Piercing the chessboard

TL;DR

This paper determines the minimum lines needed to intersect all cells of an n-by-n chessboard, strengthening the plank problem, and provides bounds for the minimum lines to pierce all cells, combining geometric and linear programming methods.

Contribution

It proves exact values for the minimum lines intersecting all chessboard cells and establishes bounds for piercing all cells, advancing understanding of geometric covering problems.

Findings

Exact value of h_n = ⌈n/2⌉ for all n ≥ 1

Bounds for p_n: 0.7n ≤ p_n ≤ n-1 for n ≥ 3

Limitations of linear programming in this context

Abstract

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ 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 $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, 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.