than each step greedily. The calculation gives the random floor; understanding information theory is what raises your Wordle play above it.