<< problem 40 - Champernowne's constant | Coded triangle numbers - problem 42 >> |
Problem 41: Pandigital prime
(see projecteuler.net/problem=41)
We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once.
For example, 2143 is a 4-digit pandigital and is also prime.
What is the largest n-digit pandigital prime that exists?
My Algorithm
The largest pandigital number is 987654321. In order to find out whether a number x <= 987654321 is prime, I precompute all primes up to sqrt{987654321} approx 31426.
These "small" prime numbers will be kept in smallPrimes
.
My second step is to generate all pandigital numbers: I create all permutations of the string "123456789"
and perform a simple primality test (using smallPrimes
).
Due to Hackerrank's variable number of digits, not only the 9-pandigitals numbers but also the 8-, 7-, 6-, ..., 2-pandigital numbers are checked, too.
The set panPrimes
will contain all 2-, ..., 9-pandigital primes after those two precomputation steps.
Each test case look ups the closest bigger pandigital prime (upper_bound
) and goes one step backwards.
Note
When looking at the results I only saw 4- and 7-pandigital primes.
Modifying my loop in step 2 accordingly would provide a 10x speed-up.
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 2500" | ./41
Output:
(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 <set>
#include <iostream>
#include <algorithm>
int main()
{
// precomputation step 1:
// find all primes below sqrt(987654321)
std::set<unsigned int> smallPrimes;
smallPrimes.insert(2);
for (unsigned int i = 3; i*i <= 987654321; i += 2)
{
bool isPrime = true;
for (auto p : smallPrimes)
{
// abort, no divisors possible
if (p*p > i)
break;
// divisor found ?
if (i % p == 0)
{
isPrime = false;
break;
}
}
// found a prime number
if (isPrime)
smallPrimes.insert(i);
}
// precomputation step 2:
// generate all permutations of the strings "12", "123", "1234", ..., "123456789"
// and test whether they are prime
std::set<unsigned int> panPrimes;
for (unsigned int digits = 2; digits <= 9; digits++)
{
std::string strNumber = "123456789";
// reduce number of digits
strNumber.erase(digits);
do
{
unsigned int number = std::stoi(strNumber);
// test whether pandigital number is prime
bool isPrime = true;
for (auto p : smallPrimes)
{
// abort, no divisors possible
if (p*p > number)
break;
// divisor found ?
if (number % p == 0)
{
isPrime = false;
break;
}
}
// found a pandigital prime ?
if (isPrime)
panPrimes.insert(number);
} while (std::next_permutation(strNumber.begin(), strNumber.end()));
}
// process input
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int limit;
std::cin >> limit;
// find next larger pandigital prime number
auto i = panPrimes.upper_bound(limit);
// smaller than the smallest pandigital prime ?
if (i == panPrimes.begin())
{
std::cout << "-1" << std::endl;
continue;
}
// upper_bound() goes one step too far
i--;
// and print it
std::cout << *i << std::endl;
}
return 0;
}
This solution contains 13 empty lines, 18 comments and 3 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.
(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 25, 2017 submitted solution
April 19, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler041
My code solves 5 out of 5 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.
Links
projecteuler.net/thread=41 - 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-41-pandigital-prime/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem41.cc (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p041.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/41 Pandigital prime.js (written by David Ernst)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p041.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler041.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/041-Pandigital-prime.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 40 - Champernowne's constant | Coded triangle numbers - problem 42 >> |