Data & Mathematics
Information Theory and Wordle: How Entropy Prunes 2,300 Words to 1
Wordle is fundamentally a game of mathematical information gain. Discover how Claude Shannon's entropy formula reveals the true power of optimal guesses.
How does information theory apply to Wordle?
Every Wordle guess acts as an informational partition. For any guess, there are 35 = 243 possible color patterns. The best guesses maximize Shannon entropy by distributing the remaining candidate answers evenly across as many buckets as possible, ensuring that whichever feedback color pattern you receive, the remaining candidate pool shrinks to the absolute minimum.
Shannon Entropy: Measuring Bits of Information
In 1948, mathematician and electrical engineer Claude Shannon introduced the concept of information entropy: a quantitative measure of uncertainty in a probability distribution, expressed in "bits." One bit of information corresponds to cutting your uncertainty in half (like answering a yes/no question where both outcomes are equally probable).
In Wordle, you begin with roughly 2,309 possible solution words. Because 211.17 ≈ 2,309, you need approximately 11.17 bits of information to pinpoint the secret answer with 100% certainty. Your objective across your six guesses is to collect those 11.17 bits as quickly and reliably as possible.
The 243 Color Permutations
When you submit a 5-letter word, each tile returns one of three colors: Green (correct), Yellow (misplaced), or Gray (absent). Because there are five independent tiles, the game board can return exactly:
3 × 3 × 3 × 3 × 3 = 35 = 243 possible feedback patterns
A mediocre guess dumps 1,500 words into a single feedback bucket (e.g., all gray), meaning if that pattern occurs, you have learned almost nothing. In contrast, an optimal entropy guess slices the 2,309 words into many small, balanced subsets. Whichever color pattern appears on the board, the remaining bucket size is guaranteed to be small.
Why SALET, TARSE, and CRANE Rank at the Top
Computer simulations simulating all possible starting guesses against the entire Wordle solution list have consistently calculated the expected information gain (measured in bits):
- SALET: 5.836 bits of expected information on turn one
- TARSE: 5.824 bits of expected information
- CRANE: 5.741 bits of expected information
- SLATE: 5.856 bits (under alternate guess lists)
- FUZZY: ~2.1 bits (a mathematically terrible opener)
Notice that an opener providing ~5.8 bits cuts your search space from 2,309 candidate words down to an average of just 40 to 60 remaining words on turn one! With over half of the necessary 11.17 bits captured immediately, your chances of solving on turn three or four skyrocket.
Practical Takeaways for Casual Players
You don't need a laptop or scientific calculator while playing your morning puzzle. Just remember these three entropy-maximizing principles:
- Avoid Duplicate Letters Early: Guessing a word with two copies of the same letter (like EERIE or SPEED) rarely yields more information than testing five unique letters. You are spending a tile to test a duplicate when that tile could test an unplayed letter.
- Pay Attention to Letter Positions: S is overwhelmingly more informative in position 1 or position 5 than in position 3. E is far more common in position 4 or 5 than position 1. High-entropy words align letters with their most likely physical slots.
- Play to Minimize Worst-Case Scenarios: Rather than gambling on an obscure word hoping to get lucky on Guess 2, pick the word that leaves the fewest remaining candidates even if all tiles come back gray.