No three counters in a line

Three points lie on one straight line when the slope between the first and second equals the slope between the second and third. Slope is rise over run: from $(x_1, y_1)$ to $(x_2, y_2)$ it is $\frac{y_2 - y_1}{x_2 - x_1}$.

To keep counters so that no three are collinear, each time you add one you check it against every line through a pair already placed. Vertical lines (same file) and horizontal lines (same rank) count too, along with every diagonal slope. The no-three-in-line problem asks how many fit; on an $n \times n$ board you can place at most $2n$.