Projective Hex

[Bill Taylor: 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.

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.

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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 Each corner still has 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.