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?

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.

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#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.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 2500" | ./41

Output:

(please click 'Go !')

(this interactive test is still under development, computations will be aborted after one second)

Benchmark

The correct solution to the original Project Euler problem was found in 0.03 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++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 never 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:

Python: www.mathblog.dk/project-euler-41-pandigital-prime/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p041.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p041.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem41.cc (written by eagletmt)
Javascript: github.com/dsernst/ProjectEuler/blob/master/41 Pandigital prime.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler041.scala (written by Michael Bayne)

Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
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