from fractions import Fraction
from functools import lru_cache


@lru_cache
def expected_singles(sheets):
    count = 0
    if len(sheets) == 1 and sheets[0] == 5:
        return 0
    elif len(sheets) == 1:
        count += 1

    weight = Fraction(1, len(sheets))
    for sheet in sheets:
        nsheets = list(sheets)
        nsheets.remove(sheet)
        for nsheet in range(sheet, 5):
            nsheets.append(nsheet + 1)
        count += weight * expected_singles(tuple(sorted(nsheets)))
    return count


def euler_151():
    return round(float(expected_singles(tuple([2, 3, 4, 5]))), 6)


if __name__ == "__main__":
    solution = euler_151()
    print("e151.py: " + str(solution))
    assert(solution == 0.464399)