Wouldn't the six pass rule fix that? Progress toward the end of the
game must be made every sixth turn so the duration of the game is
finite. Well of doesn't apply to the first turn, but there's still a
finite number of exchanges that can be made before re director ends
the game for going overtime :)
Chris Lipe
Sent from my iPhone
On Mar 6, 2009, at 15:21, Eugene Deon <[email protected]> wrote:
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.
