The Science of Information Gain: From Theory to the Grid
Information gain is the engine behind every optimal Wordle solver, yet most players use it only as a vague intuition. This article makes the science explicit — what a 'bit' of information actually is, why entropy measures uncertainty, and how to translate the theory into fast table-side judgement.
What a bit really measures
A bit is the amount of information that halves your uncertainty. If 200 candidates remain and a guess reliably cuts them to 100, that guess delivered one bit. Cut them to 50 and it delivered two bits. Information theory simply formalises this halving intuition across every possible outcome of a guess.
Each Wordle guess produces one of 243 possible colour patterns (three states across five tiles). A guess that spreads the remaining candidates evenly across many of those patterns yields more expected bits than one that funnels most candidates into a single pattern.
Expected entropy, not best case
The key word is expected. A guess might occasionally produce a brilliant result, but its value is the average over all results weighted by how likely each is. This is why solvers prefer guesses with balanced outcome distributions: balance maximises the average, even if no single outcome is spectacular.
This also explains the counterintuitive truth that an all-grey result can reflect a great guess. If that grey pattern was one of many likely outcomes and it eliminated a large slice of candidates, the guess did exactly its job.
Approximating gain at the table
You cannot compute entropy in your head, but you can approximate it. Look at the letters you are unsure about and estimate how evenly each splits your remaining list. A letter present in roughly half the candidates carries close to a full bit; a letter present in almost all or almost none carries little. Build your guess around the most balanced unknowns.
A second heuristic: count distinct live candidates after a hypothetical green/yellow result. If every plausible result of a guess leaves a small group, the guess is high-information. If one result leaves a huge group, it is risky.
Key takeaways
- A bit of information halves your uncertainty.
- Judge guesses by expected gain across all outcomes.
- Favour letters that split the candidate list evenly.