wordle

A fast solver for wordle written in Python using numba.

For the maximum information policy it takes:

• less than 3 seconds to compute the optimal next word
• 30 minutes to compute the optimal starting word considering all solutions

Test conducted on MacBook Pro (13-inch, M1, 2020)

Interactive Solver

Run demos/demo_solver.py

Optimal starting guess: reast
-----------------
Enter guess,code: reast,10000
Next guess: courd
-----------------
Enter guess,code: courd,20010
Next guess: crick
-----------------
Enter guess,code: crick,22200
Answer is crimp

Simulate a game

Run demos/demo_game.py

g = Game(word="crimp", verbose=True)
agent = StandardAgent(answers, guesses)
final_guess, _ = agent.play(g)
print(final_guess)

Game Modes

This package supports both standard and hard mode. In hard mode, any revealed hints must be used in subsequent guesses.

Use the mode parameter of the various Agent and Solver classes to change modes e.g.

agent = MaxInfoAgent(answers, guesses, mode="standard")

Bullshit Detector

Don't trust how fast someone solved wordle? Check their results by seeing all remaining words, ranked by common usage.

Run either of:

• demos/bs_plays.py
• demos/bs_tiles.py

Performance and Optimal Starting Word

I consider an optimal start word to be the one that solves the most answers and takes the fewest plays on average.

The optimal starting word depends on the policy used and the game mode. The table below outlines the optimal starting word and performance characteristics of each policy on the original wordle answer/guess list.

Policy	Mode	Optimal Starting Word	Mean Guesses	Failed Words
MaxInfo	Standard	'reast'	3.600431965442765	'goner' 'hatch' 'jaunt' 'found' 'waste' 'taunt' 'catch' 'dilly' 'boxer'
	Hard	'reast'	3.639635732870772
MaxSplits	Standard	'trace'	3.6207343412527
	Hard	'salet'	3.6259211096662334	'pound' 'hatch' 'watch' 'found' 'cover' 'blank' 'boxer' 'wight'
MaxPrune	Standard	'laten'	4.439469320066335	'sissy' 'awake' 'blush' ... 'judge' 'rower' 'shave'
	Hard	'leant'	3.8034934497816595	'dolly' 'mover' 'piper' 'water' 'foist' 'bound' 'sense' 'viper' 'rarer' 'waver' 'wreak' 'flake' 'wound' 'baste' 'tight' 'biddy' 'happy' 'fleck' 'mossy' 'hound' 'blame' 'vaunt' 'match' 'catty' 'rower'

Notes

Failed words are often due to "lookalikes". For example with the word hatch the solver will check match, batch, patch and latch first and ultimately fail.

Policies

Maximum Information

The policy plays the word from the guess list that maximises the expected information content revealed about the solution, until there is one solution remaining. Played words are not repeated as they would not reveal new information.

Let the outcome from making a guess, i.e. the code received, be a discrete random variable $X$. This random variable has 243 possible outcomes (5 letters each with 3 feedback states i.e. $3^5$). The following are examples of outcomes:

• ⬜️⬜️⬜️⬜️⬜️ - no matches,
• 🟩⬜️⬜️⬜️⬜️ - only first letter matched,
• 🟩⬜️⬜️🟨⬜️ - exact match and a partial match.

For a guessed word, $G$, the probability of an outcome is given by a Categorical distribution i.e.

$$P(X | G) \sim Cat(k, p)$$

The expected information content in the outcomes of $P(X | G)$ is given by entropy, which is defined as

$$H(X|G) = -\sum_{i=1}^{243} P(X=x_i | G) \log(P(X=x_i | G))$$

where $P(X=x_i | G)$ is the probability of outcome $i$. The value of $P(X=x_i | G)$ is the proportion of answers that fall in outcome $i$ when playing the guess word. Therefore $$H(X|G) = -\sum_{i=1}^{243} P(X=x_i | G) \log(P(X=x_i | G))$$

$$H(X | G) = - \sum_{i=1}^{243} \frac{|S_i|}{|S|} \log \left (\frac{|S_i|}{|S|} \right )$$

The word that most evenly divides the answer pool into the 243 bins necessitates the greatest number of bins and thus has highest entropy.

Maximum Splits

This policy plays the word from the guess list that results in the largest number of outcomes.

Maximum Prune

This policy plays the word that reduces the remaining answers as much as possible.

Appendix - Installation Notes

numba on Apple M1

numba requires llvmlite, which in turn requires llvm version 11. The default installed version of llvm is likely more recent than version 11.

1. Install llvm version 11

arch -arm64 brew install llvm@11

2. Install llvmlite by pointing to old llvm version

LLVM_CONFIG="/opt/homebrew/Cellar/llvm@11/11.1.0_3/bin/llvm-config" arch -arm64 pip install llvmlite

pip install numba

scipy on Apple M1

brew install openblas
pip install --no-cache --no-use-pep517 pythran cython pybind11 gast
OPENBLAS="$(brew --prefix openblas)" pip install --no-cache --no-binary :all: --no-use-pep517 scipy

Appendix - Information Gain Equivalance

The Maximum Information policy is equivalent to the Information Gain decision policy used by tree learning algorithms such as ID3, C4.5 and CART.

A decision node consists of a guess word and the leaf nodes correspond to each of the 243 possible outcomes that the answers can be placed into.

Under the information gain policy one must select the word which results in the greatest information gain, which is defined as:

$$IG(S,G) = H(S) - H(S|G)$$

where $H(S)$ is the entropy of set $S$, which is the set of all remaining answers and

$$H(S | G) = \sum_{i=1}^{243} \frac{|S_i|}{|S|} * H( S_i )$$

is the conditional entropy due to the creation 243 splits, by playing word $G$, with each split containing a set of answers called $S_i$.

Expanding

$$H( S_i ) = - \sum_{j=1}^{|S_i|} P(j) \log(P(j))$$

where $P(j)$ is the probability of answer $j$ in $S_i$ and is equal to $\frac{1}{|S_i|}$ since we treat answers as categories and the answers are unique.

We can simplify as follows:

$$H( S_i ) = - \sum_{j=1}^{|S_i|} \frac{1}{|S_i|} \log(\frac{1}{|S_i|})$$

$$H( S_i ) = - \frac{1}{|S_i|} \sum_{j=1}^{|S_i|} \log(\frac{1}{|S_i|})$$

$$H( S_i ) = - \frac{1}{|S_i|} * |S_i| * \log(\frac{1}{|S_i|})$$

$$H( S_i ) = - \log(\frac{1}{|S_i|})$$

Therefore $$H(S | G) = - \sum_{i=1}^{243} \frac{|S_i|}{|S|} - \log(\frac{1}{|S_i|})$$

The equivalence can be seen through expanding the logs of both approaches and dropping constant terms. First the criteria from the wordle policy .

$$H(X|G) = -\sum_{i=1}^{243} P(X=x_i | G) \log(P(X=x_i | G))$$

$$H(X|G) = -\sum_{i=1}^{243} \frac{|S_i|}{|S|} \log(\frac{|S_i|}{|S|})$$

$$H(X|G) = -\sum_{i=1}^{243} \frac{|S_i|}{|S|} \left(\log(|S_i|) - \log(|S|) \right)$$

$$H(X|G) = -\sum_{i=1}^{243} \frac{|S_i|}{|S|} \log(|S_i|)$$

then the decision tree case

$$H(S | G) = - \sum_{i=1}^{243} \frac{|S_i|}{|S|} - \log(\frac{1}{|S_i|})$$

$$H(S | G) = - \sum_{i=1}^{243} \frac{|S_i|}{|S|} - \left(\log(1) - \log(|S_i|) \right)$$

$$H(S | G) = - \sum_{i=1}^{243} \frac{|S_i|}{|S|} \log(|S_i|)$$