<< problem 86 - Cuboid route | Product-sum numbers - problem 88 >> |
Problem 87: Prime power triples
(see projecteuler.net/problem=87)
The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28.
In fact, there are exactly four numbers below fifty that can be expressed in such a way:
28 = 2^2 + 2^3 + 2^4
33 = 3^2 + 2^3 + 2^4
49 = 5^2 + 2^3 + 2^4
47 = 2^2 + 3^3 + 2^4
How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?
My Algorithm
A simple prime sieve is responsible to find all primes i < sqrt{50000000}.
Then three nested loops compute all sums of all combinations of such primes where a^2 + b^3 + c^4 < 50000000.
Be careful: b*b*b
and c*c*c*c
can easily exceed an unsigned int
.
Those sums are sorted and duplicate sums are removed. Now we have a nice std::vector
with all sums.
The original problem is solved now (just display sums.size()
) but the Hackerrank problem is slightly tougher.
Modifications by HackerRank
In order to find the the number of sums which are below a certain input value, I search through the sorted container with std::upper_bound
and then compute the distance to the beginning of the container.
That's extremely fast (std::upper_bound
most likely uses binary search) and easily processed thousands of test cases per second.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 50" | ./87
Output:
Note: the original problem's input 50000000
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++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#include <vector>
#include <algorithm>
#include <iostream>
int main()
{
const unsigned int MaxLimit = 100 * 1000 * 1000; // Hackerrank: 10^7 instead of 5*10^6
// prime sieve
std::vector<unsigned int> primes;
primes.push_back(2);
for (unsigned int i = 3; i*i < MaxLimit; i += 2)
{
bool isPrime = true;
// test against all prime numbers we have so far (in ascending order)
for (auto p : primes)
{
// next prime is too large to be a divisor ?
if (p*p > i)
break;
// divisible ? => not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
// yes, we have a prime number
if (isPrime)
primes.push_back(i);
}
// just three nested loops where I generate all sums
std::vector<unsigned int> sums;
for (auto a : primes)
for (auto b : primes)
for (auto c : primes)
{
auto a2 = a*a;
auto b3 = (unsigned long long)b*b*b;
auto c4 = (unsigned long long)c*c*c*c;
auto sum = a2 + b3 + c4;
// abort if too big
if (sum > MaxLimit)
break;
sums.push_back(sum);
}
// sort ascendingly
std::sort(sums.begin(), sums.end());
// a few sums occur twice, let's remove them !
auto last = std::unique(sums.begin(), sums.end());
// process test cases
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int limit = MaxLimit;
std::cin >> limit;
// find next sum which is bigger than the limit
auto pos = std::upper_bound(sums.begin(), last, limit);
// how many sums are inbetween 28 and limit ?
auto num = std::distance(sums.begin(), pos);
std::cout << num << std::endl;
}
return 0;
}
This solution contains 11 empty lines, 12 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.16 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 20 MByte.
(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
March 12, 2017 submitted solution
May 5, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler087
My code solves 9 out of 9 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 20% (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=87 - 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-87-sum-power/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/087.cs (written by Haochen Liu)
Python blog.dreamshire.com/project-euler-87/
Python github.com/hughdbrown/Project-Euler/blob/master/euler-087.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p087.py (written by Nayuki)
Python github.com/sefakilic/euler/blob/master/python/euler087.py (written by Sefa Kilic)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/87.cc (written by Meng-Gen Tsai)
C++ github.com/zmwangx/Project-Euler/blob/master/087/087.cpp (written by Zhiming Wang)
Java github.com/dcrousso/ProjectEuler/blob/master/PE087.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p087.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem87.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem087.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/087.nb
Haskell github.com/roosephu/project-euler/blob/master/87.hs (written by Yuping Luo)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_087.clj (written by Guillaume Nargeot)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler087.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler087.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/087-Prime-power-triples.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p087.rs
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 86 - Cuboid route | Product-sum numbers - problem 88 >> |