diff --git a/DIRECTORY.md b/DIRECTORY.md index 64a87dc66..f4a470c12 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -793,6 +793,8 @@ * Problem 067 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_067/sol1.py) * [Sol2](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_067/sol2.py) + * Problem 068 + * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_068/sol1.py) * Problem 069 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_069/sol1.py) * Problem 070 diff --git a/project_euler/problem_068/__init__.py b/project_euler/problem_068/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/project_euler/problem_068/sol1.py b/project_euler/problem_068/sol1.py new file mode 100644 index 000000000..772be359f --- /dev/null +++ b/project_euler/problem_068/sol1.py @@ -0,0 +1,133 @@ +""" +Project Euler Problem 68: https://projecteuler.net/problem=68 + +Magic 5-gon ring + +Problem Statement: +Consider the following "magic" 3-gon ring, +filled with the numbers 1 to 6, and each line adding to nine. + + 4 + \ + 3 + / \ + 1 - 2 - 6 + / + 5 + +Working clockwise, and starting from the group of three +with the numerically lowest external node (4,3,2 in this example), +each solution can be described uniquely. +For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3. + +It is possible to complete the ring with four different totals: 9, 10, 11, and 12. +There are eight solutions in total. +Total Solution Set +9 4,2,3; 5,3,1; 6,1,2 +9 4,3,2; 6,2,1; 5,1,3 +10 2,3,5; 4,5,1; 6,1,3 +10 2,5,3; 6,3,1; 4,1,5 +11 1,4,6; 3,6,2; 5,2,4 +11 1,6,4; 5,4,2; 3,2,6 +12 1,5,6; 2,6,4; 3,4,5 +12 1,6,5; 3,5,4; 2,4,6 + +By concatenating each group it is possible to form 9-digit strings; +the maximum string for a 3-gon ring is 432621513. + +Using the numbers 1 to 10, and depending on arrangements, +it is possible to form 16- and 17-digit strings. +What is the maximum 16-digit string for a "magic" 5-gon ring? +""" + +from itertools import permutations + + +def solution(gon_side: int = 5) -> int: + """ + Find the maximum number for a "magic" gon_side-gon ring + + The gon_side parameter should be in the range [3, 5], + other side numbers aren't tested + + >>> solution(3) + 432621513 + >>> solution(4) + 426561813732 + >>> solution() + 6531031914842725 + >>> solution(6) + Traceback (most recent call last): + ValueError: gon_side must be in the range [3, 5] + """ + if gon_side < 3 or gon_side > 5: + raise ValueError("gon_side must be in the range [3, 5]") + + # Since it's 16, we know 10 is on the outer ring + # Put the big numbers at the end so that they are never the first number + small_numbers = list(range(gon_side + 1, 0, -1)) + big_numbers = list(range(gon_side + 2, gon_side * 2 + 1)) + + for perm in permutations(small_numbers + big_numbers): + numbers = generate_gon_ring(gon_side, list(perm)) + if is_magic_gon(numbers): + return int("".join(str(n) for n in numbers)) + + raise ValueError(f"Magic {gon_side}-gon ring is impossible") + + +def generate_gon_ring(gon_side: int, perm: list[int]) -> list[int]: + """ + Generate a gon_side-gon ring from a permutation state + The permutation state is the ring, but every duplicate is removed + + >>> generate_gon_ring(3, [4, 2, 3, 5, 1, 6]) + [4, 2, 3, 5, 3, 1, 6, 1, 2] + >>> generate_gon_ring(5, [6, 5, 4, 3, 2, 1, 7, 8, 9, 10]) + [6, 5, 4, 3, 4, 2, 1, 2, 7, 8, 7, 9, 10, 9, 5] + """ + result = [0] * (gon_side * 3) + result[0:3] = perm[0:3] + perm.append(perm[1]) + + magic_number = 1 if gon_side < 5 else 2 + + for i in range(1, len(perm) // 3 + magic_number): + result[3 * i] = perm[2 * i + 1] + result[3 * i + 1] = result[3 * i - 1] + result[3 * i + 2] = perm[2 * i + 2] + + return result + + +def is_magic_gon(numbers: list[int]) -> bool: + """ + Check if the solution set is a magic n-gon ring + Check that the first number is the smallest number on the outer ring + Take a list, and check if the sum of each 3 numbers chunk is equal to the same total + + >>> is_magic_gon([4, 2, 3, 5, 3, 1, 6, 1, 2]) + True + >>> is_magic_gon([4, 3, 2, 6, 2, 1, 5, 1, 3]) + True + >>> is_magic_gon([2, 3, 5, 4, 5, 1, 6, 1, 3]) + True + >>> is_magic_gon([1, 2, 3, 4, 5, 6, 7, 8, 9]) + False + >>> is_magic_gon([1]) + Traceback (most recent call last): + ValueError: a gon ring should have a length that is a multiple of 3 + """ + if len(numbers) % 3 != 0: + raise ValueError("a gon ring should have a length that is a multiple of 3") + + if min(numbers[::3]) != numbers[0]: + return False + + total = sum(numbers[:3]) + + return all(sum(numbers[i : i + 3]) == total for i in range(3, len(numbers), 3)) + + +if __name__ == "__main__": + print(solution())