Add new solution for the euler project problem 9 (TheAlgorithms#12771)

* Add new solution for the euler project problem 9 - precompute the squares.

* Update sol4.py

* updating DIRECTORY.md

* Update sol4.py

* Update sol4.py

* Update sol4.py

---------

Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>
Co-authored-by: MaximSmolskiy <MaximSmolskiy@users.noreply.github.com>

DIRECTORY.md

@@ -956,6 +956,7 @@
     * [Sol1](project_euler/problem_009/sol1.py)
     * [Sol2](project_euler/problem_009/sol2.py)
     * [Sol3](project_euler/problem_009/sol3.py)
+    * [Sol4](project_euler/problem_009/sol4.py)
   * Problem 010
     * [Sol1](project_euler/problem_010/sol1.py)
     * [Sol2](project_euler/problem_010/sol2.py)

project_euler/problem_009/sol4.py

@@ -0,0 +1,60 @@
+"""
+Project Euler Problem 9: https://projecteuler.net/problem=9
+
+Special Pythagorean triplet
+
+A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
+
+    a^2 + b^2 = c^2.
+
+For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
+
+There exists exactly one Pythagorean triplet for which a + b + c = 1000.
+Find the product abc.
+
+References:
+    - https://en.wikipedia.org/wiki/Pythagorean_triple
+"""
+
+
+def get_squares(n: int) -> list[int]:
+    """
+    >>> get_squares(0)
+    []
+    >>> get_squares(1)
+    [0]
+    >>> get_squares(2)
+    [0, 1]
+    >>> get_squares(3)
+    [0, 1, 4]
+    >>> get_squares(4)
+    [0, 1, 4, 9]
+    """
+    return [number * number for number in range(n)]
+
+
+def solution(n: int = 1000) -> int:
+    """
+    Precomputing squares and checking if a^2 + b^2 is the square by set look-up.
+
+    >>> solution(12)
+    60
+    >>> solution(36)
+    1620
+    """
+
+    squares = get_squares(n)
+    squares_set = set(squares)
+    for a in range(1, n // 3):
+        for b in range(a + 1, (n - a) // 2 + 1):
+            if (
+                squares[a] + squares[b] in squares_set
+                and squares[n - a - b] == squares[a] + squares[b]
+            ):
+                return a * b * (n - a - b)
+
+    return -1
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")