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63 lines
1.3 KiB
Python
63 lines
1.3 KiB
Python
"""
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Problem 78
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Url: https://projecteuler.net/problem=78
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Statement:
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Let p(n) represent the number of different ways in which n coins
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can be separated into piles. For example, five coins can be separated
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into piles in exactly seven different ways, so p(5)=7.
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OOOOO
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OOOO O
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OOO OO
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OOO O O
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OO OO O
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OO O O O
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O O O O O
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Find the least value of n for which p(n) is divisible by one million.
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"""
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import itertools
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def solution(number: int = 1000000) -> int:
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"""
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>>> solution(1)
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1
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>>> solution(9)
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14
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>>> solution()
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55374
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"""
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partitions = [1]
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for i in itertools.count(len(partitions)):
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item = 0
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for j in itertools.count(1):
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sign = -1 if j % 2 == 0 else +1
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index = (j * j * 3 - j) // 2
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if index > i:
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break
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item += partitions[i - index] * sign
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item %= number
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index += j
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if index > i:
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break
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item += partitions[i - index] * sign
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item %= number
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if item == 0:
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return i
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partitions.append(item)
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return 0
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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print(f"{solution() = }")
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