Information Entropy in Wordle: Algorithmic Solving Secrets

Wordle took the world by storm because of its elegant simplicity: guess a 5-letter word in 6 attempts. However, underneath the game lies a fascinating mathematical problem. By applying information theory and Shannon entropy, we can design algorithms that solve any Wordle puzzle in less than four guesses on average.

The Math of Feedback Patterns

For every guess, Wordle returns one of $3^5 = 243$ possible color feedback patterns (each letter can be green, yellow, or gray). A good guess is one that divides the remaining list of possible words into as many small, equal-sized groups as possible, regardless of which color feedback pattern is returned.

To measure this mathematically, we use Shannon's information entropy formula:

$H(X) = - \sum_{i=1}^{n} P(x_i) \log_2 P(x_i)$

Where $P(x_i)$ is the probability of receiving color pattern $x_i$ for a given guess. The entropy $H(X)$ represents the expected amount of information (measured in bits) that the guess will provide. A higher entropy value means the word is better at cutting down the list of possibilities.

Picking the Optimal Starting Word

Using this formula, we can evaluate all 12,972 valid Wordle words as starters. The word **SALET** provides approximately 5.83 bits of information, making it the mathematically superior starting word according to entropy calculations. Other high-performing starters include **CRATE** (5.80 bits) and **TRACE** (5.78 bits).

If you use **SALET** as your starter, an average feedback will reduce the list of 2,309 possible target words down to just 15. The puzzle is practically solved in your second and third steps.

Hard Mode vs. Normal Mode

In normal mode, you can guess words that you know are not the target just to gather information (e.g., guessing **NYMPH** to check uncommon letters). In hard mode, you must use all revealed hints in subsequent guesses. Entropy algorithms show that hard mode actually increases the average solve rate because it prevents you from playing high-information words that eliminate trap letter patterns (like **-IGHT** words: BIGHT, FIGHT, LIGHT, NIGHT, TIGHT).