Monotonicity

In Hex, monotonicity is the property that an extra stone on the board can only help the stone's owner. More precisely, if there is a position that is winning for Red, and we change some number of cells from Blue to Empty and/or from Empty to Red, then the resulting position is still winning for Red. Of course, the analogous property holds from Blue's point of view as well.

This property is sometimes paraphrased as "an extra red stone can only help Red".

Proof of monotonicity

If Red has a winning strategy in the original position, then Red can win in the modified position by pretending that the modifications have not been made and following the same strategy. Should the opponent move in a cell that is empty but that Red pretended to contain a blue stone, then Red can simply ignore Blue's move and move in some arbitrary empty cell. Should Red' strategy call for Red to move in a cell that Red pretends to be empty but that actually contains a red stone, then Red can simply move anywhere. When the board is filled, the final position is what Red pretended it to be, only perhaps with some Blue stones changed to Red. Since Red has a winning path in Red's pretended position, Red has a winning path in the actual position as well.

Consequences of monotonicity

• Making an arbitrary move is at least as good as passing. This show that the game of Hex does not change if passing is allowed, since no player would ever have a reason to pass.

• Hex without swap is a first-player win. Because if the second player had a winning strategy, the first player could simply pass and then follow the second-player winning strategy.