ANTIPOD

Copyright (c) 2003 Cameron Browne

Antipod is played on two 6x6 hexagonal boards (modeling a sphere) with the following setup:

 

EQUATOR - The 30 cells around the edge of each board are identical, just appearing twice. So, there are only 6 white stones onboard, not 12. The corners only have 4 neighbors rather than 6. TURN - On each turn, each player drops a stone on an empty cell. Black starts. GOAL  Black wins if he connects the south and north poles of the sphere. White wins if he prevents Black's goal.

 

An opening example Black dropped a stone at the equator (which appears at both boards). White replied by starting to block one of the paths to the south pole.

Some comments from the author: Antipod models the connection of antipodal points upon a sphere, where the left and right boards represent the two halves of the sphere. Duplicated edge pieces are really just two views of the same piece from different sides of the sphere, and are not necessarily any stronger than non-duplicate moves. This is not a perfect hexagonal mapping on the sphere; the six equatorial corner points only have four unique neighbours each. Smaller board sizes favour Black while larger board sizes favour White. We are still experimenting to determine the optimal board size.

Antipod was invented following discussions with Bill Taylor regarding projective Hex games (games of connection which continue across board edges). Antipod's method of edge piece duplication has the advantage of explicitly showing edge crossings on both boards, making the game more comprehensible to the player than some others of this class. It's possible to play Antipod at Rognlies' PbM server.

Here is another Cameron's note explaining why the six handicap stones are not only a nice balancing mechanism, they are in fact essential for the game:

Alpern & Beck (1991) "Hex Games and Twist Maps on the Annulus" describe a winning strategy for the player joining top and bottom in cylindrical Hex. This strategy involves Vert taking the point on the same row but 180 degrees around from the point just played by Horz. This is only proven for even boards, but is probably true for odd boards as well. Since Antipod is equivalent to cylindrical Hex with the top and bottom edges contracted to single points, I was wondering whether this winning strategy (tactic?) could also be applied to Antipod. The equivalent strategy on the Antipod board is for X to play at the same latitude as O's last move but 180 degrees around from it. For instance:

      - L M -          o . . o
     K F G H N        . . . . .
    J E B C I O      . . . . . .
   - D A - a d -    o . . x . . o
    o i c b e j      . . . . . .
     n h g f k        . . . . .
      - m l -          o . . o

If O plays at cell 'a' then X plays at cell 'A' (and vice versa), if O plays at cell 'b' then X plays at cell 'B' (and vice versa) etc. The same goes for the right hemisphere. However it turns out that O's six equatorial handicap stones save the day (phew!) The game below shows how O can defeat X's cylindrical point-pairing strategy to win. In fact, I believe that only two equatorial O stones placed antipodally are required to defeat the cylindrical point-pairing strategy.

      o . . o          o . . o
     . . o . .        . . x . .
    . x . . o .      . o . . x .
   o . . x . . o    o . . x . . o
    . x . . o .      . o . . x .
     . . x . .        . . o . .
      o . . o          o . . o

Without the equatorial handicap stones then the game appears to be a trivial win for X. Unfortunately, this point pairing strategy defeats Escape Hex (annular Hex with the interior edge contracted to a point). For O's handicap stones to be effective in Escape Hex they must form an unbeatable connection from the start of the game, which defeats the purpose.