<< problem 117 - Red, green, and blue tiles | Digit power sum - problem 119 >> |
Problem 118: Pandigital prime sets
(see projecteuler.net/problem=118)
Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed.
Interestingly with the set {2,5,47,89,631}, all of the elements belonging to it are prime.
How many distinct sets containing each of the digits one through nine exactly once contain only prime elements?
My Algorithm
First, I create a prime sieve for all prime numbers up to 100000000.
My isPrime
can handle larger number, too, but has to revert to trial division (which is much, much slower).
The core routine is search
which takes a vector digits
and looks at all digits starting at position firstPos
.
It appends all digits step-by-step to a local variable current
and checks whether it is prime.
If yes, then the routine appends that prime number to merged
and calls itself recursively.
If there are no more digits left in digits
(that means firstPos == digits.size()
) then the numbers in merged
are a valid solution.
It took me some time to figure out that all numbers in merged
must be in ascending order to avoid finding the same solution multiple times.
Modifications by HackerRank
You are given a certain set of digits and have to find the sum of all solutions.
Update November 2017: Kevin McShane helped me to fix a bug in my "premature optimization" in line 114 - now the code solves all test cases.
Note
100,000,000 seems to be the "sweet spot" for my prime sieve. Higher values increase the time needed to fill the sieve significantly,
while lower values lead to a slower isPrime
(because it has to use trial division more often).
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 12345" | ./118
Output:
Note: the original problem's input 987654321
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.
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>
#include <string>
#include <vector>
#include <algorithm>
#define ORIGINAL
// odd prime numbers are marked as "true" in a bitvector
std::vector<bool> sieve;
// return true, if x is a prime number
bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// trial division for large numbers
if (x >= sieve.size() * 2)
{
for (unsigned int i = 3; i*i <= x; i += 2)
if (x % i == 0)
return false;
return true;
}
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size
void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = size >> 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2 * i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3 * i + 1;
while (current < half)
{
sieve[current] = false;
current += 2 * i + 1;
}
}
}
typedef std::vector<unsigned int> Digits;
std::vector<std::vector<unsigned int>> solutions;
void search(const Digits& digits, std::vector<unsigned int>& merged, size_t firstPos = 0)
{
// no more digits left => found a solution
if (firstPos == digits.size())
{
solutions.push_back(merged);
return;
}
// process one more digit at a time
unsigned int current = 0;
while (firstPos < digits.size())
{
// next digit
current *= 10;
current += digits[firstPos++];
// must be larger than its predecessor
if (!merged.empty() && current < merged.back())
continue;
// ... and prime, of course !
if (isPrime(current))
{
merged.push_back(current);
search(digits, merged, firstPos);
merged.pop_back();
}
}
}
int main()
{
// precompute primes (bigger primes are tested using trial division)
fillSieve(100000000);
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
// read digits
std::string strDigits = "123456789";
std::cin >> strDigits;
// convert to a sorted array/vector
Digits digits;
for (auto x : strDigits)
digits.push_back(x - '0');
std::sort(digits.begin(), digits.end());
// discard solutions from previous tests
solutions.clear();
do
{
// simple speed optimization: last digit must be odd (or it's 2)
if (digits.back() % 2 == 0 && (digits.size() > 1 || digits.back() != 2))
continue;
// let's go !
std::vector<unsigned int> merged;
search(digits, merged);
} while (std::next_permutation(digits.begin(), digits.end()));
#ifdef ORIGINAL
std::cout << solutions.size() << std::endl;
#else
// compute sum of each solution
std::vector<unsigned long long> sorted;
for (auto merged : solutions)
{
unsigned long long sum = 0;
for (auto x : merged)
sum += x;
sorted.push_back(sum);
}
// sort ascendingly
std::sort(sorted.begin(), sorted.end());
// remove duplicates (needed ?)
//auto garbage = std::unique(sorted.begin(), sorted.end());
//sorted.erase(garbage, sorted.end());
// and print all of them
for (auto x : sorted)
std::cout << x << std::endl;
std::cout << std::endl;
#endif
}
return 0;
}
This solution contains 21 empty lines, 31 comments and 8 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.23 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 11 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
May 18, 2017 submitted solution
May 18, 2017 added comments
November 10, 2017 fixed bug which prevented me from getting 100% on Hackerrank (line 114, thx to Kevin McShane)
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler118
My code solves 12 out of 12 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Hackerrank describes this problem as hard.
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=118 - 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-118-sets-prime-elements/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/118.cs (written by Haochen Liu)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-118.py (written by Hugh Brown)
Python github.com/LaurentMazare/ProjectEuler/blob/master/e118.py (written by Laurent Mazare)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p118.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/118.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/118.cpp (written by Yuping Luo)
C++ github.com/zmwangx/Project-Euler/blob/master/118/118.cpp (written by Zhiming Wang)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p118.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem118.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem118.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/118.nb
Perl github.com/gustafe/projecteuler/blob/master/118-Pandigital-prime-sets.pl (written by Gustaf Erikson)
Perl github.com/shlomif/project-euler/blob/master/project-euler/118/euler-118.pl (written by Shlomi Fish)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p118.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 117 - Red, green, and blue tiles | Digit power sum - problem 119 >> |