Let P={(a,b)/a,b∈{1,2,...,n},n∈N} be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length 1 or 2, so that both ends of the segment are in P and do not intersect each other. Furthermore, for each point (a,b) it is true that
i) if a+b is a multiple of 3, then it is an endpoint of exactly 3 segments.
ii) if a+b is an even not multiple of 3, then it is an endpoint of exactly 2 segments.
iii) if a+b is an odd not multiple of 3, then it is endpoint of exactly 1 segment.
a) Check that with n=6 it is possible to satisfy all the conditions.
b) Show that with n=2015 it is not possible to satisfy all the conditions. combinatoricslattice points