Antipod is a connection game with unequal goals played on the two halves of a sphere.

Rules

Antipod is played on two equally sized hex hex boards, typically with six cells per side. The board is set up with a black piece in the center of each half, forming Black’s antipodal goals, and white pieces placed at the six corners of each board, forming White’s handicap pieces (Figure A).

Figure A. The Antipod board and its initial setup.

Two players, Black and White, take turns placing a piece of their color on an empty cell. If any piece is placed on an edge cell of one board, then a duplicate piece of the same color is automatically placed at the same cell on the other board (which must be empty). Such duplicate pairs actually represent the same piece, offering continuous connections between the two boards.

Figure B. A game won by Black.

Black wins by forming a chain of black pieces between the antipodal goals of each board half. For instance, Figure A shows a win by Black who has completed a chain between the antipodal goal pieces a and e., via edge crossings at b, c and d.

White wins by preventing a Black win. This amounts to completing a cycle of white pieces around at least one of Black’s antipodal goals. Exactly one player must win.

Notes

Antipod is closely related to Cylindrical Hex. Looking at Figure C, we see that the Cylindrical Hex board (left) can be turned into a punctured sphere (right) by pinching the cylinder at each end. In both cases Black aims to establish a chain between top and bottom, while White aims to establish a cycle around the circumference.

Figure C. A cylindrical board pinched to form a spherical board.

The two Antipod boards each correspond to a hemisphere of this pinched sphere, and Black’s two antipodal goal points correspond to the two poles of the pinched sphere. To show this explicitly, we can reformat the Antipod board as if it were projected onto two hemispheres (Figure D).

Figure D. Antipod is played on two hemispheres.

It now becomes clear why edge pieces must be duplicated. They actually represent a single piece played on the equator but viewed from both the front hemisphere (left) and the rear hemisphere (right).

A completely hexagonal tiling of the sphere is not possible, and six of the equatorial cells are of necessity four-sided rather than six-sided. The six White handicap pieces are placed at these four-sided equatorial cells for a number of reasons:

This last point is the most important. Since the goals of Antipod are equivalent to those of Cylindrical Hex, then without the handicap pieces the game is solved by the same symmetry strategy that solves Cylindrical Hex.

Figure E. Adjacency graph of a small Antipod board.

Figure E shows how the Antipod board gives a planar adjacency graph. Note that pieces on the right hemisphere must be reflected about its vertical axis to achieve the correct adjacencies. This is equivalent to viewing the right hemisphere from behind (that is, from the inside of the sphere looking out) rather than viewing it from in front.

Figure F. A completed game and its game graph.

For instance, Figure F shows a completed game won by White. There can be no doubt that White has won since the game graph has been cut into two disconnected components with an antipodal goal in each; Black cannot possibly connect these now.

Antipod allows some interesting strategic and tactical play due to the spherical nature of the board. For instance, Figure G shows how Black can exploit two longitudinal attacks in tandem to achieve a connection. Black move a is a forking move that wins the game; White can block one longitudinal attack with either move b, but cannot stop both at once.

Figure G. Black wins with forking move a.

Smaller board sizes favor Black, while larger board sizes favor White.

Antipod was invented by Cameron Browne in 2003.

Antipod can be played on Richard's PBeM server - check out the help file for more details.
Please challenge me (camb) to a game any time.