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For a given finite subset P of points of the lattice Z^2, a friendly path isa monotone (uphill or downhill) lattice path which splits points in half;points lying on the path itself are discarded. The purpose of this paper (andits sequel) is to fully describe all configurations of n points in Z^2 which donot 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 arecounted, there are \^c(n) of them. Both sequences are new entries into OEIS(A369382 and, respectively, A367783). In particular, n=27 is the first oddnumbers 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 alsoshow that inseparable n-set exist for all even numbers n>=12 and almost all oddnumbers.

Friendly paths for finite subsets of plane integer lattice. I