mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-18 01:00:15 +00:00
4700297b3e
* Enable ruff RUF002 rule * Fix --------- Co-authored-by: Christian Clauss <cclauss@me.com>
60 lines
1.5 KiB
Python
60 lines
1.5 KiB
Python
"""
|
|
Problem 45: https://projecteuler.net/problem=45
|
|
|
|
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
|
|
Triangle T(n) = (n * (n + 1)) / 2 1, 3, 6, 10, 15, ...
|
|
Pentagonal P(n) = (n * (3 * n - 1)) / 2 1, 5, 12, 22, 35, ...
|
|
Hexagonal H(n) = n * (2 * n - 1) 1, 6, 15, 28, 45, ...
|
|
It can be verified that T(285) = P(165) = H(143) = 40755.
|
|
|
|
Find the next triangle number that is also pentagonal and hexagonal.
|
|
All triangle numbers are hexagonal numbers.
|
|
T(2n-1) = n * (2 * n - 1) = H(n)
|
|
So we shall check only for hexagonal numbers which are also pentagonal.
|
|
"""
|
|
|
|
|
|
def hexagonal_num(n: int) -> int:
|
|
"""
|
|
Returns nth hexagonal number
|
|
>>> hexagonal_num(143)
|
|
40755
|
|
>>> hexagonal_num(21)
|
|
861
|
|
>>> hexagonal_num(10)
|
|
190
|
|
"""
|
|
return n * (2 * n - 1)
|
|
|
|
|
|
def is_pentagonal(n: int) -> bool:
|
|
"""
|
|
Returns True if n is pentagonal, False otherwise.
|
|
>>> is_pentagonal(330)
|
|
True
|
|
>>> is_pentagonal(7683)
|
|
False
|
|
>>> is_pentagonal(2380)
|
|
True
|
|
"""
|
|
root = (1 + 24 * n) ** 0.5
|
|
return ((1 + root) / 6) % 1 == 0
|
|
|
|
|
|
def solution(start: int = 144) -> int:
|
|
"""
|
|
Returns the next number which is triangular, pentagonal and hexagonal.
|
|
>>> solution(144)
|
|
1533776805
|
|
"""
|
|
n = start
|
|
num = hexagonal_num(n)
|
|
while not is_pentagonal(num):
|
|
n += 1
|
|
num = hexagonal_num(n)
|
|
return num
|
|
|
|
|
|
if __name__ == "__main__":
|
|
print(f"{solution()} = ")
|