Sunday Times Teaser 2732 – Prime Meat

by Graham Smithers

Published: 1 February 2015 (link)

Mark has recently converted from vegetarianism. John sent him a coded message consisting of a list of prime numbers. Mark found that by systematically replacing each digit by a letter the list became the message

EAT BEEF AT TIMES

IT IS A PRIME MEAT

What number became PRIME?

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Brian Gladman permalink


from itertools import permutations
from number_theory import Primes

words = ('EAT', 'BEEF', 'AT', 'TIMES', 'IT', 'IS', 'A', 'PRIME', 'MEAT')

# since primes above 10 must end in 1, 3, 7 or 9, we know
# that E, F, S and T must be a permutation of these digits
# so A can only be 2 or 5
for A in ('2', '5'):
  # permute the digits 1, 3, 7 and 9 for E, F, S and T 
  for p in permutations('1379'):
    E, F, S, T = p    
    # check that AT and EAT are primes
    AT, EAT = int(A + T), int(E + A + T) 
    if AT in Primes() and EAT in Primes():
    
      # now permute remaining digits for the remaining letters
      for q in permutations(set('024685').difference((A,))):                
        # create a map from letters to digits
        d = dict(zip('AEFSTBIMPR', (A,) + p + q))        
        # B and P are leading digits so cannot be '0'
        if '0' not in (d['B'], d['P']):
          # evaluate all the words and check that they are all prime
          word_vals = tuple(int(''.join(d[c] for c in w)) for w in words)
          if all(w in Primes() for w in word_vals):
            print(word_vals[-2], 'became PRIME', word_vals)

from itertools import permutations

from number_theory import Primes

words = ('EAT', 'BEEF', 'AT', 'TIMES', 'IT', 'IS', 'A', 'PRIME', 'MEAT')

# since primes above 10 must end in 1, 3, 7 or 9, we know

# that E, F, S and T must be a permutation of these digits

# so A can only be 2 or 5

for A in ('2', '5'):

# permute the digits 1, 3, 7 and 9 for E, F, S and T

for p in permutations('1379'):

E, F, S, T = p

# check that AT and EAT are primes

AT, EAT = int(A + T), int(E + A + T)

if AT in Primes() and EAT in Primes():

# now permute remaining digits for the remaining letters

for q in permutations(set('024685').difference((A,))):

# create a map from letters to digits

d = dict(zip('AEFSTBIMPR', (A,) + p + q))

# B and P are leading digits so cannot be '0'

if '0' not in (d['B'], d['P']):

# evaluate all the words and check that they are all prime

word_vals = tuple(int(''.join(d[c] for c in w)) for w in words)

if all(w in Primes() for w in word_vals):

print(word_vals[-2], 'became PRIME', word_vals)

geoffrounce permalink


from itertools import permutations

def isprime(n): 
  for x in range(2, int(n ** 0.5) + 1): 
    if n % x == 0: 
      return False 
  return True

for e, a, t in permutations('987654321', 3):
  A, AT, EAT,  = int(a), int(a + t), int(e + a + t)
  if not (isprime(A) and isprime(AT) and isprime(EAT)): 
    continue
  
  dig_left1 = set('9876543210').difference((e, a, t))
  for i, m, s in permutations(dig_left1, 3):
    if '0' in (i, m):
      continue

    IS, IT  = int(i + s), int(i + t)
    if not (isprime(IS) and isprime(IT)): 
      continue
    TIMES, MEAT = int(t + i + m + e + s), int(m + e + a + t)
    if not (isprime(TIMES) and isprime(MEAT)): 
      continue
 
    dig_left2 = dig_left1.difference((i, m, s))
    for b, p, r, f in permutations(dig_left2):
      if '0' in (b, p):
        continue

      BEEF, PRIME = int(b + e + e + f), int(p + r + i + m + e)
      if not (isprime(BEEF) and isprime(PRIME)): 
        continue

      print('EAT BEEF AT TIMES IT IS A PRIME MEAT')
      print(EAT, BEEF, AT, TIMES, IT, IS, A, PRIME, MEAT)

from itertools import permutations

def isprime(n):

for x in range(2, int(n ** 0.5) + 1):

if n % x == 0:

return False

return True

for e, a, t in permutations('987654321', 3):

A, AT, EAT, = int(a), int(a + t), int(e + a + t)

if not (isprime(A) and isprime(AT) and isprime(EAT)):

continue

dig_left1 = set('9876543210').difference((e, a, t))

for i, m, s in permutations(dig_left1, 3):

if '0' in (i, m):

continue

IS, IT = int(i + s), int(i + t)

if not (isprime(IS) and isprime(IT)):

continue

TIMES, MEAT = int(t + i + m + e + s), int(m + e + a + t)

if not (isprime(TIMES) and isprime(MEAT)):

continue

dig_left2 = dig_left1.difference((i, m, s))

for b, p, r, f in permutations(dig_left2):

if '0' in (b, p):

continue

BEEF, PRIME = int(b + e + e + f), int(p + r + i + m + e)

if not (isprime(BEEF) and isprime(PRIME)):

continue

print('EAT BEEF AT TIMES IT IS A PRIME MEAT')

print(EAT, BEEF, AT, TIMES, IT, IS, A, PRIME, MEAT)

Sunday Times Teaser 2732 – Prime Meat

by Graham Smithers

Published: 1 February 2015 (link)

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