<< 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?

# 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.

At the moment I have no idea why some test cases fail.

## 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).

# 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
if (digits.back() % 2 == 0)
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.

# Interactive test

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

This is equivalent to`echo "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)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.21 seconds on a 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=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

May 18, 2017 submitted solution

May 18, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler118

My code solves **9** out of **12** test cases (score: **60%**)

I failed **3** test cases due to wrong answers and **0** because of timeouts

# 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 never 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:

Python: www.mathblog.dk/project-euler-118-sets-prime-elements/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p118.java (written by Nayuki)

# 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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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

<< problem 117 - Red, green, and blue tiles | Digit power sum - problem 119 >> |