## Non-bouncy numbers

### Problem 113

Published on Friday, 10th February 2006, 06:00 pm; Solved by 7625; Difficulty rating: 30%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?