Sunday Times Teaser 2583 – Playing Cards

by Graham Smithers

Published: 25 March 2012 (link)

Oliver wrote down some numbers and then used a letter-for-digit code to make them into words. In this way his numbers became JACK, QUEEN, KING and ACE. In fact JACK and KING were perfect squares, QUEEN was odd, and ACE was a prime. Furthermore, appropriately enough, the sum of the four was divisible by 52.

What number was represented by QUEEN?

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from itertools import permutations
from functools import reduce
from number_theory import is_prime

# four digit perfect squares
squares = { x ** 2:x for x in range(32, 100) }

fs = 'QUEEN = {1:d} (JACK = {0:d}, KING = {2:d}, ACE = {3:d})'
fn = lambda x, y: 10 * x + y

# JACK is a perfect square
for JACK in squares:
  J, A, C, K = map(int, str(JACK))
  # and so is KING
  for KING in squares:
    _, I, N, G = map(int, str(KING))
    # find unused digit values
    rest = set(range(10)).difference([A, C, G, I, J, K, N])
    # check that the two 'K' are the same value, that we have
    # seven different values and that N is odd (QUEEN is odd)
    if _ == K and len(rest) == 3 and N % 2:
      # permute the remaining digits for the reemaining letters
      for Q, U, E in permutations(rest):
        QUEEN = reduce(fn, (Q, U, E, E, N))
        ACE = reduce(fn, (A, C, E))
        # ACE is prime and the sum of JACK, QUEEN, KING and ACE is
        # a multiple of 52
        if is_prime(ACE) and not sum((JACK, QUEEN, KING, ACE)) % 52:
          print(fs.format(JACK, QUEEN, KING, ACE))

from itertools import permutations

from functools import reduce

from number_theory import is_prime

# four digit perfect squares

squares = { x ** 2:x for x in range(32, 100) }

fs = 'QUEEN = {1:d} (JACK = {0:d}, KING = {2:d}, ACE = {3:d})'

fn = lambda x, y: 10 * x + y

# JACK is a perfect square

for JACK in squares:

J, A, C, K = map(int, str(JACK))

# and so is KING

for KING in squares:

_, I, N, G = map(int, str(KING))

# find unused digit values

rest = set(range(10)).difference([A, C, G, I, J, K, N])

# check that the two 'K' are the same value, that we have

# seven different values and that N is odd (QUEEN is odd)

if _ == K and len(rest) == 3 and N % 2:

# permute the remaining digits for the reemaining letters

for Q, U, E in permutations(rest):

QUEEN = reduce(fn, (Q, U, E, E, N))

ACE = reduce(fn, (A, C, E))

# ACE is prime and the sum of JACK, QUEEN, KING and ACE is

# a multiple of 52

if is_prime(ACE) and not sum((JACK, QUEEN, KING, ACE)) % 52:

print(fs.format(JACK, QUEEN, KING, ACE))

Sunday Times Teaser 2583 – Playing Cards

by Graham Smithers

Published: 25 March 2012 (link)

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