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a119005135
* Added solution for Project Euler problem 113. #2695 * Updated formatting and doctests. Reference: #3256
76 lines
1.8 KiB
Python
76 lines
1.8 KiB
Python
"""
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Project Euler Problem 113: https://projecteuler.net/problem=113
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Working from left-to-right if no digit is exceeded by the digit to its left it is
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called an increasing number; for example, 134468.
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Similarly if no digit is exceeded by the digit to its right it is called a decreasing
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number; for example, 66420.
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We shall call a positive integer that is neither increasing nor decreasing a
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"bouncy" number; for example, 155349.
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As n increases, the proportion of bouncy numbers below n increases such that there
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are only 12951 numbers below one-million that are not bouncy and only 277032
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non-bouncy numbers below 10^10.
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How many numbers below a googol (10^100) are not bouncy?
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"""
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def choose(n: int, r: int) -> int:
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"""
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Calculate the binomial coefficient c(n,r) using the multiplicative formula.
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>>> choose(4,2)
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6
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>>> choose(5,3)
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10
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>>> choose(20,6)
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38760
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"""
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ret = 1.0
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for i in range(1, r + 1):
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ret *= (n + 1 - i) / i
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return round(ret)
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def non_bouncy_exact(n: int) -> int:
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"""
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Calculate the number of non-bouncy numbers with at most n digits.
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>>> non_bouncy_exact(1)
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9
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>>> non_bouncy_exact(6)
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7998
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>>> non_bouncy_exact(10)
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136126
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"""
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return choose(8 + n, n) + choose(9 + n, n) - 10
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def non_bouncy_upto(n: int) -> int:
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"""
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Calculate the number of non-bouncy numbers with at most n digits.
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>>> non_bouncy_upto(1)
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9
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>>> non_bouncy_upto(6)
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12951
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>>> non_bouncy_upto(10)
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277032
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"""
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return sum(non_bouncy_exact(i) for i in range(1, n + 1))
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def solution(num_digits: int = 100) -> int:
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"""
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Caclulate the number of non-bouncy numbers less than a googol.
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>>> solution(6)
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12951
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>>> solution(10)
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277032
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"""
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return non_bouncy_upto(num_digits)
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if __name__ == "__main__":
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print(f"{solution() = }")
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