project_euler.problem_113.sol1¶
Project Euler Problem 113: https://projecteuler.net/problem=113
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420.
We shall call a positive integer that is neither increasing nor decreasing a “bouncy” number; for example, 155349.
As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million that are not bouncy and only 277032 non-bouncy numbers below 10^10.
How many numbers below a googol (10^100) are not bouncy?
Functions¶
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Calculate the binomial coefficient c(n,r) using the multiplicative formula. |
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Calculate the number of non-bouncy numbers with at most n digits. |
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Calculate the number of non-bouncy numbers with at most n digits. |
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Calculate the number of non-bouncy numbers less than a googol. |
Module Contents¶
- project_euler.problem_113.sol1.choose(n: int, r: int) int¶
Calculate the binomial coefficient c(n,r) using the multiplicative formula. >>> choose(4,2) 6 >>> choose(5,3) 10 >>> choose(20,6) 38760
- project_euler.problem_113.sol1.non_bouncy_exact(n: int) int¶
Calculate the number of non-bouncy numbers with at most n digits. >>> non_bouncy_exact(1) 9 >>> non_bouncy_exact(6) 7998 >>> non_bouncy_exact(10) 136126
- project_euler.problem_113.sol1.non_bouncy_upto(n: int) int¶
Calculate the number of non-bouncy numbers with at most n digits. >>> non_bouncy_upto(1) 9 >>> non_bouncy_upto(6) 12951 >>> non_bouncy_upto(10) 277032
- project_euler.problem_113.sol1.solution(num_digits: int = 100) int¶
Calculate the number of non-bouncy numbers less than a googol. >>> solution(6) 12951 >>> solution(10) 277032