Difference between revisions of "Y"

Revision as of 20:59, 20 June 2024

The game of Y is a connection game first invented in the early 1950s by John Milnor, and independently discovered in 1953 by Craige Schensted (later known as Ea Ea) and Charles Titus. In its original form, it is played on a triangular grid of hexagons, but Schensted and Titus preferred grids the include some pentagons, see Variations below. There are two players, who have one colour each, and a move consists of placing a piece of your colour in one of the hexagons on the board. The winner is the first player to complete a chain connecting all three sides of the board. Y is a kind of generalisation of Hex, perhaps the one closest to it, but there are some strategic peculiarities, such as corner templates.

No draws

Y cannot end in a draw. That is, once the board is complete there must be one and only one winner.

Less than two winners

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.

At least one winner

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.

Extension to Hex

The proof above extends to Hex because a Hex game can be seen as a subset of a Y game.

For instance consider the following Y board of size 7.

We can simulate a Hex game of size 4 on it.

Now the only way to win for Blue is to cross the board horizontally, whereas the only way for Red to do so is to cross the board vertically.

Each game of Hex on a board of size n can be played on a Y board of size 2n−1 with the rules of Y. The players just need to place some stones to "construct" the Hex board.

Y-Reduction

Given a Y board of size n filled with red or blue stones, there is an operation of replacing each of the upper triangle of size 2 with a hex at its center, and with a stone of color representing the majority of the three hexes being replaced. The result would be a Y board of size n-1. This operation is called the Y-reduction, introduced by Craige Schensted.

For example, the above Y board of size 5 can be reduced to size 4, and so on.

An important property of this operation is that, one color has a winning chain if and only if the color has a winning chain for its Y-reduction. As a consequence, one can repeatedly reduce the board until size 1 to determine the winner. This can also be seen as a proof of exactly one winner for Y.

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.

Swap

The Swap rule can be used in Y too, the corner are bad moves to be played so there may well exist average moves to begin with. Further information on where to swap.

Variations

The game is usually played with the swap rule. Alternatively, one can play Double-move Y, also known as Master Y: The first player places one piece on the board, and each subsequent move consists of placing two pieces on the board. This is a pretty challenging variant, even on small boards.

The inventors tried out a number of alternative playing grids, and eventually concluded that the most suitable one is the following "bent" version. The pieces are placed on the intersections (like in Go).

See also

You can find more boards here: Printable Y boards

Help for programming the bent Y board

Try this Y puzzle.

On the web

• http://www.gamepuzzles.com/gameofy.htm
• http://www.gamepuzzles.com/revugy.htm (Games magazine reviews)
• http://home.flash.net/~markthom/html/the_game_of_y.html (Dead link?)
• http://www.neutreeko.net/y.htm
• http://www.iggamecenter.com/ (igGameCenter - play "Y" online with other opponents from your iGoogle homepage)

References

• John F. Nash. Some games and machines for playing them. RAND Corporation Report D-1164, February 2, 1952. https://www.rand.org/pubs/documents/D1164.html Credits John Milnor as the inventor of the game.