Show HN: Purple Word – A new word puzzle game

You’ve got ~170k possible words and a pure higher/lower signal, so with perfect play it’s just binary search. Number the words 1 to 170k, guess the middle, then keep halving the range. In binary terms, each guess is basically telling you whether the next bit is 0 or 1.

Since log₂(170,000) ≈ 17.4, you need at most 18 guesses. On average it’s a bit less: about half the words take 18 guesses, a quarter take 17, an eighth take 16, etc. So the expected number of guesses works out to 17.

All of that assumes you know the exact ordering of the dictionary, of course. Once you don’t, it stops being optimal and starts being human. That’s where the difficulty actually comes from.