Summary: The Mondrian problem consists of dissecting a square of side length \(n \in \mathbb{N}\) into non-congruent rectangles with natural length sides such that the difference \(d ( n )\) between the largest and the smallest areas of the rectangles partitioning the square is minimum. In this paper, we compute some bounds on \(d ( n )\) in terms of the number of rectangles of the square partition. These bounds provide us optimal partitions for some values of \(n \in \mathbb{N} \). We provide a sequence of square partitions such that \(d ( n ) / n^2\) tends to zero for \(n\) large enough. For the case of ‘perfect’ partitions, that is, with \(d ( n ) = 0\), we show that, for any fixed powers \(s_1 , \ldots , s_m\), a square with side length \(n = p_1^{s_1} \cdots p_m^{s_m} \), can have a perfect Mondrian partition only if \(p_1\) satisfies a given lower bound. Moreover, if \(n ( x )\) is the number of side lengths \(x\) (with \(n \leq x)\) of squares not having a perfect partition, we prove that its ‘density’ \( \frac{ n ( x )}{ x}\) is asymptotic to \(\frac{ ( \log ( \log ( x ) ) )^2}{ 2 \log x} \), which improves previous results.