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 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]