Conference rusure::math

    Suppose the first player can only choose pieces containing 0 in the
    first position and the second player can only choose those containing 1
    in the first position, and the first position cannot be used to create
    a winning combination.  Is the game more or less interesting?
    
    (I've got the game, but won't open it until a games party in two weeks. 
    The above are the instructions the clerk gave; the base note contains
    the instructions on the back of the box.)
    
    
    				-- edp
    
    
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>   On each turn, a player selects an unused piece
>   and the other player puts it on the board.

    How may the piece be placed on the board?  Always horizontally?
    May it cover or partially cover a previously placed piece?

Quarto, yes:  a fun game.

	It's one of these abstract games, IMHO, which aren't much fun with paper
and pencil, and so you really have to buy the game.   What you get is a 4 by 4 
board and 16 pieces.   Each piece is different;  each is:

		white or black
		tall or short
		square or round
		solid or holed

	The players place the pieces on the board one at a time.   In the 
introductory game, you win if you make a straight line where all four pieces 
share a feature in common.   When you start to play, this game seems incredibly
boggling, but quickly one's perception adjusts, and one becomes ready for the
standard game.   In the standard game, you can win EITHER as before (by making 
a straight line where all four pieces share a feature) OR by making a square of
four adjacent pieces which share a feature.

	The key rule in Quarto is that you place on the board the piece
that your opponent selected for you.

	I've played a few times with a big set where the pieces were made out of
bed-posts.   Very nice and chunky:  the tall pieces were about a foot high.   At
the time, I wrote down a few thoughts, which I put down here.   One thing:  it's
certainly a game which you learn by playing rather than by analyzing in the
abstract.   If you've played a few times, you have a major advantage over
someone who comes to it afresh, although the learning curve is fast.

Cheers,
Andrew.

--------------------------------------------------------------------------------

SITE	A row, column, diagonal or square of four spaces on the board.

There are 19 sites.

A site is FULL if all spaces are occupied.   If the four pieces share a 
feature in common (ie the game is over) the full site is a WINNING SITE.   A
site in which the (2,3 or 4) pieces share no feature is a DUD SITE.

PRYLE	Three pieces placed on the board such that:
		(i) they occupy 3 spaces of the same site
		(ii) they have at least one feature in common
(terminology taken from the card game Brag.)

A pryle is SINGLE (DOUBLE) if the tiles have exactly one (two) features in
common.   We can DEFUSE a pryle by placing a piece in the empty space to make
it dud.

TABOO.   A piece which, if nominated by a player, would allow his opponent to
extend a pryle to a winning site.

A player loses when all the unplayed pieces are taboo (unless there are no
unplayed pieces.)

Normally, a player received a non-taboo piece from his opponent.   He can try
to play it any of the existing pryles, to defuse it.   If there are no other
non-taboo pieces, then he will *have* to defuse one of the existing pryles.
This may remove the taboo status from certain other pieces, but it may create
other pryles.

CENTRE	The site in the middle of the board.   Each space partipates in 7 sites,
while the peripheral spaces each participate in 4.   The centre site ought to
be important strategically, therefore.   Exactly how remains unclear.

THE BASIC PRINCIPLE   It's hard to figure out what's going on.   The first few
moves seems truly arbitrary.   At the end of the game, it's possible to analyze
exhaustively, and one discovers that randomly, one has lost or not.   It's
often too late by then to change it.   The key part over which the players have
some control is therefore the middle game.

One *does* have control over the complexity of the game.   If there are
lots of pryles around, and one thinks the opponent is more observant, it may be
a good idea to make some of the primes dud.

Middle Game Thinking process
============================

(1) What are the pryles that are around now?   
(2) Can I win immediately through my opponents oversight?
(3) What are the moves which don't result in me losing immediately by making all
remaining pieces taboo?
(4) Out of those moves, what aggressive ones are there which complicate the
position, and constrain the opponent (particularly those preventing him from
defusing pryles).   Look at the possible consequences of these low-branching
options to see if I can force a win.   What piece can I nominate to constrain
the opponent further, if necessary?
(5) If I can't find a winning move, decide whether to complicate or to simplify
the position, depending on estimate of relative expertise.   Find a suitable
move, and check its safety.   Nominate a piece to constrain the opponent, or to
render some non-winning strategy a potential winner for next turn.

    Re .2, .3:
    
    A piece is just placed on an empty square of the board.  They are
    selected from the set without replacement.
    
    
    				-- edp

    re .*
    
    Now that I think I understand the rules...
    
    What can we apply from game theory (of which my recollection is too
    hazy) to decide whether a winning strategy exists?
    
    Is a draw possible? (Exhibit configuration)
    
    Is a draw possible if both players play optimally?
    
    Can this game be exhaustively analysed? (Seems more likely so than
    chess but probably not practically anyway). Unfortunately I have more
    free CPU cycles than free brain cycles to write the program.

Re: -.1

>    Is a draw possible? (Exhibit configuration)

	Yes.   Imagine making eight dominoes by pairing each piece with its
"antipodal" piece.   Then tile the board with the eight dominoes.   A winning
line (or square) cannot contain a domino.   So for instance, if we label the
dominoes: Aa, Bb, Cc, Dd, ..., Hh, we could arrange them:

	AaCD
	Bbcd
	hgeE	
	HGfF

	This excludes all winning lines except the diagonals, and all
winning squares except the central one.   Nearly there.   Now for each
piece, define its chromatic antipode to be the piece which differs in
every characteristic except the colour.

	Let c & D be white chromatic antipodes.   Similarly, let b & e
be white chromatic antipodes.   Let A & g be black.   Then no winning line
exists.

> Is a draw possible with best play by each player?

	I don't know.   There is no simple symmetry-playing strategy that I
know of.   I would be surprised if one exists.   I haven't played the game
enough to get a feel for whether the game is a draw.

Cheers,
Andrew.

    Re .8:
    
    Is that for the simple game (rows, columns, and diagonals) or the
    advanced game (two-by-two squares as well)?
    
    
    				-- edp
    
    
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Let's denote the pieces by ????, where the ? are replaced by letters drawn
from:
	Black	White
	whOle	Punctured
	sQuare  Round
	Short   Tall
	
In this notation, here's a position:


	....	....	WPRS	....
	....	WORS	....	....		To play: WORT
	....	....	BPQT	BWRS
	....	WOQS	....	....

(a) Who wins?
(b) In the above position, replace WOQS by WPQS.   Who wins then?

Andrew.