From: kwhyte@arthur.uchicago.edu (Kevin Whyte)
Subject: Re: defects in abstract games
Organization: Dept. of Mathematics
Date: Wed, 6 Jan 1993 17:54:32 GMT


In article <1993Jan6.085328.28673@risky.ecs.umass.edu> trodrigu@spock.ecs.umass.edu (Tao R or Stephen L) writes:
>In my (not very sophisticated) understanding of Chess, the weaker player
>can try to reach a draw by forcing the stronger player into a situation
>where any departure from a repetitive sequence is a disaster.  If the
>repetition were simply ruled to be illegal, the idea of what constitutes
>a strong position would change.  But the rule that no captures for some
>number of turns is a draw could be unchanged.  It seems to me that this
>is fairly simple, and removes many draws.  Is it a bad idea for some
>other reason, am I missing something, or do some people consider the
>draws desirable?

  As someone else pointed out, the no repetition move in chess would make
endgames very strange.  This is not to say it wouldn't be interesting, but
if you think the endgame has a different feel now, this would far more so.

>The latest candidates for 'flawless' games seem to be Hex, Traxx and
>Twixt.  These are all games involving a sequence of placed pieces, with
>no moves or captures.  If I'm remembering 'Winning Ways' correctly,
>these games can all be reduced to Nim or Sprouts, and can be solved
>without exploring the entire game tree.  I speculate that any game which
>does not have the potential of infinite duration (thus requiring the no
>repeats rule or some equivalent) is in the general class of games that
>can be reduced to Nim.  Does anyone know of any proofs or disproofs of
>this speculation?
> 
>- Stephen Linhart
>

  This is definitely not true.  The games which can be reduced to Nim
(Sprouts can, and hasn't been solved ... another good game, but the
three lives per vertex is too arbitrary to call it perfect.) are exactly
those where 1) The game always terminates, even it the players don't
               alternate moves.
            2) From every position both players have exactly the same
               legal moves, and exactly the same winning podsitions,
               etc.  This is not true for games like Hex, Twixt, Traxx,
               or Go, since which color you are makes a difference.

  There are many games which are of this form, and still unsolved. 
The isomorphism with Nim requires knowing the entire game tree (more
or less), so it's not that helpful.  An example of this is Sylver
Coinage:  The players alternate naming positive intergers, and are
not allowed to name anything which is a positive integer linear
combination of previously named numbers.  You lose if you name 1 (or,
equivalently naming 1 is illegal, and you lose if you can't move).

Kevin

 (In fact, properly interpreted, just about any game can be put in
this form, but not usefully.  Chess, for example, where the board 
position includes whose move it is,  and a legal move is to make
a legal move for that side, and change whose move it is.  If there's
only one game going, this is chess.  It does not behave like normal
chess under "addition", and since whose move it is is global, it never
breaks up into pieces, hence Winning Ways type stuff is useless.)
We are faced with the very real possibility that "digital" and 
"manual" will be used as antonyms.
Newsgroups: rec.games.abstract