Sunday, May 15, 2022

Puzzle Difficulty in Plates Across America®

Puzzle Difficulty Comparison
If you have played Plates Across America® for even a little while, you will be well aware of the fact that not every puzzle is equal in difficulty.  

The puzzle "THS" is much easier to solve than "LKZ". If we showed only the hard puzzles, it would be a frustrating experience. Conversely, if we only selected the easiest puzzles, the game would not be very challenging and people would get easily bored.  

Therefore, the balance of the puzzle difficulty is an important part of the game play for us to get right. This article gives a few details about how the game determines which puzzles to show.

What Makes a Puzzle Difficult?

On seeing a puzzle for the first time, most people immediately are drawn to words with those letters near the beginning of the word. Secondarily, words where the puzzle letters are close together in the word tend to pop up in people's minds. Conversely, words where the letters are scattered throughout the word tend to be harder to find. For each of the game's puzzles we compute a "letter position difficulty" score which serves as one of the factors that determines a puzzle's complexity.

Among the early ideas we considered was that the more possible solutions the puzzle had, the easier it may be to solve. However, this is a flawed line of thinking because not all words are created equal. Even if the puzzle had only one possible solution, if that word was in everyone's common vocabulary, it might be relatively easy to solve. Conversely, a puzzle that has 100 solutions could be very hard to solve if all those words were relatively obscure.

We decided to use word frequency to gauge how common words are. This is based on looking at how often words appeared in large collections of text documents (e.g., how often the word appears across all the Wikipedia articles). We take these word frequencies across all possible solutions and derive a "frequency score" for the puzzle. This better represents how easy it will be for someone to come up with an answer rather than the shear number of possible solutions.  

We pre-compute the "letter position difficulty" and the frequency score for each puzzle and then blend those together to arrive at an overall difficulty score. It is this final difficulty score that determines how frequently the puzzle will appear in the game.

How Does the Game Pick the Puzzle?

With each possible of the possible 14,000+ puzzles, we assign a difficulty score to them as described above. We then group them into 6 difficulty level tiers from easiest to hardest. The puzzle selection is then done randomly, but with a two level selection scheme.

  • First, we randomly choose one of the 6 difficulty tiers. This is a weighted selection so that the easiest tier is chosen 50% of the time, the second easiest 35% of the time, then 9%, 3%, 2% and finally 1% for the most difficult puzzle tier.
  • Then, from within the tier, randomly choose one of its puzzles (uniformly selected).

In Closing

The puzzle selection scheme is an important part of the playability of the game, but it is fairly involved to determine the relative difficulties of all the game's word puzzles. We think we have struck a good balance in our game, but always looking to improve if you have suggestions for us.

If you have not already played, try our game and send us feedback about what you think:

Happy Travels!

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