<< problem 71 - Ordered fractions | Counting fractions in a range - problem 73 >> |
Problem 72: Counting fractions
(see projecteuler.net/problem=72)
Consider the fraction, dfrac{n}{d}, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get:
dfrac{1}{8}, dfrac{1}{7}, dfrac{1}{6}, dfrac{1}{5}, dfrac{1}{4}, dfrac{2}{7}, dfrac{1}{3}, dfrac{3}{8}, dfrac{2}{5}, dfrac{3}{7}, dfrac{1}{2}, dfrac{4}{7}, dfrac{3}{5}, dfrac{5}{8}, dfrac{2}{3}, dfrac{5}{7}, dfrac{3}{4}, dfrac{4}{5}, dfrac{5}{6}, dfrac{6}{7}, dfrac{7}{8}
It can be seen that there are 21 elements in this set.
How many elements would be contained in the set of reduced proper fractions for d <= 1,000,000?
My Algorithm
All reduced fraction have gcd(numerator, denominator) = 1 (gcd stands for "greatest common divisor")
therefore for a given denominator the number of suitable numerators is the same as the Euler totient of the denominator.
In project 70 I implemented phi(x) like this:
result = x * (1 - 1/prime1) * (1 - 1/prime2) * (1 - 1/prime3) * ...
I realized that something like a prime sieve can do the same job "in reverse" much faster:
- initialize phi(x) = x for all numbers
- for all multiples of a prime: phi_{new}(k * prime) = phi_{old}(k * prime) * (1 - 1/prime) = phi_{old}(k * prime) - phi_{old}(k * prime) / prime
- compute all sums phi(2) + phi(3) + ... + phi(i)
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 8" | ./72
Output:
Note: the original problem's input 1000000
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.
#include <iostream>
#include <vector>
int main()
{
// 1. initialize phi(x) = x for all numbers
unsigned int limit = 1000000;
std::vector<unsigned int> phi(limit + 1); // vectors are zero-based
for (size_t i = 0; i < phi.size(); i++)
phi[i] = i;
// 2. for all multiples of a prime:
// phi_new(k*prime) = phi_old(k*prime) * (1 - 1/prime)
// = phi_old(k*prime) - phi_old(k*prime) / prime
for (unsigned int i = 2; i <= limit; i++)
{
// prime number ? (because not modified yet by other primes)
if (phi[i] == i)
// adjust all multiples
for (unsigned int k = 1; k * i <= limit; k++)
phi[k * i] -= phi[k * i] / i;
}
// note: since we are only interested in 0 < fractions < 1
// we have to exclude phi(1) (which would yield 1/1 = 1)
// and start at phi(2)
std::vector<unsigned long long> sums(phi.size(), 0);
// 3. compute all sums phi(2) + phi(3) + ... + phi(i)
for (unsigned int i = 2; i <= limit; i++)
sums[i] = sums[i - 1] + phi[i];
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
std::cin >> limit;
std::cout << sums[limit] << std::endl;
}
}
This solution contains 5 empty lines, 10 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.03 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 13 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 13, 2017 submitted solution
May 2, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler072
My code solves 21 out of 21 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=72 - 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-72-reduced-proper-fractions/ (written by Kristian Edlund)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-072.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p072.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/72.py (written by Stephen Macke)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/072.cpp (written by Haochen Liu)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/72.cc (written by Meng-Gen Tsai)
Java github.com/dcrousso/ProjectEuler/blob/master/PE072.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p072.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem72.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem072.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p072.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/072.nb
Haskell github.com/roosephu/project-euler/blob/master/72.hs (written by Yuping Luo)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_072.clj (written by Guillaume Nargeot)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler072.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler072.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/072-Counting-fractions.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p072.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 71 - Ordered fractions | Counting fractions in a range - problem 73 >> |