New results for the Mondrian art problem

The Mondrian problem consists of dissecting a square side length n∈N into non-congruent rectangles with natural sides such that the difference d(n) between largest and smallest areas partitioning is minimum. In this paper, we compute some bounds on in terms number partition. These provide us optimal partitions for values n∈N. We sequence d(n)∕n2 tends to zero n large enough. For case ‘perfect’ partitions, is, d(n)=0, show that, any fixed powers s1,…,sm, n=p1s1⋯pmsm, can have perfect partition only if p1 satisfies given lower bound. Moreover, n(x) lengths x (with n≤x) squares not having partition, prove its ‘density’ n(x)x asymptotic (log(log(x)))22logx, which improves previous results.