Given an infinite chessboard and $6$ points on it, you can rotate or translate the six points at the same time. How many points can you guarantee to move into (include boundaries) a black cell?
I observe that any point on the boundary of cells are in a black cell because the two colors interphase. Therefore I can give a solution for $5$ points.
By rotating and translating, we could easily put two points on the boundaries ($A$ and $B$). Then we translate along $\overrightarrow{AB}$ so a third point $C$ falls on a boundary line perpendicular to $AB$. As shown below.
Now for the rest $3$ points, by the pigeonhole principle, exist $\left\lfloor\dfrac{3-1}2\right\rfloor+1=2$ in cells of the same color. If it is black, then we've got $3+2=5$, finished. If it is white, transform one unit along the grid will turn them black, finished.
So my problem is, is $6$ possible? If not, we need to find six points as a counter example.