The board gaming site has a recent question about Scrabble. What is the highest Scrabble score possible?

The question specifies that there is only 1 player and he/she is using the International Scrabble Dictionary. I think it is probably impossible to prove that any particular game has the highest possible score with today's computers.

The question made me wonder how many possible games there are. I'm not a mathematician. My estimates are as follows.

  • The highest ranked answer to How Many Starting Racks are there in Scrabble says 3,199,724. After each play, some of these letters will be replaced with newly drawn letters. The number of combinations for the drawn letters after each play will be less than 3,199,724. Assume about 20 draws to finish the game (5 letters/play). If there were on average only 1000 possible combinations per draw, then there would be $1000^{20}=10^{60}$ combinations of racks during the game.
  • Assume there are an average of 100 possible plays on the board for each of the 20 plays. That means that during the game there are $100^{20}=10^{40}$ plays for each combination of racks.
  • Using these estimates, there are $10^{60} \cdot{} 10^{40} = 10^{100}$ possible games.

I'm guessing that this is a low estimate because it assumes that 5 tiles are played each turn. I am aware that some of my estimates are probably very inaccurate.

Can you give me a better estimate?



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