<< problem 29 - Distinct powers | Coin sums - problem 31 >> |

# Problem 30: Digit fifth powers

(see projecteuler.net/problem=30)

Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:

1634 = 1^4 + 6^4 + 3^4 + 4^4

8208 = 8^4 + 2^4 + 0^4 + 8^4

9474 = 9^4 + 4^4 + 7^4 + 4^4

As 1 = 1^4 is not a sum it is not included.

The sum of these numbers is 1634 + 8208 + 9474 = 19316.

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

# My Algorithm

The sum of the digits' fifth powers is maximized when each digit is 9:

1 digit: 1 * 9^5 = 59049

2 digits: 2 * 9^5 = 118098

3 digits: 3 * 9^5 = 177147

4 digits: 4 * 9^5 = 236196

5 digits: 5 * 9^5 = 295245

6 digits: 6 * 9^5 = 354294

7 digits: 7 * 9^5 = 413343

The last line is pretty interesting: it's impossible for a seven-digit number to have a seven-digit sum of its digits' fifth powers,

because all those sums would have at most six digits.

If we analyse all numbers from 2 to 354294 (maximum sum for 6 digits) then we can solve the problem:

1. split each number into its digits

2. add all fifth powers of these digits

3. if the sum is equal to the original number then add it to our result

## Modifications by HackerRank

The exponent varies between 3 and 6. The maximum number can be computed similar to the table above:

7 digits: 7 * 9^6 = 3720087

8 digits: 8 * 9^6 = 4251528 → impossible

To simplify the code my loop always stops at 4251528 even though lower numbers would suffice for exponents < 6.

Timeouts are no issue here.

# My code

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

#include <iostream>
int main()
{
unsigned int exponent;
std::cin >> exponent;
// result
unsigned int sum = 0;
// there can't be a number with 8 digits (or more) which fulfils the condition for exponent=6
// if all digits are 9s, then
// 7 digits: 7 * 9^6 = 3720087
// 8 digits: 8 * 9^6 = 4251528
for (unsigned int i = 2; i <= 7*9*9*9*9*9*9; i++)
{
// sum of i's digits to the power of "exponent"
unsigned int thisSum = 0;
// split current number into its digit
unsigned int reduce = i;
while (reduce > 0)
{
// take the right-most digit
unsigned int digit = reduce % 10;
// and remove it
reduce /= 10;
// compute digit^exponent, could use pow() as well
unsigned int power = 1;
for (unsigned int j = 1; j <= exponent; j++)
power *= digit;
// add to thisSum
thisSum += power;
}
// sum of digits^n equal to the original number ?
if (thisSum == i)
sum += i;
}
// and we're done
std::cout << sum << std::endl;
return 0;
}

This solution contains 8 empty lines, 13 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 4 | ./30`

Output:

*Note:* the original problem's input `5`

__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 0.15 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 23, 2017 submitted solution

April 6, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler030

My code solves **4** out of **4** 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.

# Similar problems at Project Euler

Problem 34: Digit factorials

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=30 - **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-30-sum-numbers-that-can-be-written-as-the-sum-fifth-powers-digits/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p030.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p030.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p030.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem30.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem030.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/30 Digit fifth powers.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler030.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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

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

My username at Project Euler is

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

# 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 29 - Distinct powers | Coin sums - problem 31 >> |