The number of valid games of Scrabble is uncountably infinite. For any real number, I can find a unique, valid scrabble game by mapping 7 digit chunks of it's decimal representation to the alternating racks of both players as they both change 7 an infinite number of times. I won't waste people's time with the details. Someone asked for the complexity of scrabble. It's at least infinite (not just "approaching infinity"), and I would argue that it's uncountably infinite, since an infinite number of change 7s (while unplayable in practice) is not invalidated by any rulebook statement that I have read.
If you wish to argue the complexity of "significant" games of scrabble, make that a rigorous statement first. "Brightness" is also a poorly defined statement, being used to mean both luminance and radiance (incorrectly). Luminance is the closest well-defined statement, and has nothing to do with the distance to the observer. ________________________________ From: Russell Honeybun <[email protected]> To: [email protected] Sent: Thursday, March 5, 2009 8:30:14 AM Subject: Re: [quackle] Re: Scrabble Complexity Roger, a typical game does not have 490 moves. such a game would have a probability of 10^(very large negative number) of ever being played, thus it's significance is ~0. This is a dampening factor that N(number of moves) tends to infinity, the probability of playing such a game reduces dramatically. Your brief assay would be like summing the cumulative brightness of all stars in the universe as if they were the distance of our own sun away. I don't have my more detailed calcs at hand, but it's fairly simple to work out the approximation - but you have to cheat by assuming the bag is endless for X moves, for the purposes of simplicity. 1. Assume all words of eight letters or more are playable on each turn(114575) . 2. Assume the average moves per person per game is X=15. 3. N^X=114575^30= ~10^151 The true answer is significantly less than (3). 200 words or less are playable per turn (play a few games of quackle and ask it to generate all moves to calculate this for yourself) 200^30=10^69 I think in my original calcs i looked at my own stats of 14 plays per game average 200^28=2*10^ 64 and you can see how the original figure is tending to the answer Graham correctly gave you. This in itself is a rough overestimation, as on each turn available there are less than 50 (and in a lot of cases less than ten) "sane" options. unfortunately, there is no mathematical function to describe "saneness" when talking about near-infinite branching integrals. If we implement "saneness" (<50 moves/turn) the number of total games drops to around 10^40, if we implement "expert saneness" (average 5 moves playable) you are looking at a figure around 10^21. Still, this is a number so large we could play a game every hour for the age of the universe and have a never repeat. I think that when considering games of go and chess, the saneness and expert-saneness protocols reduce the number of games playable far more than scrabble is restricted. As it stands, a definite approximation is impossible to achieve, but we are absolutely sure it is more than 10^21 and less than 10^151 Regards.
