sol1.py

"""
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?
"""

frommathimportceil, sqrt


defsolution(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

forouter_widthinrange(3, (limit//4) +2):
ifouter_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

returnanswer


if__name__=="__main__":
print(f"{solution() =}")