Projective Hex.
Bill Taylor
for the possible interest in tilings of Projective Planes.
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One of the great blessings of connection games like Hex and Bridgit is,
that victory is certain for one or other side, AND the structure of the
game ensures that a victory for one is *automatically* a defeat for the
other, with no special rule needed to say so. So there is no element of
a mere "race" to do something first, where both players might achieve
this goal almost simultaneously.
Although there is no "social" defect in such races, (e.g. even chess can
be so viewed - a race to capture the opponent's king before he captures
yours), it is mathematically and game-theoretically slightly unaesthetic,
compared to the Hexlike feature of [win = not(loss)] by structure.
Hex and Bridgit both suffer from another slight unaestheticity though,
to wit, that the two players have (slightly) different tasks; one must
make a North/South connection, and the other an East/West one. Indeed,
in Bridgit they even play on different points! Again, this is no barrier
to playing the game or to its being a jolly good game, but again it seems
a very slight aesthetic defect.
One game that achieves both goals, i.e. (1) complementary winning conditions
and (2) identical tasks; is the excellent "Y" version of Hex, which really
deserves to be better known. However, I introduce yet a new variant here.
------
Some while ago, Dan Hoey and myself jointly invented a game we called
PROJECTIVE HEX, invented in this newsgroup, in fact.
It was Dan who, partly inspired by "Y", first ventured onto Projective Planar
boards for Hex-like games, but couldn't find a nice winning condition,
surprisingly. My contribution was to observe that the condition of making
a GLOBAL LOOP, (i.e. a closed path that crossed the boundary an odd number
of times) was "THE ONE" - and that it stood out "like a sore thumb".
Dan agreed about the sore thumb, and kicked himself for not having seen
it before. Dan also constructed a program to print out beautiful Hex-like
boards based on the Projective Plane, and thus having 6 pentagons amongst
a variable number of hexagons.
My latest contribution has been to change the pattern of the boards
slightly, to make them more homogeneous-looking (though not fully
homogeneous in fact), and thereby arrange it so that games can
easily be played at the keyboard, i.e. by email etc.
For the new Projective Hex, now probably the best abstract board game
in the world (ha-ha!), the boards are similar to this as follows...
A B C As you see I've had to insert a 27th alphabet letter!
D E F G Interior cells and interior-edge cells each have 6
H I J K L neighbours, as in Hex; but the 6 corner cells have 5.
M N O # P Q
R S T U V The side dimensions are always n and n+1. Each edge
W X Y Z is flipped end-to-end and laid alongside its opposite.
In this 3-&-4-sided board, there are 15 edge cells which thus connect
to their opposite cells via Projective connections as shown here...
z_y_x_w
z/A B C\w Each of the original edge/corner cells "re-appears"
v/D E F G\r on the opposite side, in lower case letters.
q/H I J K L\m
q<M N O # P Q>m Note once again that each corner has only 5 neighbors.
l\R S T U V/h
g\W X Y Z/d
c~c~b~a~a
So on the original board, cell H is connected to D I N M Q V (in order).
Whereas M is connected only to N R L Q H.
The whole collection of 21 hexagons and 6 pentagons makes up a "standard"
tiling of the Projective Plane.
To play the game, "Projective Hex", one merely plays as at Hex, filling
any one cell your own colour on your turn; and whoever makes a global
loop of adjacent cells of their own colour, is the winner. And, as
mentioned above, it is only possible for ONE colour to do so, and
at least one of them must always do so, by the time all cells have
been coloured. So complementary winning conditions, and equal tasking
have both been achieved.
Example: here is a completed game, with both having played 7 moves,
and the 2nd player (white) has won, despite his opening disadvantage.
. X O
. . X O
. . O O .
. X O X . .
X O X . .
X O . .
The loop might be more visible if "ghost" edge cells are entered as well...
___o_x
/. X O\x
/. . X O\x
/. . O O .\
<. X O X . .>
\X O X . ./
o\X O . ./
o~o~x~~~
For actually playing the game, naturally, as always, the first player
has an enormous advantage; a sure win, in fact, by the usual strategy
stealing argument. But beyond the very smallest boards it is very
hard to find.
This advantage can be left as is, giving the weaker player first move;
or (say for more formal games), one of the usual equalizing methods
can be used. Probably the simplest is the "cut-and-choose" method
of 3-move equalization (mentioned on another thread recently), whereby
one player plays 3 opening moves, black-white-black, then the other
player chooses which colour to be. It is also conceivable that even
2-move or 1-move equalization would be suitable, as e.g. the corner
cells are not quite so valuable as the central ones, so an opening
move there might well be a losing one, but only just, making 1-move
equalization a viable option.
--------
In any event, there is the basic game. I have various other variants
to disclose, but they can wait for another time. I have played several
games against myself, usually on tiny boards, though once on a big one,
and all the usual Hex style tactics seem to apply.
Give it a go!
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Bill Taylor W.Ta...@math.canterbury.ac.nz
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Every game is unique, and this one is no different to any other.
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Wei-Hwa Huang
>Hex and Bridgit both suffer from another slight unaestheticity though,
>to wit, that the two players have (slightly) different tasks; one must
>make a North/South connection, and the other an East/West one. Indeed,
>in Bridgit they even play on different points! Again, this is no barrier
>to playing the game or to its being a jolly good game, but again it seems
>a very slight aesthetic defect.
A slight nitpick: Bridgit is isomorphic to a "Hex"-like game being
played on a grid of squares. The "different points" are actually
a deliberate aesthetic assymetry.
--
Wei-Hwa Huang, whu...@ugcs.net, http://www.ugcs.net/~whuang/
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I see dead links. They don't know they're dead. I see them all the time.
James Hunter
Bill Taylor wrote:
> This article is chiefly for rec.games.abstract; but I cross-post to sci.math
> for the possible interest in tilings of Projective Planes.
> =========================================================
>
> One of the great blessings of connection games like Hex and Bridgit is,
> that victory is certain for one or other side, AND the structure of the
> game ensures that a victory for one is *automatically* a defeat for the
> other, with no special rule needed to say so. So there is no element of
> a mere "race" to do something first, where both players might achieve
> this goal almost simultaneously.
But that's OK since it was *mathematicians* who put the "race"
in such a fine game in the first place.
It obviously is a childish butchering of a fine old game,
requiring REAL intelligence, known as:
"Storm the castle, and behead the King".
Bill Taylor
based on a board in the form of a projective plane, which consisted of cells
locally in hexagon tiling pattern, but with (necessarily) a few exceptions.
The boards I suggested were (with variable side length) of the following type:
|> For the new Projective Hex, now probably the best abstract board game
|> in the world (ha-ha!), the boards are similar to this as follows...
|>
|> A B C As you see I've had to insert a 27th alphabet letter!
|> D E F G Interior cells and interior-edge cells each have 6
|> H I J K L neighbours, as in Hex; but the 6 corner cells have 5.
|> M N O # P Q
|> R S T U V The side dimensions are always n and n+1. Each edge
|> W X Y Z is flipped end-to-end and laid alongside its opposite.
|>
|> In this 3-&-4-sided board, there are 15 edge cells which thus connect
|> to their opposite cells via Projective connections as shown here...
|>
|> z_y_x_w
|> z/A B C\w Each of the original edge/corner cells "re-appears"
|> v/D E F G\r on the opposite side, in lower case letters.
|> q/H I J K L\m
|> q<M N O # P Q>m Note once again that each corner has only 5 neighbors.
|> l\R S T U V/h
|> g\W X Y Z/d
|> c~c~b~a~a
|>
|> So on the original board, cell H is connected to D I N M Q V (in order).
|> Whereas M is connected only to N R L Q H.
|>
|> The whole collection of 21 hexagons and 6 pentagons makes up a "standard"
|> tiling of the Projective Plane.
So those boards have 6 cells with only 5 neighbours.
It later occurred to me that there is another, possibly simpler, way to
almost-hexagonally tile the projective plane, which would also do well.
The following type of tiling has 3 cells with just 4 neighbours each.
a i h g I haven't bothered to letter in the 19 central cells,
b . . . f though there are almost enough letters to do so!
c . . . . e
d . . . . . d For this board, we do not need to imagine the pairs
e . . . . c of opposite sides dovetailed as before, because they
f . . . b are in fact *identified*, as indicated by the letters.
g h i a So this board has a mere 19 + 9 = 28 cells.
As always for the projective plane, opposite sides connect in reversed
directions, as shown above. All cells now have 6 neighbours, except for
cells a d g which have 4 each.
e.g. cell a has neighbours b i and 2 internals; and
e.g. cell i has neighbours a h and 4 internals, 2 "above" and two "below".
So in playing the game, when you fill an edge/corner cell you must remember
also to mark-as-filled the opposite one with the same letter (all as one move).
As before, the basic connection game to be played on such boards is like
the games "Hex" or "Y", the goal being to make a global loop of adjacent
cells of your own colour. For example here...
x x . .
x o o . . X has played 10 moves and O has played 9,
. x x . x x and X has just won with the loop-path from
. o o x . . . the top-left corner to the bottom-right,
x o x o o . thus completing the loop.
. o x o x
. . x x
As always on the P-P the formation of a loop of one colour automatically
prevents any future loop of the other colour, so there is no "mere race"
element; it IS a race, but one whose completion cripples the opponent
as well! Thus as explained earlier, the game is both equi-tasked and
mono-completable; both IMHO extremely desriable elements in an abstract game.
Of course the first player has a huge (indeed automatically winning)
advantage, and this can be dealt with in various ways, which is not
the main theme of this post, however.
It might even be possible to devise a commercial presentation that could
be played by ordinary folk with no idea of what a P-P was. One would
have a plastic sphere, with holes appropriately arranged as above,
and a move consisting of sticking one of your coloured skewers through
so that it poked out two opposite holes. Then to win you must make
a loop that includes at least two opposite points in it! The difficulty,
of course, is to find skewers that do this without all mangling up in
the centre of the ball! Though I can't think of an answer to this,
I nevertheless claim, copyright and patent all commercial rights to
this game; though naturally anyone else is at liberty to play or make it
at any time providing they don't make any money for anyone in doing so!
Cheers,
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Bill Taylor W.Ta...@math.canterbury.ac.nz
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Exam: The Projective Plane.
Candidates must write on one side of the paper only.
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