Game of Y

The game of Y is a two-player boardgame with no chance and a relative to the more popular Hex. Y is played on a triangular board formed of hexagonal cells that are empty in the beginning. Two players alternately fill empty cells with pieces of their own colour. The player who creates an unbroken chain of pieces that connect all three edges, wins. Note that corners belong to both edges. The standard board in Figure 2 is slightly bent with three of the hexagons replaced by pentagons.

Figure 1: A small filled Y board is shown to be won by black by micro reductions.

The fact that Y cannot end in a draw on a straight board can be proven using so called micro reductions described in Rijswijck02hex. Size $n$ board is reduced to a size $n-1$ board where each cell on the reduced board gets the majority colour of the three nearest cells on the original board. An example is given in Figure 1. It turns out that if a chain touches a side of the board, it does so also on the reduced board. A winning chain will thus be a winning chain on all the smaller boards including the trivial board of size $1$. Figure 2 shows how the same analysis can be done for the bent board by first transforming it into a straight one.

Figure 2: Top: The game of Y is usually played on the bent board, on points rather than cells. Bottom: The bent board can be transformed into a straight one for analysis such as micro reduction. Filling a point in the bent board fills all the cells with the same letter in the straight board. Only one corner is shown.