Python/project_euler/problem_078/sol1.py

"""
Problem 78
Url: https://projecteuler.net/problem=78
Statement:
Let p(n) represent the number of different ways in which n coins
can be separated into piles. For example, five coins can be separated
into piles in exactly seven different ways, so p(5)=7.

            OOOOO
            OOOO   O
            OOO   OO
            OOO   O   O
            OO   OO   O
            OO   O   O   O
            O   O   O   O   O
Find the least value of n for which p(n) is divisible by one million.
"""

import itertools


def solution(number: int = 1000000) -> int:
    """
    >>> solution(1)
    1

    >>> solution(9)
    14

    >>> solution()
    55374
    """
    partitions = [1]

    for i in itertools.count(len(partitions)):
        item = 0
        for j in itertools.count(1):
            sign = -1 if j % 2 == 0 else +1
            index = (j * j * 3 - j) // 2
            if index > i:
                break
            item += partitions[i - index] * sign
            item %= number
            index += j
            if index > i:
                break
            item += partitions[i - index] * sign
            item %= number

        if item == 0:
            return i
        partitions.append(item)

    return 0


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    print(f"{solution() = }")