mathematics - Strategy to beat the Casino

Two players, $A$ and $B$, and the Casino play a game. It involves showing zeros or ones: each player picks $0/1$ and also the Casino picks $0/1$.

• If the Casino and both players $A$ and $B$ show the same number ($000$ or $111$), then the two players win the game.


• But in case not all three chosen numbers are identical, the Casino wins.

Altogether there are nine rounds. Now it happens that player $A$ is going to learn some illegal information, just seconds before the game starts: $A$ is going to learn the Casino's choice for each of the coming nine rounds. Unfortunately, there is no way of communicating this information to player $B$ without the Casino noticing. The only way of communicating is via the cards chosen by $A$.

The evening before this game, $A$ and $B$ meet and agree on a common strategy. Is there a strategy that guarantees them to win at least $5$ of the $9$ rounds? And is there a strategy that guarantees at least $6$ wins?