#!/usr/bin/env python3
from functools import lru_cache
from math import isqrt
N=112
@lru_cache(None)
def possible(T):
 B=set(T)
 S={x+y for x in B for y in B}
 if 113 in S:return False
 D=S-{x*y for x in B for y in B}
 if not D:return True
 def pairs(s):
  return [(u,s//u) for u in range(1,isqrt(s)+1)
          if s%u==0 and s//u<=N and 113-u not in B
          and 113-s//u not in B and u+s//u!=113]
 s=min(D,key=lambda s:len(pairs(s)))
 return any(possible(tuple(sorted(B|{u,v}))) for u,v in pairs(s))
print(possible((1,2,3,5,7)))
print(possible.cache_info())
A=set(range(1,13))|{16,113}|set(range(13,110,2))
S={x+y for x in A for y in A};P={x*y for x in A for y in A}
print(len(A),max(A),sorted(S-P))
print(S==set(range(2,223,2))|{226}|set(range(3,126,2))|{129})