Question 20
A continent has a finite number of castles and each castle belongs to exactly one of the two countries $A$ and $B$. Each castle has one general. We say two castles are *neighboring* if there is a path between those two castles. We also say that a castle $P$ and a set of castles $Q$ is *neighboring* if $P$ is neighboring some castle in $Q$. Prove that the following two conditions are equivalent.
1\. If some of the generals from the country $B$ move to some neighboring castles and attack, generals from the country $A$ can move to some neighboring castles so that at each castle from the country $A$, the number of generals from $A$ is greater than or equal to that of generals from $B$.
2\. If $X$ is a set of castles from the country $A$, the number of generals from $A$ who are in $X$ or in castles neighboring $X$ is greater than or equal to the number of generals from $B$ who are in castles neighboring $X$.
1\. If some of the generals from the country $B$ move to some neighboring castles and attack, generals from the country $A$ can move to some neighboring castles so that at each castle from the country $A$, the number of generals from $A$ is greater than or equal to that of generals from $B$.
2\. If $X$ is a set of castles from the country $A$, the number of generals from $A$ who are in $X$ or in castles neighboring $X$ is greater than or equal to the number of generals from $B$ who are in castles neighboring $X$.