Richard Guy writes many cheerful facts about the square of n by n dots: << Given a square n by n array of dots, what is the maximum number of edges in a convex polygon whose vertices are members of the array?

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As usual, I wonder about the same question applied, instead of to the n by n array of dots, to the n by n "discrete square torus" array of dots. I.e., if all addition is mod n, then each point (K,L) of Z/nZ x Z/nZ is connectable to (K+d, L+e) where (d,e) lies in the set {-1,0,1} x {-1,0,1} - {0,0}. Another related question is to use the "discrete rhombic torus", also describable as the points Z/nZ x Z/nZ, but with the point (K,L) connectable to (K+d,L+e) only for (d,e) in {-1,0,1}x{-1,0,1} - {(-1,-1),(0,0),(1,1)} (again, addition is mod n). --Dan