<< problem 111 - Primes with runs | Non-bouncy numbers - problem 113 >> |

# Problem 112: Bouncy numbers

(see projecteuler.net/problem=112)

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%.

# Algorithm

`isBouncy`

determines whether its parameter is bouncy or not by stepping through its digits.

The rest is just a brute-force search.

## Modifications by HackerRank

Any percentage can be entered.

Even though my brute-force search finds the solution to the original problem (99%) in less than 0.1 seconds, it's way too slow for

the potentially huge search space of Hackerrank. Someone indicated that one "answer requires about 90 bits".

# My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
// return true if x is a bouncy number

bool isBouncy(unsigned long long x)
{
// figure out whether x is monotonic ascending or descending
// it's bouncy if neither ascending nor descending
bool ascending = true;
bool descending = true;
// initial digit (the right-most digit)
auto previous = x % 10;
x /= 10;
// still digits left ?
while (x > 0)
{
// current digit
auto current = x % 10;
// compare two digits
descending &= previous >= current;
ascending &= previous <= current;
// bouncy ?
if (!ascending && !descending)
return true;
// keep going ...
x /= 10;
previous = current;
}
// not bouncy (either ascending, descending or all digits are equal)
return false;
}
int main()
{
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
// original problem: 99%
unsigned long long p = 99;
unsigned long long q = 100;
std::cin >> p >> q;
// brute-force ...
unsigned long long current = 100; // no bouncy numbers below 100
unsigned long long numBouncy = 0;
do
{
// check next number if bouncy
current++;
if (isBouncy(current))
numBouncy++;
} while (numBouncy * q < current * p); // same as numBouncy/current == p/q (=99%)
// print result
std::cout << current << std::endl;
}
return 0;
}

This solution contains 13 empty lines, 14 comments and 1 preprocessor command.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 90 100" | ./112`

Output:

*Note:* the original problem's input `99 100`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.02 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

May 11, 2017 submitted solution

May 11, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler112

My code solves **1** out of **16** test cases (score: **0%**)

I failed **0** test cases due to wrong answers and **15** because of timeouts

# Difficulty

Project Euler ranks this problem at **15%** (out of 100%).

Hackerrank describes this problem as **advanced**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Links

projecteuler.net/thread=112 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-112-density-bouncy-numbers/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p112.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler112.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

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