Problem $1$

For which integers $n$ between $2$ and $21$ is every configuration on an $nn$ game board winnable? $($Hint: this is related to the dimension of the null space of $A_n).$

Problem $2$

Use Theorem $1$ to determine whether or not the configuration below is winnable.

Problem $3$

Suppose vec$(x)=(0,0,1,0,0,0,0,1,1,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,1{)}^T$ is the winning strategy for a configuration vec$(b).$ Use Theorem $2$ to write down all four winning strategies for vec$(b),$