def main():
"""
>>> main()
16695334890
"""
base = frozenset(xrange(10))
result = 0
for d4 in [0, 2, 4, 6, 8]: # to be divisible by 2
d6 = 5 # otherwise d_7 and d_8 must be equal to be divisible by 11
for div11 in range(506, 605, 11): # divisible by 11
_, d7, d8 = int_to_list(div11)
number = set([d4, d6])
if d7 in number or d8 in number:
continue # not pandigital
for div17 in range(102, 1003, 17): # divisible by 17
if div17 < d8 * 100 or div17 > (d8 + 1) * 100:
continue # d8 doesn't match
_, d9, d10 = int_to_list(div17)
number = set([d4, d6, d7, d8])
if d9 in number or d10 in number:
continue # not pandigital
if list_to_int([d7, d8, d9]) % 13 != 0:
continue # not divisible by 13
number = set([d4, d6, d7, d8, d9, d10])
for d3 in base - number:
number = set([d3, d4, d6, d7, d8, d9, d10])
for d5 in base - number:
if list_to_int([d3, d4, d5]) % 3 != 0:
continue # not divisible by 3
if list_to_int([d5, d6, d7]) % 7 != 0:
continue # not divisible by 7
number = set([d3, d4, d5, d6, d7, d8, d9, d10])
for d2 in base - number:
number = set([d2, d3, d4, d5, d6, d7, d8, d9,
d10, 0])
for d1 in base - number:
result += list_to_int([
d1, d2, d3, d4, d5, d6, d7, d8, d9, d10])
print(result)
def main():
"""
>>> main()
932718654
"""
max_len_number = 9
max_pandigital_number = 0
# since the multiplier must be at least 2 we can stop when the
# multiplicand * 1 and multiplicand * 2 together have more than 9 numbers
for multiplicand in range(2, 10000):
product_as_list = int_to_list(multiplicand)
for multiplier in count(2):
product_as_list += int_to_list(multiplicand * multiplier)
product = list_to_int(product_as_list)
if is_int_pandigital(product) and max_pandigital_number < product:
max_pandigital_number = product
elif len(product_as_list) > max_len_number:
break
print(max_pandigital_number)
def main():
"""
>>> main()
972
"""
max_digit_sum = 0
for a in range(1, 100):
for b in range(1, 100):
digit_sum = sum(int_to_list(a ** b))
if digit_sum > max_digit_sum:
max_digit_sum = digit_sum
print(max_digit_sum)
def main():
"""
>>> main()
8581146
"""
chains_that_arrive_at_89 = 0
for number in combinations_with_replacement(xrange(10), 7):
n = list_to_int(number)
while n != 1 and n != 89 and n != 0:
n = sum([d ** 2 for d in int_to_list(n)])
if n == 89:
chains_that_arrive_at_89 += count_permutations_of(number)
print(chains_that_arrive_at_89)
def main():
"""
>>> main()
40730
"""
factorials = [factorial(n) for n in range(10)]
summe = -3 # 1 & 2 which are not counted
for num_digits in range(8): # upper limit: 7 * 9! < 10**7
for number in combinations_with_replacement(xrange(10), num_digits):
factorial_sum = sum((factorials[digit] for digit in number))
if sorted(int_to_list(factorial_sum)) == list(number):
summe += factorial_sum
print(summe)
def main():
"""
>>> main()
55
"""
even_digits = frozenset([d for d in range(0, 10, 2)])
primes = Primes(10 ** 6)
count_circular_primes = 1 # for the prime 2
for prime in primes:
# Exculde all primes that have an even digit in it.
if not even_digits.isdisjoint(int_to_list(prime)):
continue
if all((rotated_prime in primes for rotated_prime in rotate(prime))):
count_circular_primes += 1
print(count_circular_primes)
def main():
"""
>>> main()
121313
"""
amount_of_primes = 8
primes = Primes(10 ** 6)
for prime in primes:
prime_as_list = int_to_list(prime)
list_of_indices = [] # indices of reoccuring digits
for i in range(len(prime_as_list)):
for j in range(i + 1, len(prime_as_list)):
for k in range(j + 1, len(prime_as_list)):
if prime_as_list[j] == prime_as_list[k]:
list_of_indices.append((i, j, k))
for indices in list_of_indices:
prime_family = []
for d in range(10):
prime_candidate = prime_as_list[:]
for i in indices:
prime_candidate[i] = d
# ensure that we don't look at n-1-digit numbers
if prime_candidate[0] == 0:
continue
prime_candidate = list_to_int(prime_candidate)
if prime_candidate in primes:
prime_family.append(prime_candidate)
# check if we still can get the neccasry amount of primes
if (10 - d) + len(prime_family) < amount_of_primes:
break
if len(prime_family) == amount_of_primes:
print(prime)
return
def rotate(number):
""" return all rotations of this number"""
digits = int_to_list(number)
for _ in range(len(digits)):
yield list_to_int(digits)
digits.append(digits.pop(0))
