Sunday Times Teaser 2515 – No Title

by Des MacHale

Published December 5 2010 (link)

Using each digit just once I wrote down two five-figure numbers. One was a cube and the other was a square. Then I combined these two five-figure numbers to give me a ten-figure number by using one of them to occupy the odd positions (from left to right) and the other to occupy the even positions (again from left to right). Then the four-figure number made up in order from the digits in the prime positions turned out to be prime.

What was the ten-figure number?

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Brian Gladman permalink


from itertools import product
from number_theory import is_prime

# filter a sequence of strings, removing those with duplicate letters
filter = lambda seq: (x for x in seq if len(set(x)) == len(x))

# create a list of five digit squares with no duplicate digits
squares = list(filter(str(x ** 2) for x in range(100, 317)))

# create a list of five digit cubes with no duplicate digits
cubes = list(filter(str(x ** 3) for x in range(21, 47)))

# consider all combinations of squares and cubes
for s, c in product(squares, cubes):
  # they must have no common digits
  if not set(s) & set(c):
    # interleave them in two orders
    for s, c in ((s, c), (c, s)):
      # the interleaved string
      n = ''.join(x + y for x, y in zip(s, c))
      # form the potential prime
      p = int(''.join(n[i - 1] for i in (2, 3, 5, 7)))
      # and check for a prime
      if(is_prime(p)):
        # create a list of five digit squares with no duplicate digits
        print(n, (int(s), int(c), p))

from itertools import product

from number_theory import is_prime

# filter a sequence of strings, removing those with duplicate letters

filter = lambda seq: (x for x in seq if len(set(x)) == len(x))

# create a list of five digit squares with no duplicate digits

squares = list(filter(str(x ** 2) for x in range(100, 317)))

# create a list of five digit cubes with no duplicate digits

cubes = list(filter(str(x ** 3) for x in range(21, 47)))

# consider all combinations of squares and cubes

for s, c in product(squares, cubes):

# they must have no common digits

if not set(s) & set(c):

# interleave them in two orders

for s, c in ((s, c), (c, s)):

# the interleaved string

n = ''.join(x + y for x, y in zip(s, c))

# form the potential prime

p = int(''.join(n[i - 1] for i in (2, 3, 5, 7)))

# and check for a prime

if(is_prime(p)):

# create a list of five digit squares with no duplicate digits

print(n, (int(s), int(c), p))

Sunday Times Teaser 2515 – No Title

by Des MacHale

Published December 5 2010 (link)

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