Y is an
abstract strategy
Abstract may refer to:
*"Abstract", a 2017 episode of the animated television series ''Adventure Time''
* ''Abstract'' (album), 1962 album by Joe Harriott
* Abstract algebra, sets with specific operations acting on their elements
* Abstract of ti ...
board game
A board game is a type of tabletop game that involves small objects () that are placed and moved in particular ways on a specially designed patterned game board, potentially including other components, e.g. dice. The earliest known uses of the ...
, first described by
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
in the early 1950s. The game was independently invented in 1953 by
Craige Schensted and Charles Titus. It is a member of the
connection game
A connection game is a type of abstract strategy game in which players attempt to complete a specific type of connection with their pieces. This could involve forming a path between two or more endpoints, completing a closed loop, or connecting all ...
family inhabited by
Hex,
Havannah,
TwixT
TwixT is a two-player Abstract strategy game, strategy board game, an early entrant in the 1960s 3M bookshelf game series. It became one of the most popular and enduring games in the series. It is a connection game where players alternate tu ...
, and others; it is also an early member in a long line of games Schensted has developed, each game more complex but also more generalized.
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. The
pie rule
The pie rule, sometimes referred to as the swap rule, is a rule used to balance abstract strategy games where a first-move advantage has been demonstrated. After the first move is made in a game that uses the pie rule, the second player must sel ...
can be used to mitigate any first-move advantage.
Rules
The rules are as follows:
* Players take turns placing one stone of their color on the board.
* Once a player connects all three sides of the board, the game ends and that player 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 tend towards pure
tactical play, whereas larger boards tend to make the game more
strategic
Strategy (from Greek στρατηγία ''stratēgia'', "troop leadership; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the "art o ...
.
Relation to other connection games
Schensted and Titus argue that Y is a superior game to Hex because Hex can be seen as a subset of Y. Consider a board subdivided by a line of white and black pieces into three sections. The portion of the board at the bottom-right can then be considered a 5×5 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
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
and
*Star.
Criticism
Y, like Hex, yields a strong first-player advantage. The standard approach to solving this difficulty is the "pie" rule: one player chooses where the first move will go and the other player then chooses who will be the first player.
Y's chief criticism is that on the standard hexagonal board a player controlling center can easily reach any edge no matter what the other player does. This is because the distance from the center to an edge is only approximately 1/3 the distance along the edge from corner to corner. As a result, defending an edge against a center attack is very difficult.
Schensted and Titus attacked this problem with successive versions of the game board, culminating in the present "official" board with three pentagons inserted among the hexagons. They noted that were players to play on a hemisphere rather than a plane with hexagons, with the equator divided into three "sides" (each 1/3 the circumference of the hemisphere), the distance from the "north pole" of the hemisphere to the equator was 1/4 the circumference, and thus the distance ratio improved from 1/3 to 3/4. This made defending a side from a center attack much more plausible. Thus the present "official" board is essentially a geodesic dome hemisphere squashed flat into a triangle to provide this effect.
No draws
It has been formally shown that Y cannot end in a draw.
Y Can't End in a Draw
/ref> That is, once the board is complete there must be one and only one winner.
The first player wins
In Y the strategy-stealing argument
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game ...
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 choose a random first move and then pretend that she is the second player and apply the strategy. An important point is that an extra stone on the board is never 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. It is nevertheless possible for the first player to lose by making a sufficiently bad move, since although that stone has value, it may have significantly less value than the second move—an important consideration for understanding the nature of the pie rule.
If the "pie rule" is in force, however, the second player wins, because the second player can in principle evaluate whether or not the first move is a winning move and choose to invoke the pie rule if it is (thereby effectively becoming the first player).
In practice, assuming the pie rule is in force and the official Schensted/Titus board is being used, Y is a very well balanced game giving essentially equal chances for any two players of equal strength. The balance is achieved because the first player will intentionally make a move that is sufficiently "bad" that it is not clear to the second player whether it is a winning move or a losing move. It is up to the judgement of the second player to make this difficult determination and invoke the pie rule accordingly.
See also
* Hex
*Connection game
A connection game is a type of abstract strategy game in which players attempt to complete a specific type of connection with their pieces. This could involve forming a path between two or more endpoints, completing a closed loop, or connecting all ...
s
References
Bibliography
* Browne, Cameron. ''Hex Strategy: Making the Right Connections''.
* Schensted, Craige and Titus, Charles. ''Mudcrack Y & Poly-Y''.
External links
Y on HexWiki
*{{bgg, 5242, The Game of Y
Board games introduced in 1953
Abstract strategy games
Connection games