  We report some novel properties of a square lattice filled with white sites,
randomly occupied by black sites (with probability $p$). We consider
connections up to second nearest neighbours, according to the following rule.
Edge-sharing sites, i.e. nearest neighbours of similar type are always
considered to belong to the same cluster. A pair of black corner-sharing sites,
i.e. second nearest neighbours may form a 'cross-connection' with a pair of
white corner-sharing sites. In this case assigning connected status to both
pairs simultaneously, makes the system quasi-three dimensional, with
intertwined black and white clusters. The two-dimensional character of the
system is preserved by considering the black diagonal pair to be connected with
a probability $q$, in which case the crossing white pair of sites are deemed
disjoint. If the black pair is disjoint, the white pair is considered
connected. In this scenario we investigate (i) the variation of the Euler
number $\chi(p) \ [=N_B(p)-N_W(p)]$ versus $p$ graph for varying $q$, (ii)
variation of the site percolation threshold with $q$ and (iii) size
distribution of the black clusters for varying $p$, when $q=0.5$. Here $N_B$ is
the number of black clusters and $N_W$ is the number of white clusters, at a
certain probability $p$. We also discuss the earlier proposed 'Island-Mainland'
transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90:
213) and show mathematically that the proposed transition is not, in fact, a
critical phase transition and does not survive finite size scaling. It is also
explained mathematically why clusters of size 1 are always the most numerous.
