Is there an upper limit to the potential geometric complexity of the Wordle's puzzle algorithm?

 Yes, there are practical upper limits to the potential geometric complexity of the Wordle puzzle algorithm, but these limits are constrained by several factors.

1. Word List Size

The complexity of a Wordle puzzle is fundamentally tied to the size of the word list. In Wordle, the puzzle involves selecting a hidden 5-letter word from a predetermined list of possible words (in the official version, this list typically consists of 2,300+ words). The geometric complexity of the puzzle relates to the number of possible combinations of guesses, feedback, and subsequent eliminations based on those guesses.

• Size of the word list directly limits the number of possible solutions and therefore the number of possible configurations of the puzzle state.
• If you were to increase the word list size (e.g., using a larger set of words or longer words), the number of possible configurations and guesses would increase, but the underlying algorithm remains finite because the puzzle is constrained by the word list size.

2. Guess and Feedback Space

After each guess in Wordle, feedback is provided in the form of colored tiles (yellow, green, or gray), indicating whether letters are correct and in the right position. This feedback reduces the space of possible solutions, but the feedback itself operates in a discrete space:

• Number of possible feedback configurations for each guess is limited by the number of positions (5 in Wordle) and the three possible feedback colors (green, yellow, or gray). This gives $3^5 = 243$ possible feedback patterns per guess.
• The number of feedback possibilities limits how "complex" the puzzle's progression can become. After each guess, the set of valid possibilities shrinks according to the feedback, and while the feedback space might seem large, it’s still bounded by the number of valid words in the list.

3. Decision Tree Depth

If we consider Wordle as a decision tree, where each guess splits the possible solution set based on the feedback, the tree depth will be constrained by how quickly the solution space can be narrowed down:

• Worst-case scenario: In theory, you could have a decision tree where each guess reduces the search space in an optimal way. The upper bound for the number of guesses needed to solve a puzzle is typically considered to be around 6 guesses in the optimal case (e.g., when using a strategy like minimax or other heuristic approaches).
• Tree branching factor: With a large word list, each guess might result in a different set of potential words, but there’s still a finite branching factor defined by the number of valid words and feedback options.

4. Combinatorial Explosion

The combinatorial complexity of solving Wordle increases with the size of the word list and the number of guesses available. However, even in the case of a very large word list, there are still practical limits on how complex the puzzle can get because each guess eliminates a subset of possibilities based on feedback. The depth and breadth of the decision tree are not arbitrarily large due to the information provided by the feedback mechanism, which reduces the complexity of the problem step by step.

5. Information-theoretic Bound

From an information-theoretic perspective, Wordle can be viewed as a game of narrowing down the space of possible answers based on feedback. The feedback from each guess is like a "bit" of information. The upper limit on the number of guesses required to solve the puzzle (in an optimal strategy) is determined by how many bits of information are needed to distinguish among all possible solutions.

• With a word list of size $N$, you need at least $\log_2(N)$ bits of information to uniquely identify the answer. Given that each guess provides feedback with up to $\log_2(243)$ bits of information (since there are 243 possible feedback patterns), the number of guesses required to guarantee a solution is bounded by the information entropy of the problem.

Conclusion

While there is no strict theoretical "upper limit" to the geometric complexity of the puzzle algorithm itself, practical limits arise from the size of the word list, the structure of feedback, and the combinatorial nature of the guessing process. These limits ensure that the game's complexity remains manageable despite the growing number of possible solutions or guesses. In the official Wordle game, these factors contribute to the puzzle's solvability within six guesses, which is a practical upper bound in terms of complexity, even if the underlying word list were expanded.