We consider the minimum number of lines hn and pn needed to intersect or pierce, respectively, all the cells of the n×n chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that hn=⌈n2⌉ for each n≥1. Studying the piercing problem, we show that 0.7n≤pn≤n−1 for n≥3, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated.
