S2T2

Jim Randell 8:38 am on 8 June 2023 Permalink Reply
Tags: by: Peter Morris ( 4 )   

• 

  Jim Randell 8:39 am on 8 June 2023 Permalink | Reply

  We can use the [[ SubstitutedExpression ]] solver from the enigma.py library to solve this puzzle.

  The following run file executes in 75ms. (Internal runtime of the generated program is 5.7ms).

  Run: [ @replit ]

  #! python3 -m enigma -rr
  
  #     C B A -> ABC is prime
  #  +  F E D -> DEF is prime
  #   -------
  #   J I H G -> GHIJ is square
  #   =======
  
  SubstitutedExpression
  
  "CBA + FED = JIHG"
  
  "is_prime(ABC)"
  "is_prime(DEF)"
  "A < D"
  "is_square(GHIJ)"
  
  --answer="(ABC, DEF)"
  

  Solution: The two primes are: 467 and 523.

  And the sum is:

  764 + 325 = 1089

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  Frits 1:19 pm on 8 June 2023 Permalink | Reply

     
  from enigma import is_square_p
  from itertools import combinations
  
  # prime numbers between 100 and 1000 with different digits
  # without using digit 1
  P = [3, 5, 7]
  P += [x for x in range(11, 33, 2) if all(x % p for p in P)]
  P = [s for x in range(203, 1000, 2) if all(x % p for p in P) and 
       '1' not in (s := str(x)) and len(set(s)) == 3]
  
  # pick two primes 
  for p1, p2 in combinations(P, 2):
    # different digits
    if len(s := set(p1 + p2)) != 6: continue
    
    # sum of reversed primes should be a 4-digit number using different digits 
    sm = int(p1[::-1]) + int(p2[::-1])
    if sm < 1000 or s.difference(str(sm)) != s: continue
    # and a square
    if not is_square_p(sm): continue
    
    print(f"the two primes are {p1} and {p2}")
  

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• 

  GeoffR 7:46 am on 9 June 2023 Permalink | Reply

  
  from enigma import is_prime
  from math import isqrt
  
  def is_sq(x):
     return isqrt(x) ** 2 == x
  
  digits = set('1234567890')
  
  from itertools import permutations
  for p1 in permutations(digits, 3):
      A, B, C = p1
      if '0' in (A, C):continue
      # 1st prime
      ABC = A + B + C
      if not is_prime(int(ABC)):continue
      q1 = digits.difference([A, B, C])
      
      # find 2nd prime
      for p2 in permutations(q1, 3):
          D, E, F = p2
          if D < A: continue  # 2nd prime > 1st prime
          if '0' in (D, F):continue
          DEF = D + E + F
          if not is_prime(int(DEF)):continue
          q2 = q1.difference(p2)
          for p3 in permutations(q2):
              G, H, I, J = p3
              if '0' in (G, J):continue
              GHIJ = G + H + I + J
              rGHIJ = GHIJ[::-1]
              
              # check sum of two reversed primes equals the reversed square
              if int(ABC[::-1]) + int(DEF[::-1]) == int(rGHIJ):
                  if is_sq(int(rGHIJ)):
                      print(f"Two primes are {ABC} and {DEF}.")
                                              
  # Two primes are 467 and 523.
  
  

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