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Span |
Span is a two-player connection game for the Shibumi set. |
Rules Players: White and Black. Start: The board is initially empty. White owns the top and bottom sides of the board. Black owns the left and right sides of the board. Play: Players take turns adding a piece of their colour to an empty playable point (board point or 2x2 platform), until all 30 pieces have been placed to form a complete pyramid. End: The winner is the player with a visibly connected path of their pieces spanning their sides of the board. |
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Notes For example, the game on the right has been won by White, who has completed a visible path of White pieces that spans the top and bottom sides of the board (marked '+'). Players can use the extra ball of their colour provided with the Shibumi set as markers, to indicate their target direction. The five interior (non-visible) points constitute cold moves that neither player wants to make, as pieces placed there do not contribute to any connection and allow the opponent to stack on top. This helps balance out any first move advantage; White has the advantage of first move, but can be forced by Black to make the first interior move, with typical play. "Typical" play involves occupying the outer points first before someone is forced to play in an interior cell; this will occur on the 13th move, which is White's turn. Black also has the advantage of making the last (30th) move at the pyramid's apex, which is the last say in any connection if the game has not already been decided before then.
The final piece can be the deciding move. So it is difficult to say which player, if any, has the advantage. |
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Topology
Span is Hex on the Shibumi board. Like Hex, exactly one player must win! This may seem odd given that connection games played on the square grid are prone to deadlock problems. However, stacking changes this, and the connectivity of the visible pieces in a complete pyramid is trivalent in nature, which avoids deadlocks.
The following figures show another interesting aspect of Span topology. Consider the following 5x5 game played to completion (left), and the graph formed by the connectivity of touching visible balls (right).
The dual of this graph (left) is topologically equivalent to the ConHex board (right)! ConHex is one of the best connection games, and its author Michail Antonow points out that his board design had no mathematical basis - it was simply the design that produced the best game out of many that he tried. This coincidental link between Span and ConHex is another case in which a mathematical construct that works well in one context also works well in another.
History
Span by Cameron Browne (c) 2011, designed for the Shibumi set.
The "SP" prefix denotes that Span is a Square Pyramidal 4x4 or SP4 game.
The winning condition is similar to Sphex and identical to Sponnect, but Span is simpler than these games.