## Project Euler 113: Count how many numbers below a googol (10^{100}) are not "bouncy"

#### Problem Description

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?

#### Analysis

Whenever a problem asks for the number of elements in a set, as does this one, then the solution in typically a counting problem solved with combinometrics, dynamic programming or a generating function.

Our solution uses the general formula that counts the monotonically increasing/decreasing digits ignoring leading zeros.

To remove confusion surrounding the terms *increasing* and *non-decreasing*, Rudin, in his *Principles of Mathematical Analysis* (Def. 3.13), defines a sequence
of real numbers to be:

(a) *monotonically increasing* if
[i.e., non-decreasing].

(b) *monotonically decreasing* if
[i.e., non-increasing].

To avoid ambiguity one can specify that a monotonic sequence is “strictly increasing” or that it is non-decreasing, or non-increasing, or strictly decreasing.

#### Project Euler 113 Solution

Runs < 0.001 seconds in Python 2.7.Use this link to get the Project Euler 113 Solution Python 2.7 source.

#### Answer

Slowly swipe from either end beginning with the white vertical bar to get an idea of the starting or ending digits. For less drama, just double click the answer area. The distance between the two bars will give you an idea of the magnitude. Touch devices can tap and hold the center of the box between the two bars and choose*define*to reveal the answer.

#### Afterthoughts

- Function
`binomial`

is listed in Common Functions and Routines for Project Euler - See also, Project Euler 112 Solution: Investigating the density of "bouncy" numbers.

*Project Euler 113 Solution last updated*

Could you please provide more information about that general formula for counting increasing/decreasing digits? Is there a paper about it?

Thanks.

I’m going to have to revisit this problem. I know it’s an application of the Inclusionâ€“exclusion principle that works for limits of any power of ten. Give me a few days to look at it.