How do you play the Mabry Dart & Kite game, a two-player game of 3-coloring a Penrose tiling of kites and...

1. Penrose Rhombus:

It's the rhombus with angles 72 and 108 degrees.  Dividing the long
diagonal in golden ratio (phi) and joining the point to the obtuse
corners, we get two shapes: the (so called by John H. Conway) KITE
[that one with angles 72,72,72,2*72] and the DART.  This division was
discovered by Roger Penrose, and is very interesting since Kites and
Darts tile the plane non-periodically (only).

Martin Gardner, in his book _Penrose Tiles [Editor: "Tile" in the
original] to Trapdoor Ciphers..._ (New York : W.H. Freeman, 1989), has two long chapters on Penrose Tiling
(pp. 1-18 and 19-29).

2. Mabry Dart & Kite Game:

The game was invented by Dr. Rick Mabry and is unpublished.  He says he
devised it as a way to think about 3-coloring the kites and darts and
simultaneously not totally ignore his children.  Rick kindly sent it to
me using "web mail", a good idea of his (writing it in a web page,
where he was able to insert pictures).  The page is located at:

      http://www.lsus.edu/sc/math/rmabry/antreas/antreas2.htm

Rick's home page is at:

      http://www.lsus.edu/sc/math/rmabry/rmabry.htm

Interested readers may try it using appropriate programs.  ("Paint"
for Win95, for example).  Unfortunately, I myself have no any such
program, so I haven't tried it yet.

Rick welcomes any comments etc. at his e-mail address:
rmabry@pilot.lsus.edu

Following is the text Rick has in his page, for those who have no web
access.

------ http://www.lsus.edu/sc/math/rmabry/antreas/antreas2.htm ------

Hi Antreas,

Here is a portion of the K&D that the kids and I start with.  We play
only within the boundary, but one could also expand the boundary
dynamically as needed.  (As I mentioned, we play on the computer by
clicking colors with a mouse, so that is easy.)

Player 1 goes first (now THAT is original) by filling any available
tile with one of 3 agreed upon colors.  I'll use red, yellow and green.
Player 1 goes first below with a red kite.  Subsequent legal moves are
to color only tiles that are edge-adjacent to those already colored,
and never to color two adjacent tiles the same color.  So the second
move must be yellow or green on any of the 4 tiles adjacent to the
first red kite.

I'll let Player 2 take a green kite.  Player 1 then has many choices,
but chooses a yellow dart.  So here are the first three moves:

Not very interesting so far.  Notice though, that there is a kite
adjacent only to the green and yellow tiles that can only be colored
red.  Maybe Player 2 will stumble near this trap --- you win if you
create an uncolored tile that is surrounded by 3 different colors,
thereby destroying the 3-coloring of the tiling.  (Remember, JHC
conjectures that all of these can be 3-colored.)

Player 2 fills the "ace" --- a novice!  But that is not a bad move.
Player 1 tries to sucker his opponent and colors a green kite:

But Player 2 is not such a novice after all and traps Player 1 by
coloring an adjacent red kite and claiming a dart (which is then
colored blue --- it easier for us to color our captures than to
initial them):

At this point we could either say that Player 2 has won the game and
begin again (which I like --- it is more "pure" a game), or continue,
leaving the capture tiles on the board.  When doing the latter, we have
decided that after capturing a tile, the victor may move again, which
occasionally results in a few consecutive captures.  Also, we don't
allow the captured tile to be considered "colored", so a play adjacent
to a captured tile must also be adjacent to a colored tile.  The former
game is interesting despite appearing to be short.  When I play with
Katy (15), the games are sometimes long --- a few dozen or so moves.
(She wins about 40%, I'd say!)  With 10-year-old Patrick, we usually
play the 2nd method.  I am not particularly good at this sort of thing,
but I still win about 90% with Patrick!  I have yet to play with a
mathematician!

So does it make sense?  This is the first time I have tried to explain
it, and I haven't tried to give an indication of how subtle and pretty
a game I have found it to be.  But as I said, I have yet to really test
it.

Let me know what you think.  Want to play a quick transcontinental
game?

Rick

------ http://www.lsus.edu/sc/math/rmabry/antreas/antreas2.htm ------

[ Editor: the pictures of the moves, referred to by the colons above,
  were on the web page and are not in the posting. ]

Thanks to Rick Mabry for his interesting game!

-- Antreas P. Hatzipolakis <xpolakis@hol.gr>, January 1998