Add solution for Project Euler problem 173. (#3075)

* Added solution for Project Euler problemm problem 173. #2695

* Added docstring

* Update formatting, doctest and annotations. Reference: #3256

Author: fpringle

project_euler/problem_173/__init__.py

@@ -0,0 +1,41 @@
+"""
+Project Euler Problem 173: https://projecteuler.net/problem=173
+
+We shall define a square lamina to be a square outline with a square "hole" so that
+the shape possesses vertical and horizontal symmetry. For example, using exactly
+thirty-two square tiles we can form two different square laminae:
+
+With one-hundred tiles, and not necessarily using all of the tiles at one time, it is
+possible to form forty-one different square laminae.
+
+Using up to one million tiles how many different square laminae can be formed?
+"""
+
+
+from math import ceil, sqrt
+
+
+def solution(limit: int = 1000000) -> int:
+    """
+    Return the number of different square laminae that can be formed using up to
+    one million tiles.
+    >>> solution(100)
+    41
+    """
+    answer = 0
+
+    for outer_width in range(3, (limit // 4) + 2):
+        if outer_width ** 2 > limit:
+            hole_width_lower_bound = max(ceil(sqrt(outer_width ** 2 - limit)), 1)
+        else:
+            hole_width_lower_bound = 1
+        if (outer_width - hole_width_lower_bound) % 2:
+            hole_width_lower_bound += 1
+
+        answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1
+
+    return answer
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")