  For a given finite subset P of points of the lattice Z^2, a friendly path is
a monotone (uphill or downhill) lattice path which splits points in half;
points lying on the path itself are discarded. The purpose of this paper (and
its sequel) is to fully describe all configurations of n points in Z^2 which do
not admit a friendly path. We say that such an n-set is inseparable. There are,
up to the lattice symmetry, exactly c(n) such sets. If only lattice shifts are
counted, there are \^c(n) of them. Both sequences are new entries into OEIS
(A369382 and, respectively, A367783). In particular, n=27 is the first odd
numbers with c(n)=1. No example was known so far. This solves problem 11484(b)*
posed in American Mathematical Monthly (February 2010). In this paper we also
show that inseparable n-set exist for all even numbers n>=12 and almost all odd
numbers.
