OPM 2019

On a board with $3$ columns and $4$ rows, each of the $12$ squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?

A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be dispersedi f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is dispersed, but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be correctable. Indicates all integers $n$ for which there is a network with $n$ stations, dispersedand not correctable.