Covering rectangles by few monotonous polyominoes

Abstract

A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $$f:[a,b] \to \mathbb{R}$$ with $$f(k) \notin \mathbb{Z}$$ whenever $$k \in \mathbb{Z}$$ . Our main result says that the least cardinality of a covering of a lattice $$(m \times n)$$ -rectangle by monotonous polyominoes is $$\big\lceil \frac{2}{3}(m+n-\sqrt{m^2+n^2-mn})\big\rceil.$$ The paper is motivated by a problem on arrangements of straight lines on chessboards.