Add Project Euler Problem 180 (TheAlgorithms#4017)

* Added solution for Project Euler problem 180

* Fixed minor details in Project Euler problem 180

* updating DIRECTORY.md

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Author: 2 people

Diff for: DIRECTORY.md

@@ -245,6 +245,7 @@
   * [Max Sum Contiguous Subsequence](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/max_sum_contiguous_subsequence.py)
   * [Minimum Cost Path](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/minimum_cost_path.py)
   * [Minimum Partition](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/minimum_partition.py)
+  * [Minimum Steps To One](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/minimum_steps_to_one.py)
   * [Optimal Binary Search Tree](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/optimal_binary_search_tree.py)
   * [Rod Cutting](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/rod_cutting.py)
   * [Subset Generation](https://github.com/TheAlgorithms/Python/blob/master/dynamic_programming/subset_generation.py)
@@ -754,6 +755,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_173/sol1.py)
   * Problem 174
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_174/sol1.py)
+  * Problem 180
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_180/sol1.py)
   * Problem 188
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_188/sol1.py)
   * Problem 191

Diff for: project_euler/problem_180/__init__.py

@@ -0,0 +1,174 @@
+"""
+Project Euler Problem 234: https://projecteuler.net/problem=234
+
+For any integer n, consider the three functions
+
+f1,n(x,y,z) = x^(n+1) + y^(n+1) - z^(n+1)
+f2,n(x,y,z) = (xy + yz + zx)*(x^(n-1) + y^(n-1) - z^(n-1))
+f3,n(x,y,z) = xyz*(xn-2 + yn-2 - zn-2)
+
+and their combination
+
+fn(x,y,z) = f1,n(x,y,z) + f2,n(x,y,z) - f3,n(x,y,z)
+
+We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers
+of the form a / b with 0 < a < b ≤ k and there is (at least) one integer n,
+so that fn(x,y,z) = 0.
+
+Let s(x,y,z) = x + y + z.
+Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples
+(x,y,z) of order 35.
+All the s(x,y,z) and t must be in reduced form.
+
+Find u + v.
+
+
+Solution:
+
+By expanding the brackets it is easy to show that
+fn(x, y, z) = (x + y + z) * (x^n + y^n - z^n).
+
+Since x,y,z are positive, the requirement fn(x, y, z) = 0 is fulfilled if and
+only if x^n + y^n = z^n.
+
+By Fermat's Last Theorem, this means that the absolute value of n can not
+exceed 2, i.e. n is in {-2, -1, 0, 1, 2}. We can eliminate n = 0 since then the
+equation would reduce to 1 + 1 = 1, for which there are no solutions.
+
+So all we have to do is iterate through the possible numerators and denominators
+of x and y, calculate the corresponding z, and check if the corresponding numerator and
+denominator are integer and satisfy 0 < z_num < z_den <= 0. We use a set "uniquq_s"
+to make sure there are no duplicates, and the fractions.Fraction class to make sure
+we get the right numerator and denominator.
+
+Reference:
+https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
+"""
+
+
+from fractions import Fraction
+from math import gcd, sqrt
+from typing import Tuple
+
+
+def is_sq(number: int) -> bool:
+    """
+    Check if number is a perfect square.
+
+    >>> is_sq(1)
+    True
+    >>> is_sq(1000001)
+    False
+    >>> is_sq(1000000)
+    True
+    """
+    sq: int = int(number ** 0.5)
+    return number == sq * sq
+
+
+def add_three(
+    x_num: int, x_den: int, y_num: int, y_den: int, z_num: int, z_den: int
+) -> Tuple[int, int]:
+    """
+    Given the numerators and denominators of three fractions, return the
+    numerator and denominator of their sum in lowest form.
+    >>> add_three(1, 3, 1, 3, 1, 3)
+    (1, 1)
+    >>> add_three(2, 5, 4, 11, 12, 3)
+    (262, 55)
+    """
+    top: int = x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
+    bottom: int = x_den * y_den * z_den
+    hcf: int = gcd(top, bottom)
+    top //= hcf
+    bottom //= hcf
+    return top, bottom
+
+
+def solution(order: int = 35) -> int:
+    """
+    Find the sum of the numerator and denominator of the sum of all s(x,y,z) for
+    golden triples (x,y,z) of the given order.
+
+    >>> solution(5)
+    296
+    >>> solution(10)
+    12519
+    >>> solution(20)
+    19408891927
+    """
+    unique_s: set = set()
+    hcf: int
+    total: Fraction = Fraction(0)
+    fraction_sum: Tuple[int, int]
+
+    for x_num in range(1, order + 1):
+        for x_den in range(x_num + 1, order + 1):
+            for y_num in range(1, order + 1):
+                for y_den in range(y_num + 1, order + 1):
+                    # n=1
+                    z_num = x_num * y_den + x_den * y_num
+                    z_den = x_den * y_den
+                    hcf = gcd(z_num, z_den)
+                    z_num //= hcf
+                    z_den //= hcf
+                    if 0 < z_num < z_den <= order:
+                        fraction_sum = add_three(
+                            x_num, x_den, y_num, y_den, z_num, z_den
+                        )
+                        unique_s.add(fraction_sum)
+
+                    # n=2
+                    z_num = (
+                        x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
+                    )
+                    z_den = x_den * x_den * y_den * y_den
+                    if is_sq(z_num) and is_sq(z_den):
+                        z_num = int(sqrt(z_num))
+                        z_den = int(sqrt(z_den))
+                        hcf = gcd(z_num, z_den)
+                        z_num //= hcf
+                        z_den //= hcf
+                        if 0 < z_num < z_den <= order:
+                            fraction_sum = add_three(
+                                x_num, x_den, y_num, y_den, z_num, z_den
+                            )
+                            unique_s.add(fraction_sum)
+
+                    # n=-1
+                    z_num = x_num * y_num
+                    z_den = x_den * y_num + x_num * y_den
+                    hcf = gcd(z_num, z_den)
+                    z_num //= hcf
+                    z_den //= hcf
+                    if 0 < z_num < z_den <= order:
+                        fraction_sum = add_three(
+                            x_num, x_den, y_num, y_den, z_num, z_den
+                        )
+                        unique_s.add(fraction_sum)
+
+                    # n=2
+                    z_num = x_num * x_num * y_num * y_num
+                    z_den = (
+                        x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
+                    )
+                    if is_sq(z_num) and is_sq(z_den):
+                        z_num = int(sqrt(z_num))
+                        z_den = int(sqrt(z_den))
+                        hcf = gcd(z_num, z_den)
+                        z_num //= hcf
+                        z_den //= hcf
+                        if 0 < z_num < z_den <= order:
+                            fraction_sum = add_three(
+                                x_num, x_den, y_num, y_den, z_num, z_den
+                            )
+                            unique_s.add(fraction_sum)
+
+    for num, den in unique_s:
+        total += Fraction(num, den)
+
+    return total.denominator + total.numerator
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")