  We report site percolation thresholds for square lattice with neighbor
interactions at various increasing ranges. Using Monte Carlo techniques we
found that nearest neighbors (N$^2$), next nearest neighbors (N$^3$), next next
nearest neighbors (N$^4$) and fifth nearest neighbors (N$^6$) yield the same
$p_c=0.592...$. At odds, fourth nearest neighbors (N$^5$) give $p_c=0.298...$.
These results are given an explanation in terms of symmetry arguments. We then
consider combinations of various ranges of interactions with (N$^2$+N$^3$),
(N$^2$+N$^4$), (N$^2$+N$^3$+N$^4$) and (N$^2$+N$^5$). The calculated associated
thresholds are respectively $p_c=0.407..., 0.337..., 0.288..., 0.234...$. The
existing Galam--Mauger universal formula for percolation thresholds does not
reproduce the data showing dimension and coordination number are not sufficient
to build a universal law which extends to complex lattices.
