Newsgroups: geometry.research
From: haoyuep@aol.com (Dan Hoey)
Date: Mon, 23 Feb 2004 20:11:19 +0000 (UTC)
Subject: Re: Brussels sprouts

Joe Wasserman <MymilRi...@ameritech.net> wrote:
> Brussels Sprouts is the same as the original Sprouts game, but
> the maximum degree of every node in Brussels Sprouts is 4, rather
> than the traditional 3 in Sprouts.

No.  That is not a correct description of Brussels Sprouts.  I
quote the summary of Brussels Sprouts from David Molnar's game
page:

  Several plus signs are drawn to start with, and a move consists
  of drawing an arc connecting two "branches" of plus signs, and
  then crossing that arc at some point with a small dash,
  effectively constructing a new plus sign, two of whose branches
  are already used.

This accords with my recollection of the game described in Martin
Gardner's column.

The key distinction between Brussels Sprouts and Degree-4 Sprouts
is that in Brussels sprouts, every spot has predetermined sites
(the arms of the plus signs) at which connections can be made.
This seemingly minor change makes the game completely different
 from the usual kind of Sprouts.

A true Degree-4 Sprouts game can never end, since each move
creates a new spot with two lives, and the next player can connect
that spot to itself.

We might imagine a game in which the initial spots are small T
shapes whose three arms are available for connections, and in
which a move consists of connecting two arms with an arc and
adding a new arm to the arc (on one side or the other).  I
imagine that game would have some strategy, though it wouldn't
have the stark elegance of regular Sprouts.

Another possibility would be for the first player to draw the
arcs and the second player to add the arms.  That would be a
(normal play) win for the second player with one spot; I suspect
the first player might win all the other sizes.

Dan Hoey
haoyuep@aol.com