added problem 60 solution

project_euler/problem_060/sol1.py

@@ -0,0 +1,170 @@
+"""
+Project Euler Problem 60: https://projecteuler.net/problem=60
+
+Prime Pair Sets
+
+Problem: The primes 3, 7, 109, and 673, are quite remarkable. By taking any two
+primes and concatenating them in any order the result will always be prime.
+For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four
+primes, 792, represents the lowest sum for a set of four primes with this property.
+
+Question: Find the lowest sum for a set of five primes for which
+any two primes concatenate to produce another prime.
+
+Explanation of the solution: the idea in finding a set of primes that satisfy the
+concatenation condition is to build a graph of primes where a node would represent a
+prime and an edge between two nodes would indicate that the two primes form a valid
+pair. Hence, the solution to the problem can be computed more efficiently since
+it does not require brute-forcing all possible combinations.
+"""
+
+from itertools import combinations
+
+
+def sieve_of_eratosthenes(limit: int) -> set[int]:
+    """Generate primes up to a limit using the Sieve of Eratosthenes and return them
+        as a set.
+
+    Parameters
+    ----------
+    limit : int
+        The upper limit to generate primes up to.
+
+    Returns
+    ----------
+    set[int]
+        A set of prime numbers up to the limit.
+    """
+    sieve: list = [True] * (limit + 1)
+    primes_set = set()
+    for p in range(2, limit + 1):
+        if sieve[p]:
+            primes_set.add(p)
+            for i in range(p * p, limit + 1, p):
+                sieve[i] = False
+    return primes_set
+
+
+def is_prime(n: int) -> bool:
+    """Checks whether a number is prime or not.
+
+    Parameters
+    ----------
+    n : int
+        The number to be checked.
+
+    Returns
+    ----------
+    bool
+        True if the number is prime, False otherwise.
+    """
+    if n < 2:
+        return False
+    return all(n % i != 0 for i in range(3, int(n**0.5) + 1, 2))
+
+
+def valid_pair(p1: int, p2: int) -> bool:
+    """Checks whether a pair of primes concatenated both ways is prime or not.
+
+    Parameters
+    ----------
+    p1 : int
+        The first prime number.
+    p2 : int
+        The second prime number.
+
+    Returns
+    ----------
+    bool
+        True if the pair is valid, False otherwise.
+    """
+    return is_prime(int(str(p1) + str(p2))) and is_prime(int(str(p2) + str(p1)))
+
+
+def build_graph(prime_set: set[int]) -> dict[int, set[int]]:
+    """Builds a graph of primes where each prime is a node and an edge exists
+        between two primes if they form a valid pair.
+
+    Parameters
+    ----------
+    prime_set : set[int]
+        A set of prime numbers.
+
+    Returns
+    ----------
+    dict[int, set[int]]
+        A graph of primes.
+    """
+    graph: dict[int, set[int]] = {p: set() for p in prime_set}
+    for p1, p2 in combinations(prime_set, 2):
+        if valid_pair(p1, p2):
+            graph[p1].add(p2)
+            graph[p2].add(p1)
+    return graph
+
+
+def find_cliques(
+    node: int, graph: dict[int, set[int]], clique: set[int], depth: int
+) -> list[set[int]]:
+    """The problem of finding a set of n primes that all satisfy
+        the concatenation condition can be reduced to that of finding
+        a clique of size n in the graph of primes.
+
+    Parameters
+    ----------
+    node : int
+        The node to be considered.
+    graph : dict[int, set[int]]
+        The graph of primes.
+    clique : set[int]
+        The set of primes that form a clique.
+    depth : int
+        The size of the clique.
+
+    Returns
+    ----------
+    list[set[int]]
+        A list of sets of primes that form cliques of size n.
+    """
+    if depth == 1:
+        return [clique]
+    cliques: list[set[int]] = []
+    for neighbour in graph[node]:
+        if all(neighbour in graph[member] for member in clique):
+            cliques.extend(
+                find_cliques(neighbour, graph, clique | {neighbour}, depth - 1)
+            )
+    return cliques
+
+
+def solution(n: int = 5) -> int:
+    """Aims at finding the lowest sum for a set of five primes for which any two primes
+        concatenate to produce another prime.
+
+    Parameters
+    ----------
+    n : int
+        The size of the set of primes (5 by default)
+
+    Returns
+    ----------
+    int
+        The lowest sum for a set of five primes for which any two primes concatenate to
+        produce another prime.
+
+    >>> solution()
+    26033
+    """
+    limit: int = 10000  # the limit is arbitrary but is sufficient for the problem
+    prime_set: set[int] = sieve_of_eratosthenes(limit)
+    prime_graph: dict[int, set[int]] = build_graph(prime_set)
+
+    for prime in prime_set:
+        cliques = find_cliques(prime, prime_graph, {prime}, depth=n)
+        if cliques:
+            return min(sum(clique) for clique in cliques)
+    return -1
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")