  The Mondrian problem consists of dissecting a square of side length $n\in
\NN$ 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 $n \in \NN$. 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 $n\le x$) 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.
