## Project Euler 112: Investigating the density of "bouncy" numbers.

#### 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.

Clearly there cannot be any bouncy numbers below one-hundred, but just over half of the numbers below one-thousand (525) are bouncy. In fact, the least number for which the proportion of bouncy numbers first reaches 50% is 538.

Surprisingly, bouncy numbers become more and more common and by the time we reach 21780 the proportion of bouncy numbers is equal to 90%.

Find the least number for which the proportion of bouncy numbers is exactly 99%.

#### Analysis

It was a surprise how long this program took to run. The *is_bouncy()* routine seemed to slow things down. We begin at the 90% point specified in the problem description and wait for the ratio to hit 99%.

#### Project Euler 112 Solution

Runs < 2.8 seconds in Python 2.7.```
def is_bouncy(n):
inc, dec, s = False, False, str(n)
for i in range(len(s)-1):
if s[i+1] > s[i]: inc = True
elif s[i+1] < s[i]: dec = True
if inc and dec: return True
return False
n, p = 21780, 0.90
b = n * p
while p != 0.99:
n += 1
if is_bouncy(n): b += 1
p = b / n
print "Project Euler 112 Solution =", n
```

Use this link to get the Project Euler 112 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

- See also, Project Euler 113 Solution: Count how many numbers below a googol (10**100) are not "bouncy"

*Project Euler 112 Solution last updated*

A better way would be generate all the increasing(symmetrically decreasing) numbers. So the rest are bouncy numbers. No need to check all the numbers one by one