Y (game)
From Wikipedia, the free encyclopedia
Y is an abstract strategy game which was first described by Claude Shannon in the early 1950s. Y was independently reinvented around the same time by Craige Schensted (now Ea Ea) and Charles Titus. It is a member of the connection game family inhabited by Hex, Havannah, TwixT, and others; it is also an early member in a long line of games that Ea Ea has developed, each game more complex but also more generalised.
Contents |
[edit] Gameplay
Y is typically played on a triangular board with hexagonal spaces; the "official" Y board has three points with five-connectivity instead of six-connectivity, but it is just as playable on a regular triangle. Schensted and Titus' book Mudcrack Y & Poly-Y has a large number of boards for play of Y, all hand-drawn; most of them seem irregular but turn out to be topologically identical to a regular Y board.
As in most games of this type, one player takes the part of Black and one takes the part of White; they place stones on the board one at a time, neither removing nor moving any previously-placed stones, and the pie rule can be used to mitigate any first-move advantage.
[edit] Rules
The rules are as follows:
- Players take turns placing one stone of their color on the board.
- The first player to connect all three sides of the board wins; the corners count as belonging to both sides of the board to which they are adjacent.
As in most connection games, the size of the board changes the nature of the game; small boards lend towards pure tactical play, whereas larger boards tend to make the game more strategic.
The portion of the board at the bottom-right can now be considered a 5x5 Hex board, and played identically. However, this sort of artificial construction on a Y board is extremely uncommon, and the games have different enough tactics (outside of constructed situations) to be considered separate, though related.
Mudcrack Y & Poly-Y also describes Poly-Y, the next game in the series of Y-related games; after that come Star and *Star.
The simple (regular) form of Y can be played by email, using Richard Rognlie's Play-By-eMail Server.
[edit] No draws
Y cannot end in a draw. That is, once the board is complete there must be one and only one winner.
There cannot be two winners at the same time. If there were, each player would have a region of the board touching all three sides of the triangle as well as the opponent's region. Considering the three sides as regions themselves, this gives a map of five regions, each of which is adjacent to the other four. However, this is impossible, as the graph K5 is non-planar.
It can be proved by an algorithm that once a board is complete there is at least one player linking the 3 sides. Let the "state" of a board refer to the answer to the question "Is there at least one winner?" We want to prove that the state of every board is "Yes".
First step: if there is a pawn group (red for instance) completely surrounded by the opponent (blue for instance) let's consider the board with this pawn group replaced by opponent's pawns (blue ones). The new board has the same status as the older one as the remote group was not winning and the new big (blue) one is winning iff it was in the former board. Also note that there is still at least one group left.
Repeat this step until there is no completely surrounded group more (of either color). The board obtained has the same state as the original.
Second step: if there is a pawn group surrounded by the opponent and a side, removing it does not change the state of the board (for similar reasons as in step 1) and there is still at least one group left.
Repeat this step.
Third step: if there is a pawn group surrounded by the opponent and two sides removing it does not change the state of the board (for similar reasons as in step 1) and there is still at least one group left.
Repeat this step.
It is quite clear that after applying this algorithm there is no group connected to more than 1 opponent's groups. No group is connected to zero sides and one opponent's group, no group is connected to one side and one opponent's group, no group is connected to two sides and one opponent's group. No group can be connected to 0 1 or 2 sides without connecting an opponent group. Moreover there is at least one group left. Hence this group left is connected to 3 sides.
So the state of the board is "yes"; as it is the same as the state of the beginning board, there was a winner to begin with.
Note that this algorithm ends because the number of different groups is finite.
[edit] The first player wins
In Y the strategy-stealing argument can be applied. It proves that the second player has no winning strategy. The argument is that if the second player had a winning strategy, then the first player could chose a random first move and then pretend that she is the second player and apply the strategy. An important point is that an extra pawn is not a disadvantage in Y. Y is a complete and perfect information game in which no draw can be conceived, so there is a winning strategy for one player. The second player has no winning strategy so the first player has one.
[edit] See also
[edit] References
- Browne, Cameron. Hex Strategy: Making the Right Connections. ISBN 1-56881-117-9
- "Game of Y." http://www.gamerz.net/~pbmserv
- Schensted, Craige and Titus, Charles. Mudcrack Y & Poly-Y.
