Yes I too do not understand the incredible amount of faff about not making a move. It is a completely worthless endeavour. The question is how many games can be played. Not how many moves can we stall for no reason. It is like chess with infinite passes. Sure it's inifintie, but it doesn't answer the question. The solution can be estimated using montecarlo modelling and other approximation methods. I don't see where Analytical method (nitpickers method) has any place in approximation whatsoever. I wonder how many of those arguing how many nonmoves should be allowed would even bother to think what they argue is futile considering the approximations of 10^60, 10^40 and 10^21 Here's a simple case-in-point for the "non-move" geniuses and how it extends to N^(no one cares) plays Using approximation "no play" counts as a move As I used the general form of average number of plays per move=200, "no play"=1 move. So they can just use N=200+1. That is the significance of such a discussion, or, for the nitpickers delight, it means that always incorporating a blank move increases the number of total moves by a factor of (1.005)^30 (16%). Pretty insignificant when you are looking at a range of values between 10^21 and 10^60 Now If the nitpickers would prefer to spend their energy doing something useful, they can design a program that calls from quackle the total number of plays possible per move for a typical 11,12,13,14,15 or whichever length game they desire, sample size=10^6 for each, and they will get a clearer answer. Hell you could probably get a clear picture with 10^4 or even 10^3. If they want to put rack leave complexity in, they can simply factor that as a subset of rackleave, even though such an endeavour is equally pointless (rackleave determinism has properties that are too complex to bother with, for the purposes of simplicity I assume that it's nett value when combined with rackleave variability is 1 (could be more, could be less, I have no clue, nor any desire to investigate such mathematical anomalies as it simply doesn't answer the question)- it's a PhD for anyone who can find a regular expression for finite determinism in a dynamic environment. Too many words about nothing. :p I look forward to some results.
--- On Sun, 3/8/09, [email protected] <[email protected]> wrote: From: [email protected] <[email protected]> Subject: Re: [quackle] Re: Scrabble Complexity To: [email protected] Date: Sunday, March 8, 2009, 9:22 AM 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 >>
