<< problem 62 - Cubic permutations | Odd period square roots - problem 64 >> |

# Problem 63: Powerful digit counts

(see projecteuler.net/problem=63)

The 5-digit number, 16807=7^5, is also a fifth power. Similarly, the 9-digit number, 134217728=8^9, is a ninth power.

How many n-digit positive integers exist which are also an nth power?

# My Algorithm

`check(n)`

finds all such powers with `n`

digits:

- first it generates the smallest n-digit number `from`

and the largest n-digit number `to`

- e.g. for `n=5`

we have `from=10000`

and `to=99999`

- then all numbers `1^n`

to `9^n`

are computed, if they are between `from`

and `to`

we have a match

While experimenting I saw no number with more than 21 digits to fulfilled the problem's conditions.

A minor headache was that 21 digits don't fit into C++ `unsigned long long`

anymore.

Therefore I switched to `double`

which has a few rounding issues but they don't affect the original problem.

## Alternative Approaches

It's easy to determine the number of digits using `log10`

.

And as mentioned in the code, `pow(a,b)`

returns a^b.

Both functions are available in the C++ standard library.

## Modifications by HackerRank

Hackerrank wants you to print all powers with a certain number of digits - instead of finding the number of all such powers.

However, the rounding issues of `double`

now come into play and the right-most digits are a bit off for large `n`

.

Fortunately, Hackerrank wants us only to find the powers with at most 19 digits ... and a 64 bit `unsigned long long`

is sufficient for this task.

That's why you find the `#ifdef`

construct where the type of `Number`

is defined.

# Interactive test

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

This live test is based on the Hackerrank problem.

This is equivalent to`echo 5 | ./63`

Output:

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

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

#include <iostream>
//#define ORIGINAL

// find all numbers where x^digits has digits digits (I'm loving that comment ...)

unsigned int check(unsigned int digits)
{
// numbe rof matches
unsigned int count = 0;
// unsigned long long isn't sufficient for the highest digits
// a double has some rounding issues but they don't affect to result
#ifdef ORIGINAL
typedef double Number;
#else
typedef unsigned long long Number;
#endif
// range of valid numbers
// from = 10^(digits-1)
// to = 10^digits - 1
Number to = 1;
for (unsigned int i = 1; i <= digits; i++)
to *= 10;
Number from = to / 10;
to--;
// try all single-digit base numbers
for (unsigned int base = 1; base <= 9; base++)
{
// compute power = base ^ digits
Number power = base;
for (unsigned int i = 1; i < digits && power <= to; i++)
power *= base;
// could use C++'s pow(), too
// right number of digits ?
if (power >= from && power <= to)
{
count++;
#ifndef ORIGINAL
std::cout << std::fixed << power << std::endl;
#endif
}
}
return count;
}
int main()
{
#ifdef ORIGINAL
// check all digits
unsigned int count = 0;
for (unsigned int digits = 1; digits <= 21; digits++) // I observed no results with more than 21 digits
count += check(digits);
std::cout << count << std::endl;
#else
// check only certain digits
unsigned int digits = 9;
std::cin >> digits;
check(digits);
#endif
return 0;
}

This solution contains 11 empty lines, 14 comments and 9 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 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

February 28, 2017 submitted solution

April 26, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler063

My code solves **10** out of **10** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **easy**.

*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 rarely an option.

# Links

projecteuler.net/thread=63 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

C#: www.mathblog.dk/project-euler-63-n-digit-nth-power/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p063.java (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem063.go (written by Frederick Robinson)

Scala: github.com/samskivert/euler-scala/blob/master/Euler063.scala (written by Michael Bayne)

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.

You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result.

# Heatmap

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

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

yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |

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

blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |

orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |

red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too |

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I scored 13,386 points (out of 15600 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is

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

Look at my progress and performance pages to get more details.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort !!!

<< problem 62 - Cubic permutations | Odd period square roots - problem 64 >> |