from itertools import combinations
+
+def concatenate_all(numbers):
+ result = []
+ for a, b in combinations(numbers, 2):
+ result.append([a, b])
+ result.append([b, a])
+ return result
+
+print(concatenate_all([3, 7, 109, 673]))
+def sieve_of_eratosthenes(limit):
+ primes = []
+ prospects = [n for n in range(2, limit)]
+ while prospects:
+ p = prospects[0]
+ prospects = [x for x in prospects if x % p != 0]
+ primes.append(p)
+ if p * p > limit:
+ break
+ primes += prospects
+ return primes
+
+primes = sieve_of_eratosthenes(100000)
+primes_set = set(primes)
+def is_prime(n):
+ if n % 2 == 0:
+ return False
+ while n % 2 == 0:
+ n //= 2
+ f = 3
+ while f * f <= n:
+ if n % f == 0:
+ return False
+ else:
+ f += 2
+ return True
+
+def concatentation_is_prime(a, b):
+ ab = int(str(a) + str(b))
+ ba = int(str(b) + str(a))
+ if is_prime(ab) and is_prime(ba):
+ return True
+ else:
+ return False
+
+assert(concatentation_is_prime(673, 3))
+assert(concatentation_is_prime(673, 1) == False)
+potentials = []
+new_potentials = []
+done = False
+for p in primes:
+ if done:
+ break
+ potentials += new_potentials
+ new_potentials = []
+ for potential in potentials:
+ all_concatenations_prime = True
+ for prime in potential:
+ if not concatentation_is_prime(p, prime):
+ all_concatenations_prime = False
+ break
+ if all_concatenations_prime:
+ new_potential = list(potential)
+ new_potential.append(p)
+ if len(new_potential) > 4:
+ print(new_potential)
+ s = sum(new_potential)
+ done = True
+ break
+ new_potentials.append(new_potential)
+ if p < 15:
+ potentials.append([p])
+
+print(s)
+assert(s == 26033)
+def get_four_digit_numbers(function):
+ r = []
+ n = 1
+ f = function
+ while f(n) < 10000:
+ if f(n) > 999:
+ r.append(f(n))
+ n += 1
+ return r
+
+triangles = get_four_digit_numbers(lambda n: n * (n + 1) // 2)
+squares = get_four_digit_numbers(lambda n: n**2)
+pentas = get_four_digit_numbers(lambda n: n * (3 * n - 1) // 2)
+hexas = get_four_digit_numbers(lambda n: n * (2 * n - 1))
+heptas = get_four_digit_numbers(lambda n: n * (5*n - 3) // 2)
+octas = get_four_digit_numbers(lambda n: n * (3*n - 2))
+def is_cyclic(a, b):
+ return str(a)[-2:] == str(b)[:2]
+
+assert(is_cyclic(3328, 2877))
+assert(is_cyclic(3329, 2877) == False)
+def search_solution(aggregator, polygonals):
+ if not polygonals:
+ if is_cyclic(aggregator[-1], aggregator[0]):
+ return aggregator
+ else:
+ return []
+
+ if not aggregator:
+ for polygonal in polygonals:
+ for number in polygonal:
+ aggregator.append(number)
+ s = search_solution(aggregator, [p for p in polygonals if p != polygonal])
+ if s:
+ return s
+ aggregator.pop()
+
+ for polygonal in polygonals:
+ for number in polygonal:
+ if is_cyclic(aggregator[-1], number) and not number in aggregator:
+ aggregator.append(number)
+ s = search_solution(aggregator, [p for p in polygonals if p != polygonal])
+ if s:
+ return s
+ aggregator.pop()
+ return []
+
+s = search_solution([], [triangles, squares, pentas, hexas, heptas, octas])
+print(s)
+s = sum(s)
+print(s)
+assert(s == 28684)
+cubes = [n * n * n for n in range(1, 5000)]
+cubes_set = set(cubes)
+solutions = {}
+
+for n in range(1, 10000):
+ cube = n * n * n
+ try:
+ key = "".join(sorted(str(cube)))
+ solutions[key].append(cube)
+ if len(solutions[key]) > 4:
+ print(solutions[key])
+ s = solutions[key][0]
+ break
+ except KeyError:
+ solutions[key] = [cube]
+
+print(s)
+assert(s == 127035954683)
+