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Consider the following Scrabble board:

enter image description here

We can play NO for 1 point, AI for 2 points, NOSE for 3 points etc as indicated on the right. All this obviously depends on having the correct letters on your rack (not shown). Eventually we will reach a number N such that it is impossible to score exactly N points on our next turn.

Let us say the "smallest unattainable score" of a game-state is the smallest positive number N such that we cannot score exactly N points on our next turn. The game-state includes both our rack and the board (we ignore the opponent's rack for simplicity). Can you construct a game-state with the largest smallest unattainable score?

Assume that the CSW19 dictionary is being used.

EDIT: The game state must be reachable by valid plays (no phonies)

For partial credit: to simplify the problem assume that the board state contains only one word.

SOURCE: The Scrabble board was cut-n-pasted from Wikipedia.

Thanks to FlanMan for a recent Scrabble-related question

happystar
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    Does the game state have to have a valid history? e.g. if this was a game state: https://imgur.com/FTB2L6b

    If "SOFT" was the last word played, "NCE" would've previously on the board, which is not a valid word. etc. So there is no valid 'previous history'

     – thesilican Aug 02 '20 at 18:39
  • How is "NO" worth just 1 point? 1 for the N and 1 for the O, making 2, right? Or are we using a blank for the O? – JLee Jun 01 '22 at 14:40
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    @JLee the N inside a square is a blank. I've seen this convention used when annotating actual games. – happystar Jul 25 '22 at 04:09
  • I got curious about the algorithmic complexity of this problem: There are 4,114,349 scrabble racks according to this video. There are probably more than 50,000 words with 7 letters or fewer. Meaning even the simplified version has 200 billion things to check (naively). Oh there’s also the 2 to 7 different positions of the first word. Still, it’s an interesting problem. Some pruning methods: assume we have two blanks; assume the opponent played a 7-letter word; etc – Benjamin Wang Aug 20 '23 at 21:05

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