KGS math club/solution 11 17

Five, or if we count rotations and reflections separately, 310.

Iceweasel gave this list

  [[0,1,3,8,12,18],[0,1,3,10,14,26],[0,1,4,6,13,21],[0,1,4,10,12,17],[0,1,8,11,13,17]] 

and the Haskell code that generated it (and its reflections):

 import Data.List 
 import Control.Monad 
 npts = 31 
 n' = npts-1 
 dists l = group . sort $ [min (y-x) (npts+x-y)| (x:ys) <- tails l, y <- ys] 
 sols = [pts| c<-[3..n'],d<-[c+2..n'],e<-[d+2..n'],f<-[e+2..n'],c-1<31-f,let pts=[0,1,c,d,e,f],length (dists pts)==15]