You crazy lunkheads. You make it much too difficult with the infinite passing, etc.
Try figuring out the total number of opening moves if no phoneys and you have to touch the center star and all passes and exchanges are considered equal and count as one. Geesh, just because it is hard... Zax PS I won the Westinghouse Prize. For having the cleanest fridge. --- On Sat, 3/7/09, Steven Gordon <[email protected]> wrote: From: Steven Gordon <[email protected]> Subject: Re: [quackle] Re: Scrabble Complexity To: [email protected] Date: Saturday, March 7, 2009, 8:16 PM If the number of total moves was bounded (as are the number of racks and legal moves at each point in the game), the number of distinct sequences of moves must be bounded, making the total number of distinct games bounded. You can enumerate them all, but you will run out of numbers before you run out of legal move sequences as long as the N-pass rule applies for some finite N. Steven Gordon On Sat, Mar 7, 2009 at 3:10 PM, G. Vincent Castellano <g...@ocsystems. com> wrote: > Ok, point, I was being too clever. Can I save face by claiming that the > number > of games is countably infinite as long as you require that all the games be > of > finite length? > > I concede that if you allow for games which never terminate, then my method > will > not enumerate them. > > I should probably stop now before I annoy the Real Mathematicians among us. > --gvc > > Eugene Deon wrote: >> *That's like saying you can enumrate the real numbers by starting with > >> the first decimal digit and building a tree with 10 children per node >> off into infinity. * >> >> *I'm quite sure I can find a unique game of scrabble for every real >> number. And the decision tree is trivial. Both players exchange 7 >> every turn. One decision. It's what's on the rack that makes it >> different.* >> >> >> Even allowing a game with an unbounded number of consecutive exchanges, >> it's >> still countably infinite, because you can enumerate not just all the >> games, but >> all the infinite moves in these possibly-infinitely -long games by a >> breadth-first scan of the decision tree starting with turn one. >> >> Now, if you want to figure in the length of time taken for each move, >> you're >> getting somewhere, maybe, until you run into the quantization of time (the >> nature of which has not yet been plausibly postulated to date). >> --gvc >>
