## Obstruction

28th October 2012

The latest game to be added to Pencil and Paper Games is Obstruction, a territory game invented by László Kozma, a Romanian mathematician.

The game is played on a grid of squares. One player is 'O' and the other is 'X'.

The players take turns in writing their symbol in a square. The restriction is that you can only play in a square if all its neighbours are empty (shown shaded in the following diagrams). The first player unable to move loses.

On boards of 6 x 5, 6 x 6, 7 x 6, 8 x 7, or 8 x 8 the game is sufficiently complex to make it a challenging game.

### Winning strategies

Although the beginning of the game is difficult to analyse, as the game proceeds the board soon gets divided up into islands of unblocked squares, and then some tactics become possible.

#### Pairing strategy

If you can play to create one or more pairs of identical islands you are guaranteed a win. For example, after O's first move X can play as follows:

The two islands are the same shape, so after any move that O makes, X can make an equivalent move in the other island, maintaining the symmetry. X will have the last move, winning the game.

Here's another example. After two moves each the position is the one on the left:

Now O can play in the bottom row, leaving two identical islands as shown on the right, ensuring a win.

#### Neutral islands

Some island shapes are neutral - they don't affect the outcome of the game; here are three simple examples (obviously their orientation doesn't matter):

When analysing a position you can ignore any neutral islands, because wherever your opponent makes a move in a neutral island, there will be exactly one move left in the island that you can take in return.

Here's an example using a neutral island. After two moves by each player the position is the one on the left:

The next player, O, can ignore the neutral island on the bottom row. So the winning move is to play in the top left corner, leaving a pair of identical islands as shown on the right.

Happy playing!