<< 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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 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)
My code
… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#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.
Benchmark
The correct solution to the original Project Euler problem was found in 0.15 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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:
C# www.mathblog.dk/project-euler-30-sum-numbers-that-can-be-written-as-the-sum-fifth-powers-digits/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/30-39/problem30.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p030.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/30 Digit fifth powers.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem030.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p030.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p030.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler030.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/030-Digital-fifth-powers.pl (written by Gustaf Erikson)
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 and/or exceed time/memory limits.
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 | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 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 29 - Distinct powers | Coin sums - problem 31 >> |