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