Add project euler problem 51 (TheAlgorithms#3018)

* Add project euler problem 51

* Apply review suggestions

Author: Hyftar

project_euler/problem_51/__init__.py

@@ -0,0 +1,111 @@
+"""
+https://projecteuler.net/problem=51
+Prime digit replacements
+Problem 51
+
+By replacing the 1st digit of the 2-digit number *3, it turns out that six of
+the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
+
+By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit
+number is the first example having seven primes among the ten generated numbers,
+yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993.
+Consequently 56003, being the first member of this family, is the smallest prime
+with this property.
+
+Find the smallest prime which, by replacing part of the number (not necessarily
+adjacent digits) with the same digit, is part of an eight prime value family.
+"""
+
+from collections import Counter
+from typing import List
+
+
+def prime_sieve(n: int) -> List[int]:
+    """
+    Sieve of Erotosthenes
+    Function to return all the prime numbers up to a certain number
+    https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
+
+    >>> prime_sieve(3)
+    [2]
+
+    >>> prime_sieve(50)
+    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
+    """
+    is_prime = [True] * n
+    is_prime[0] = False
+    is_prime[1] = False
+    is_prime[2] = True
+
+    for i in range(3, int(n ** 0.5 + 1), 2):
+        index = i * 2
+        while index < n:
+            is_prime[index] = False
+            index = index + i
+
+    primes = [2]
+
+    for i in range(3, n, 2):
+        if is_prime[i]:
+            primes.append(i)
+
+    return primes
+
+
+def digit_replacements(number: int) -> List[List[int]]:
+    """
+    Returns all the possible families of digit replacements in a number which
+    contains at least one repeating digit
+
+    >>> digit_replacements(544)
+    [[500, 511, 522, 533, 544, 555, 566, 577, 588, 599]]
+
+    >>> digit_replacements(3112)
+    [[3002, 3112, 3222, 3332, 3442, 3552, 3662, 3772, 3882, 3992]]
+    """
+    number = str(number)
+    replacements = []
+    digits = ["0", "1", "2", "3", "4", "5", "6", "7", "8", "9"]
+
+    for duplicate in Counter(number) - Counter(set(number)):
+        family = [int(number.replace(duplicate, digit)) for digit in digits]
+        replacements.append(family)
+
+    return replacements
+
+
+def solution(family_length: int = 8) -> int:
+    """
+    Returns the solution of the problem
+
+    >>> solution(2)
+    229399
+
+    >>> solution(3)
+    221311
+    """
+    numbers_checked = set()
+
+    # Filter primes with less than 3 replaceable digits
+    primes = {
+        x for x in set(prime_sieve(1_000_000)) if len(str(x)) - len(set(str(x))) >= 3
+    }
+
+    for prime in primes:
+        if prime in numbers_checked:
+            continue
+
+        replacements = digit_replacements(prime)
+
+        for family in replacements:
+            numbers_checked.update(family)
+            primes_in_family = primes.intersection(family)
+
+            if len(primes_in_family) != family_length:
+                continue
+
+            return min(primes_in_family)
+
+
+if __name__ == "__main__":
+    print(solution())