Add euler project problem 142 solution

project_euler/problem_142/sol1.py

@@ -0,0 +1,80 @@
+"""
+Project Euler Problem 142: https://projecteuler.net/problem=142
+
+Perfect Square Collection
+
+Find the smallest x + y + z with integers x > y > z > 0  such that
+x + y, x - y, x + z, x - z, y + z, y - z are all perfect squares.
+
+
+Change the variables to a, b, c, so that 3 requirements are satisfied automatically:
+a^2 = y - z
+b^2 = x - y
+c^2 = z + x
+
+and the rest of requirements for perfect squares are:
+z + y = c^2 - b^2
+y + x = a^2 + c^2
+x - z = a^2 + b^2
+
+Then iterate over a^2, b^2 and c^2 to check if the combination satisfies all
+3 requirements.
+
+The total sum x + y + z = (a^2 - b^2 + 3c^2) / 2, so we break loop for c^2 if
+the sum is already bigger than found sum.
+
+"""
+
+
+def solution(number_of_terms: int = 3) -> int | None:
+    """
+
+    Iterate over combinations of a, b, c and save min sum.
+    In case only one term x = 1 is solution.
+    In case of two terms, x = 5, y = 4 is the solution.
+
+    >>> solution(1)
+    1
+    >>> solution(2)
+    9
+    """
+
+    if number_of_terms == 1:
+        return 1
+    if number_of_terms == 2:
+        return 9
+
+    n_max = 2500
+    squares: list[int] = []
+
+    for a in range(n_max + 1):
+        squares += [a * a]
+    squares_set = set(squares)
+
+    min_sum = None
+    for a in range(1, len(squares)):
+        a_sq = squares[a]
+        for b in range(1, len(squares)):
+            b_sq = squares[b]
+            if a_sq + b_sq not in squares_set:
+                continue
+            for c in range(max(a, b) + 1, len(squares)):
+                c_sq = squares[c]
+                # break if x + y + z is already bigger than min_sum:
+                if min_sum is not None and (a_sq - b_sq + 3 * c_sq) // 2 > min_sum:
+                    break
+                if (c_sq - b_sq in squares_set) and (a_sq + c_sq in squares_set):
+                    x2, y2, z2 = (
+                        a_sq + b_sq + c_sq,
+                        a_sq - b_sq + c_sq,
+                        c_sq - a_sq - b_sq,
+                    )
+                    if z2 > 0 and x2 % 2 == 0 and y2 % 2 == 0 and z2 % 2 == 0:
+                        sum_ = (x2 + y2 + z2) // 2
+                        min_sum = sum_ if min_sum is None else min(min_sum, sum_)
+
+    return min_sum
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")