doddle 1.0.0

doddle: Make Wordle A Doddle

Features

• Run the solver to see how the game is optimally played
• Solve a game in realtime using Doddle's solver
• Play a variation of the game where the solver attempts to hide the answer from you for as long as possible (inspired by Absurdle)
• Play using words of length 4-9 (inclusive) by adding the optional --size argument (default is 5).
• Choose your solver using the --solver=ENTROPY --solver=MINIMAX argument (default is minimax)
• Run deep searches using the depth argument (default is 1)
• Doddle can solve multiple games of Wordle at the same time. This mode is inspired by popular spin-offs such as Dordle, Quordle and Octordle. Playing multiple games with Doddle is easy: just add more solutions to the run doddle run --answer=ULTRA,QUICK,SOLVE and Doddle will solve them all at the same time.

Install

pip install doddle

Commands

Run

doddle run --answer=SALTY
doddle run --answer=SPEED --guess=SOLVE
doddle run --answer=FABULOUS --guess=SOLUTION --solver=ENTROPY
doddle run --answer=DEEP --guess=MIND --solver=ENTROPY --depth=2

Run a simulation with an answer of your choosing to see how Doddle solves the problem. You can optionally provide your own starting guess to see how the game plays out. With every guess, Doddle acquires more information and prunes the list of possible solutions. The output (far right column) shows you how many possible solutions still exist at each step of the solve.

Doddle provides multiple ways to solve the game. You can choose between a minimax solver (the default), or an entropy reduction approach. An entropy based approach optimises for best average play; a minimax approach tries to avoid worst-case scenarios and never goes above 5 moves.

Doddle supports deep searches for all of its solvers 🧠🧠🧠. These are slower, more exhaustive searches for those that want to go deeper. By default, Doddle plays the move that will yield the 'best' outcome on its next turn. In the case of minimax, that means playing the move that will result in the fewest number of possible words to search in the worst-case scenario (it choose a word that minimises the outcome that provides maximum uncertainty). In the case of --solver=ENTROPY, it plays the move that results in the greatest expected reduction in Shannon entropy. With deep searches, it thinks n steps ahead. So for --depth=2, the minimax solver plays the word that minimses the number of possible solutions in the worst case scenario on the turn after next taking into consideration all sensible first moves. The performance is still decent - a few seconds per game - but it is noticably slower as the search space explodes exponentially with depth.

Solve

doddle solve
doddle solve --size=7
doddle solve --guess=RAISE
doddle solve --guess=MIND --solver=ENTROPY
doddle solve --guess=MIND --solver=MINIMAX --depth=2

Work smarter not harder. Use Doddle's solver to solve Wordle as fast as possible. If you're playing Wordle and need some... ahem... divine inspiration, fire up Doddle's solver. Doddle will give you the optimal word to use. Type the response back into Wordle to generate the next guess. Doddle represents answers as ternary numbers! Doddle uses 2 for exact matches, 1 for partial matches and 0 for unmatched letters (e.g. 10202).

>>> doddle solve --guess=TORCH
Enter score for TORCH:
>>> 10000

The best guess is SNAIL
Enter score for SNAIL:
>>> 20101

The best guess is SALTY
Enter score for SALTY:
>>> 22222

Great success! ✨ 🔮 ✨

Half way through a game and just want to know the next best move? Doddle's got you covered. With solve, you're not forced to play the move Doddle suggests. Doddle accepts two syntaxes when entering a score: 1) just a score (as above e.g. 20101); or 2) a WORD=SCORE pair (SNAKE=20100). If you're half way through and need the best move, fill in what you've done so far using the second input method and let Doddle take over from there.

The best guess is SNAIL
Enter score for SNAIL:
>>> SNAKE=20100

The best guess is SALTY
...

Hide

doddle hide --guess=SALTY

Hide is a spin on the conventional Wordle game. Here, Doddle uses its solver to hide the final answer for as long as possible. Doddle doesn't choose an answer before the game starts - instead it always presents you with the score that results in maximum ambiguity. You'll get there in the end, but the game might take a while. 😈

Similar to the original Wordle game, a keyboard is rendered to display what characters have been guessed so far.

Algorithm

Doddle offers two choices of algorithms for solving Wordle: Minimax and Entropy.

Minimax

By default, Doddle uses a minimax algorithm to solve the game. The easiest way to explain the algorithm is through example.

Suppose you are half way through a game and have narrowed the solution down to one of four possibilties: SKILL, SPILL, SWILL, STILL.

Clearly, if we work our way through these words sequentially, the worst case scenario will be a further four guesses. To make things precise, let's create a histogram of all the scores that Wordle could return for each guess. We will consider the case where we naïvely choose the word SKILL.

Guess	Score	Partition Size	Possible Words
SKILL	🟩🟨🟩🟩🟩	3	{ SPILL, SWILL, STILL }
SKILL	🟩🟩🟩🟩🟩	1	{ SKILL }

The histogram is a great way to see how any guess partitions the remaining words. In the case above, there are two partitions with the worst case scenario being three (because three is the size of the largest partition).

Minimax works by considering all possible words in the dictionary and choosing the word that minimises the size of its largest partition. So, in searching through all possible words, minimax would stumble upon a word like 💥 KAPOW 💥.

Guess	Score	Partition Size	Possible Words
KAPOW	⬜⬜⬜⬜⬜	1	{ STILL }
KAPOW	🟨⬜⬜⬜⬜	1	{ SKILL }
KAPOW	⬜⬜🟨⬜⬜	1	{ SPILL }
KAPOW	⬜⬜⬜⬜🟨	1	{ SWILL }

In this case, each word is partitioned perfectly into its own bucket of length one and the game can be immediately solved on the next move. It's simple enough to compute this histogram for every possible word and the approach generalises all the way through the game.

Entropy

As an alternative to minimax, it is possible to play the game using an entropy based approach. Here, the solver always chooses the word that, on average, lowers the Shannon entropy of the game. To see how this works, let's assume we have reduced the game down to 20 possible words and decide to play the (excellent) move THURL. We shall construct a histogram as before - they're very useful.

Guess	Score	Partition Size	Probability	Possible Words
THURL	⬜⬜⬜⬜⬜	3	0.15	{ SNAKE, SPACE, SPADE }
THURL	⬜⬜⬜⬜🟨	1	0.05	{ SCALE }
THURL	⬜⬜⬜🟩⬜	3	0.15	{ SCARE, SNARE, SPARE }
THURL	⬜🟩⬜⬜⬜	5	0.25	{ SHADE, SHAKE, SHAME, SHAPE, SHAVE }
THURL	⬜🟩⬜⬜🟨	1	0.05	{ SHALE }
THURL	⬜🟩⬜🟩⬜	2	0.10	{ SHARE, SHARK }
THURL	🟨⬜⬜⬜⬜	3	0.15	{ SKATE, STAGE, STAGE }
THURL	🟨⬜⬜⬜🟨	2	0.10	{ SLATE, STALE }

Under minimax, we would simply look at the largest bucket and assign a score of 5 to the word THURL. However, with an entropy based approach, we take into consideration how much each guess cuts down the entire problem on average. To do that, we need to look at all possible outcomes in the histogram and calculate the expected value of the number of bits of entropy that each guess provides.

The probability of any outcome is calculated simply as the Partition Size / Total Number of Words. The number of bits associated with any outcome is calculates as -log(probability, base=2) and, hence, the expected number of bits is simply the sum of the bits multiplied by their respective probabilities.

In the example above, the expected number of Shannon bits is 2.83 which tells us that the guess THURL roughly cuts the problem size in half 2.83 times. To be explicit, cutting the problem in half once would leave 10 words left to search on average. Cutting the problem in half twice would leave 5. And cutting the problem in half 2.83 times would leave 2.82 words on average which looks eminently sensible when we look at the partition sizes remaining.

The guess with the highest information content, as measured in Shannon bits, is picked. In this case, THURL is pretty optimal.

Simultaneous Play

Documentation on playing mutliple games at once coming soon!

Coming soon

• An explain feature to show what the solver is thinking
• A facade layer - for those who want to call a simple API in their code
• Best pair mode - Find the best pair of moves for those who like to take it easy
• Rate my guess - see how your guess compares against the optimal move
• Proper setup of CI/CD and containerisation to check deployment on UNIX based systems
• More algorithms for solving (DNNs, RL, MIPs, decision trees!)
• Post match analysis
• Documentation
• More views! For snazzy reporting. e.g.:

Wordle 242 3/6

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