Assume we have a board game with rules similar to chess, where :
- A set of pieces $\mathcal P = \{\mathcal p_1,p_2,\cdots,p_m\}$,
- These can move "relative vectors" according to the set (of sets) $\mathcal V = \{\mathcal V_1,\mathcal V_2,\cdots,\mathcal V_m\}$.
- We have a number of each kind of piece $\{n_1,n_2,\cdots,n_m\}$
- And finally a board of size $N_1\times N_2\times\cdots\times N_l$.
Example a pawn in chess has $\mathcal V_{pawn} = \{(n,m+1),(n,m-1)\}$ given that it stands at $(n,m)$. $$\cases{N_1=N_2=8\\N_3=\cdots=N_l = 1}$$ etc.
How can we calculate a setting where a player controlling these pieces can attack all free spots?