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* Rename all Project Euler directories: Reason: The change was done to maintain consistency throughout the directory and to keep all directories in sorted order. Due to the above change, some config files had to be modified: 'problem_22` -> `problem_022` * Update scripts to pad zeroes in PE directories
60 lines
1.5 KiB
Python
60 lines
1.5 KiB
Python
"""
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Problem 45: https://projecteuler.net/problem=45
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Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
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Triangle T(n) = (n * (n + 1)) / 2 1, 3, 6, 10, 15, ...
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Pentagonal P(n) = (n * (3 * n − 1)) / 2 1, 5, 12, 22, 35, ...
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Hexagonal H(n) = n * (2 * n − 1) 1, 6, 15, 28, 45, ...
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It can be verified that T(285) = P(165) = H(143) = 40755.
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Find the next triangle number that is also pentagonal and hexagonal.
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All trinagle numbers are hexagonal numbers.
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T(2n-1) = n * (2 * n - 1) = H(n)
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So we shall check only for hexagonal numbers which are also pentagonal.
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"""
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def hexagonal_num(n: int) -> int:
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"""
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Returns nth hexagonal number
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>>> hexagonal_num(143)
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40755
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>>> hexagonal_num(21)
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861
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>>> hexagonal_num(10)
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190
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"""
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return n * (2 * n - 1)
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def is_pentagonal(n: int) -> bool:
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"""
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Returns True if n is pentagonal, False otherwise.
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>>> is_pentagonal(330)
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True
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>>> is_pentagonal(7683)
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False
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>>> is_pentagonal(2380)
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True
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"""
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root = (1 + 24 * n) ** 0.5
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return ((1 + root) / 6) % 1 == 0
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def solution(start: int = 144) -> int:
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"""
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Returns the next number which is traingular, pentagonal and hexagonal.
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>>> solution(144)
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1533776805
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"""
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n = start
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num = hexagonal_num(n)
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while not is_pentagonal(num):
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n += 1
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num = hexagonal_num(n)
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return num
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if __name__ == "__main__":
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print(f"{solution()} = ")
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