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Blocking A Wormhole



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Blocking A Wormhole

Wormhole is played in a 2x2x2x2 hypercube. The object is to connect five of your pieces in an orthogonal 4D path. When Zillions plays itself on the default settings, Black wins in five moves - as soon as possible - about half the time, and White forces a draw the other half. (It may vary, of course, depending on your computer.) As the thinking time is increased, draws become more frequent. By studying the draws, you can divine two reflection strategies for White that force a draw.

The first strategy is to always play to the north or south of the Black piece just played. A winning path must connect all four dimensions and consequently must have one north-south connection. If White always plays to the north or south, Black will never be able to make the north-south connection. This strategy works along any axis.

The second strategy is to always play to the opposite corner. Each space in the Wormhole board has a unique opposite space that is four orthogonal moves away. The opposite corner for the bottom-left-most square, for example, is the upper-right-most square. To move from one to the other, you must make a move east, north, up, and 4-east, in any order. It is a property of wormholes that they must start and end in opposite corners. As a result, if White systematically plays to the opposite corner, Black will be thwarted from completing a winning path.

This strategy results in an interesting optical effect, similar to a camera obscura. If Black creates a 3D path of four pieces, White will create a reverse image of the path in the remaining dimension, as illustrated:

The two ends of the Black wormhole cannot connect into the fourth dimension because both 4-east orthogonal spaces are occupied by White. The "front" of the wormhole (pick an end) is blocked by the "back" of the White reverse wormhole, and the "back" of the wormhole is blocked by the "front" of the reverse wormhole. No matter what Black does, its move into the fourth dimension will always be blocked by a reverse 3D image, resulting in a draw.

2000 W. D. Troyka

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